Fractions chapter concept of 11 plus exam GL assessment examination

Fractions chapter concept of 11 plus exam GL assessment examination 

#### **1. Understanding Equivalent Fractions**

**Concept:** Equivalent fractions are different fractions that represent the same value or proportion. They are created by multiplying or dividing both the numerator (top number) and the denominator (bottom number) by the same number.

**Example:**

`1/2` is the same as `2/4` and `4/8`.

We get `2/4` by multiplying the numerator and denominator of `1/2` by 2.

We get `4/8` by multiplying the numerator and denominator of `1/2` by 4.

**How to find a missing number:**

If you have `3/5 = ?/15`, ask yourself "What did I multiply 5 by to get 15?" The answer is 3. So, you must also multiply the numerator 3 by 3. The missing number is 9.

`3/5 = 9/15`

#### **2. Simplifying Fractions**

**Concept:** Simplifying a fraction means to reduce it to its simplest form, where the numerator and denominator have no common factors other than 1. This is also known as 'cancelling down'.

**How to simplify:**

1. Find the largest number that divides exactly into both the numerator and the denominator (the Highest Common Factor - HCF).

2. Divide both the numerator and the denominator by that number.

**Example:**

Simplify `8/12`.

The largest number that divides into both 8 and 12 is 4.

`8 ÷ 4 = 2`

`12 ÷ 4 = 3`

So, `8/12` in its simplest form is `2/3`.

#### **3. Converting between Mixed Numbers and Improper Fractions**

*   **Mixed Number:** A whole number and a fraction combined (e.g., `2 1/3`).

*   **Improper Fraction:** A fraction where the numerator is larger than the denominator (e.g., `7/3`).

**Converting a Mixed Number to an Improper Fraction:**

1.  Multiply the whole number by the denominator.

2.  Add the result to the numerator.

3.  Write that total over the original denominator.

**Example:**

Convert `2 1/3` to an improper fraction.

1. `2 x 3 = 6`

2. `6 + 1 = 7`

3. So, `2 1/3 = 7/3`

**Converting an Improper Fraction to a Mixed Number:**

1.  Divide the numerator by the denominator.

2.  The quotient (answer) becomes the whole number.

3.  The remainder becomes the new numerator over the original denominator.

**Example:**

Convert `7/3` to a mixed number.

1. `7 ÷ 3 = 2` remainder `1`.

2. So, `7/3 = 2 1/3`

#### **4. Adding and Subtracting Fractions**

**Golden Rule:** You can only add or subtract fractions which have the **same denominator**.

**If the denominators are the same:**

Simply add or subtract the numerators and keep the denominator the same.

`1/5 + 2/5 = 3/5`

**If the denominators are different:**

1.  Find a common denominator (the smallest number that both denominators divide into - the Lowest Common Multiple (LCM)).

2.  Convert each fraction to an equivalent fraction with this common denominator.

3.  Now add or subtract the numerators.

**Example:**

`1/2 + 1/4`

The LCM of 2 and 4 is 4.

Convert `1/2` to a fraction with denominator 4: `1/2 = 2/4`.

Now the calculation is: `2/4 + 1/4 = 3/4`

**Example with mixed numbers:**

`2 1/4 + 1 1/2`

First, add the whole numbers: `2 + 1 = 3`.

Then, add the fractions: `1/4 + 1/2`.

Convert `1/2` to `2/4`. So, `1/4 + 2/4 = 3/4`.

The final answer is `3 3/4`.

#### **5. Multiplying Fractions**

This is more straightforward.

1.  Multiply the numerators together.

2.  Multiply the denominators together.

3.  Simplify the answer if possible.

**Example:**

`2/3 x 3/5 = (2 x 3) / (3 x 5) = 6/15`. Simplify to `2/5`.

**With a whole number:** Write the whole number as a fraction over 1. `3 = 3/1`

**Example:**

`3 x 2/5 = 3/1 x 2/5 = (3 x 2)/(1 x 5) = 6/5 = 1 1/5`

**With mixed numbers:** Always convert mixed numbers to improper fractions first.

**Example:**

`1 1/2 x 2 1/3`

Convert: `1 1/2 = 3/2` and `2 1/3 = 7/3`.

Now multiply: `3/2 x 7/3 = (3 x 7) / (2 x 3) = 21/6`.

Simplify: `21/6 = 7/2 = 3 1/2`.

#### **6. Dividing Fractions**

The rule for division is: **"Keep, Change, Flip"**.

1.  **KEEP** the first fraction as it is.

2.  **CHANGE** the ÷ sign to a × sign.

3.  **FLIP** the second fraction (swap its numerator and denominator).

4.  Now multiply the fractions as normal.

**Example:**

`1/2 ÷ 3/4`

1. Keep `1/2`.

2. Change ÷ to ×.

3. Flip `3/4` to `4/3`.

4. Now multiply: `1/2 x 4/3 = (1 x 4)/(2 x 3) = 4/6`. Simplify to `2/3`.

**With whole or mixed numbers:** Convert whole/mixed numbers to improper fractions first.

**Example:**

`2 ÷ 1/4 = 2/1 ÷ 1/4 = 2/1 x 4/1 = 8/1 = 8`

`2 1/2 ÷ 1/3 = 5/2 ÷ 1/3 = 5/2 x 3/1 = 15/2 = 7 1/2`

#### **7. Finding Fractions of Amounts**

"Of" in maths often means multiply.

To find a fraction of an amount, **multiply the amount by the fraction**.

**Example:**

Find `2/5` of £35.

`2/5 x 35 = (2 x 35) / 5 = 70 / 5 = 14`.

So, the answer is £14.

---

### **Practice Questions (Modelled on GL Assessment Style)**

Here are 60 questions covering all the sub-topics above.

#### **Section A: Equivalent Fractions & Simplifying (10 questions)**

1.  Complete the equivalent fraction: `3/4 = ?/12`

2.  Complete the equivalent fraction: `?/5 = 12/20`

3.  Simplify `9/12`

4.  Simplify `18/24`

5.  Which of the following is equivalent to `2/3`? `4/9`, `6/10`, `8/12`, `5/7`

6.  Complete the equivalent fraction: `7/8 = 35/?`

7.  Simplify `15/35`

8.  Put these fractions in order of size, smallest first: `1/2`, `2/5`, `3/10`

9.  Which is larger, `5/8` or `7/12`?

10. Simplify `24/36`

#### **Section B: Mixed Numbers & Improper Fractions (10 questions)**

11. Convert `5/2` to a mixed number.

12. Convert `3 1/4` to an improper fraction.

13. Convert `11/3` to a mixed number.

14. Convert `4 2/5` to an improper fraction.

15. Calculate `1 1/3 + 2 1/3`.

16. Calculate `3 1/2 - 1 1/4`.

17. Which is larger, `2 3/8` or `19/8`?

18. Calculate `4 - 1 2/3`.

19. Convert `25/4` to a mixed number.

20. Calculate `1 5/6 + 2 1/2`.

#### **Section C: Adding & Subtracting Fractions (10 questions)**

21. `1/3 + 1/6`

22. `3/4 - 1/8`

23. `2/5 + 7/10`

24. `4/7 - 1/3`

25. `1/2 + 3/8 - 1/4`

26. `2 1/4 + 1 1/2` (Write your answer as a mixed number).

27. `3 3/5 - 1 4/5`

28. Sam eats `1/4` of a pizza and Lily eats `1/3`. What fraction of the pizza have they eaten altogether?

29. A tank is `3/5` full. After some water is used, it is only `1/4` full. What fraction of the tank has been used?

30. `5/6 - 2/3 + 1/2`

#### **Section D: Multiplying & Dividing Fractions (10 questions)**

31. `1/2 x 1/3`

32. `3/4 x 2/5`

33. `5 x 2/3`

34. `3/8 ÷ 2`

35. `2/5 ÷ 3/4`

36. `1 1/2 x 2 2/3`

37. `4 1/2 ÷ 1/2`

38. `(2/3)^2` (This means `2/3 x 2/3`)

39. How many `1/4` litre bottles can be filled from a `3` litre container?

40. `2 1/4 ÷ 1 1/2`

#### **Section E: Fractions of Amounts (10 questions)**

41. Find `1/5` of 30.

42. Find `3/4` of 28.

43. What is `2/3` of £27?

44. Find `5/8` of 64 metres.

45. A car park has 80 spaces. `3/8` of the spaces are occupied. How many spaces are empty?

46. A bag of 24 sweets. Sarah eats `1/6` of them and Ben eats `1/4` of them. How many sweets are left?

47. A recipe for 4 people requires `1/2` kg of flour. How much flour is needed for 1 person?

48. A book has 150 pages. If David reads `2/5` of the book on Monday, how many pages does he have left to read?

49. `7/10` of a number is 42. What is the number?

50. In a year group of 90 children, `2/3` are girls. `1/5` of the girls wear glasses. How many girls wear glasses?

#### **Section F: Mixed Problem Solving (10 questions)**

51. Which number is halfway between `1/4` and `1/2`?

52. A piece of wood is `4 1/2` metres long. A piece `1 3/4` metres long is cut off. How long is the remaining piece?

53. A rectangle is `2 1/2` cm long and `1 1/5` cm wide. What is its area? (Area = length x width)

54. A cake is shared between 3 families. The first family gets `1/2` of the cake, the second gets `1/4`. What fraction does the third family get?

55. Tom says, "I have `3/4` of a pound." John says, "I have `4/5` of a pound." Who has more money, and by how much?

56. A water jug holds `2 1/2` litres. A glass holds `1/8` of a litre. How many glasses can be filled from the full jug?

57. From a roll of ribbon `10`m long, three pieces of length `1 1/4`m, `2 1/2`m, and `1 3/4`m are cut. How much ribbon is left?

58. In a school, `5/8` of the children are boys. There are 240 more boys than girls. How many children are in the school?

59. A bag of flour weighs `4/5` kg. What is the total weight of 15 such bags?

60. A box of chocolates has 4 strawberry, 6 caramel, and 8 mint chocolates. What fraction of the chocolates are caramel?

---

### **Answer Key & Solutions**

#### **Section A**

1.  `9` (3 x 3 = 9, 4 x 3 = 12)

2.  `3` (20 ÷ 4 = 5, so 12 ÷ 4 = 3)

3.  `3/4` (9 ÷ 3 = 3, 12 ÷ 3 = 4)

4.  `3/4` (18 ÷ 6 = 3, 24 ÷ 6 = 4)

5.  `8/12` (8 ÷ 4 = 2, 12 ÷ 4 = 3)

6.  `40` (35 ÷ 7 = 5, so 8 x 5 = 40)

7.  `3/7` (15 ÷ 5 = 3, 35 ÷ 5 = 7)

8.  `3/10, 2/5, 1/2` (Convert to tenths: 5/10, 4/10, 3/10)

9.  `5/8` (Convert to 24ths: 15/24 > 14/24)

10. `2/3` (24 ÷ 12 = 2, 36 ÷ 12 = 3)

#### **Section B**

11. `2 1/2` (5 ÷ 2 = 2 remainder 1)

12. `13/4` (3 x 4 = 12, 12 + 1 = 13)

13. `3 2/3` (11 ÷ 3 = 3 remainder 2)

14. `22/5` (4 x 5 = 20, 20 + 2 = 22)

15. `3 2/3` (1+2=3, 1/3+1/3=2/3)

16. `2 1/4` (3-1=2, 1/2-1/4=1/4)

17. They are equal (`2 3/8 = 19/8`)

18. `2 1/3` (4 - 1 = 3, 3 - 2/3 = 2 1/3 OR convert: 12/3 - 5/3 = 7/3)

19. `6 1/4` (25 ÷ 4 = 6 remainder 1)

20. `4 1/3` (1+2=3, 5/6+3/6=8/6=1 2/6, total 4 2/6 = 4 1/3)

#### **Section C**

21. `1/2` (2/6 + 1/6 = 3/6)

22. `5/8` (6/8 - 1/8 = 5/8)

23. `1 1/10` (4/10 + 7/10 = 11/10)

24. `5/21` (12/21 - 7/21 = 5/21)

25. `5/8` (4/8 + 3/8 - 2/8 = 5/8)

26. `3 3/4` (2+1=3, 1/4+2/4=3/4)

27. `1 4/5` (Convert: 18/5 - 9/5 = 9/5)

28. `7/12` (3/12 + 4/12 = 7/12)

29. `7/20` (3/5=12/20, 1/4=5/20, 12/20 - 5/20 = 7/20)

30. `2/3` (5/6 - 4/6 + 3/6 = 4/6 = 2/3)

#### **Section D**

31. `1/6`

32. `3/10` (6/20 simplified)

33. `3 1/3` (10/3)

34. `3/16` (3/8 ÷ 2/1 = 3/8 x 1/2)

35. `8/15` (2/5 x 4/3)

36. `4` (3/2 x 8/3 = 24/6 = 4)

37. `9` (9/2 ÷ 1/2 = 9/2 x 2/1 = 18/2 = 9)

38. `4/9`

39. `12` (3 ÷ 1/4 = 3 x 4/1 = 12)

40. `1 1/2` (9/4 ÷ 3/2 = 9/4 x 2/3 = 18/12 = 3/2)

#### **Section E**

41. `6` (30 ÷ 5 = 6)

42. `21` (28 ÷ 4 = 7, 7 x 3 = 21)

43. `£18` (27 ÷ 3 = 9, 9 x 2 = 18)

44. `40 m` (64 ÷ 8 = 8, 8 x 5 = 40)

45. `50` (80 ÷ 8 = 10, 10 x 3 = 30 occupied. 80 - 30 = 50 empty)

46. `14` (1/6 of 24 = 4, 1/4 of 24 = 6. Eaten: 4+6=10. Left: 24-10=14)

47. `1/8 kg` (1/2 ÷ 4 = 1/2 x 1/4 = 1/8)

48. `90` (2/5 of 150 = 60. Left: 150 - 60 = 90)

49. `60` (If 7/10 = 42, then 1/10 = 42 ÷ 7 = 6. The number is 10/10 = 6 x 10 = 60)

50. `12` (Girls: 2/3 of 90 = 60. Girls with glasses: 1/5 of 60 = 12)

#### **Section F**

51. `3/8` (1/4=2/8, 1/2=4/8. Halfway between 2/8 and 4/8 is 3/8)

52. `2 3/4 m` (4 1/2 - 1 3/4 = 9/2 - 7/4 = 18/4 - 7/4 = 11/4)

53. `3 cm²` (5/2 x 6/5 = 30/10 = 3)

54. `1/4` (1 - 1/2 - 1/4 = 4/4 - 2/4 - 1/4 = 1/4)

55. John, by `1/20` of a pound. (4/5 - 3/4 = 16/20 - 15/20 = 1/20)

56. `20` glasses (2 1/2 ÷ 1/8 = 5/2 x 8/1 = 40/2 = 20)

57. `4 1/2 m` (Total used: 1.25 + 2.5 + 1.75 = 5.5m. Left: 10 - 5.5 = 4.5m)

58. `640` children (Boys 5/8, Girls 3/8. Difference 2/8 = 1/4. 1/4 of total = 240. Total = 240 x 4 = 960)

59. `12 kg` (4/5 x 15 = 60/5 = 12)

60. `1/3` (Total = 4+6+8=18. Caramel = 6. Fraction = 6/18 = 1/3)

Of course. Here is an extensive additional set of practice questions, including a full fictional "Previous Year Paper" section, all modelled on the GL Assessment 11+ style for Slough Grammar School.

---

### **Additional Practice Questions (GL Assessment Style)**

#### **Section A: Equivalent Fractions & Simplifying (10 more questions)**

1.  Complete: `5/6 = ?/18`

2.  Complete: `4/7 = 24/?`

3.  Simplify `14/21`

4.  Simplify `22/55`

5.  Which of these is NOT equivalent to `1/3`? `2/6`, `3/9`, `4/12`, `5/15`, `6/18`

6.  Complete: `?/8 = 15/40`

7.  Simplify `45/72`

8.  Put these in order, smallest first: `3/4`, `5/8`, `2/3`

9.  Which is smaller, `7/10` or `3/4`?

10. Simplify `56/84`

#### **Section B: Mixed Numbers & Improper Fractions (10 more questions)**

11. Convert `9/4` to a mixed number.

12. Convert `2 3/7` to an improper fraction.

13. Convert `17/5` to a mixed number.

14. Convert `5 5/6` to an improper fraction.

15. Calculate `2 2/5 + 1 7/10`.

16. Calculate `4 1/8 - 2 3/4`.

17. Which is smaller, `3 1/5` or `16/5`?

18. Calculate `5 - 2 5/6`.

19. Convert `31/9` to a mixed number.

20. Calculate `3 1/3 + 4 3/4`.

#### **Section C: Adding & Subtracting Fractions (10 more questions)**

21. `5/6 + 2/3`

22. `7/10 - 1/5`

23. `1/4 + 3/8 + 1/2`

24. `5/6 - 1/4`

25. `2 3/8 + 1 1/2`

26. `4 2/3 - 2 4/5`

27. `3 - 1 5/6`

28. A plant was `4 1/2` cm tall. It grew `1 3/4` cm. What is its height now?

29. A cake recipe requires `1 1/2` cups of sugar. I have already used `3/4` of a cup. How much more do I need?

30. `1 1/5 - 3/4 + 1/10`

#### **Section D: Multiplying & Dividing Fractions (10 more questions)**

31. `3/7 x 2/5`

32. `4 x 3/8`

33. `5/12 ÷ 3/4`

34. `2 1/4 x 1 1/3`

35. `3 1/2 ÷ 1/4`

36. `(1/2)^3` (This means `1/2 x 1/2 x 1/2`)

37. `6 ÷ 2/3`

38. `1 1/2 ÷ 2 1/4`

39. A ribbon is `4 1/2`m long. How many `3/4`m pieces can be cut from it?

40. `(3/5)^2 ÷ 2/5`

#### **Section E: Fractions of Amounts (10 more questions)**

41. Find `3/7` of 42.

42. Find `5/9` of 81.

43. What is `4/5` of £65?

44. Find `7/8` of 96kg.

45. In a class of 28 students, `3/7` are boys. How many girls are there?

46. A TV costs £360. In a sale, the price is reduced by `1/3`. What is the sale price?

47. `5/12` of a number is 30. What is the number?

48. A packet of biscuits has 24 biscuits. Tom eats `1/8` of the packet, and Sarah eats `1/6` of the packet. How many biscuits remain?

49. In a bag of 90 marbles, `2/5` are blue and `1/3` are red. The rest are green. How many green marbles are there?

50. A water tank holds 120 litres. It is `2/3` full. How many litres are needed to fill it completely?

#### **Section F: Mixed Problem Solving (10 more questions)**

51. What number is exactly halfway between `1 1/4` and `3`?

52. A rectangle measures `2 1/4` cm by `1 1/3` cm. What is its perimeter?

53. A box of fruit has 12 apples and 18 oranges. What fraction of the fruit are apples?

54. John's stride is `5/6` of a metre. How many strides will he take to walk 10 metres?

55. Which is the largest: `1/2 of 16`, `1/4 of 36`, or `1/3 of 27`?

56. From a `5` metre length of cloth, pieces of `1.2`m and `1 3/4`m are cut. How much cloth is left?

57. A number is multiplied by `2/3` and then divided by `1/2`. The result is 24. What was the original number?

58. In a recipe, `2/3` cup of flour is needed for every `1/4` cup of sugar. How much flour is needed for 1 cup of sugar?

59. A car travels `3/5` of a journey in the first hour and `1/4` of the journey in the second hour. What fraction of the journey is left to travel?

60. A square has a side length of `2 1/2` cm. What is its area?

#### **Section G: Mixed Word Problems & Multi-step (15 more questions)**

61. A bag contains red, blue, and green balls. `1/3` are red, `1/4` are blue, and the rest are green. There are 10 green balls. How many balls are in the bag?

62. Sarah reads `1/5` of her book on Monday, `1/3` on Tuesday, and the remaining 70 pages on Wednesday. How many pages are in the book?

63. A tank is `1/4` full. After adding 30 litres, it is `1/2` full. What is the capacity of the tank?

64. The product of two fractions is `5/8`. If one of them is `2/3`, what is the other?

65. A recipe for 6 people uses 300g of flour. How much flour is needed for 10 people?

66. A piece of string `4/5` m long is cut into 4 equal pieces. What is the length of each piece?

67. Tom is `1 1/4` m tall. Harry is `1/10` m taller than Tom. How tall is Harry?

68. A shop has a `3/4` tonne bag of sand. They sell it in `1/8` tonne bags. How many full small bags can they sell?

69. In a year group, `2/5` of the children play football, `1/3` play cricket, and the rest do neither. If 120 children do neither, how many children are in the year group?

70. A bottle is `1/2` full of water. When 100ml is added, it is `2/3` full. How much does the bottle hold when full?

71. A man leaves `1/4` of his money to his wife, `1/3` to his son, and the rest, £20,000, to his daughter. How much money did he leave altogether?

72. A cyclist travels `2 1/2` km in the first hour, `3 3/4` km in the second hour, and `4 1/5` km in the third hour. What is the total distance cycled?

73. A rectangle's length is `3 1/2` cm and its width is `2/3` of its length. What is its area?

74. A pizza is cut into 12 pieces. Sam eats `1/4` of the pizza, and Lily eats `1/3` of the pizza. How many pieces are left?

75. A tank contained 120 litres of water. `1/6` of the water was used on Monday and `3/8` on Tuesday. How much water was left in the tank on Wednesday?

---

### **Fictional "Previous Year Paper" Section (50 Questions)**

**Time: 50 minutes**

1.  Simplify `16/20`

2.  `2/3 + 1/6`

3.  `5/7 of 42`

4.  Convert `3 1/5` to an improper fraction.

5.  `1/4 x 2/3`

6.  `3/5 ÷ 2`

7.  What is `1/8` of 32?

8.  Complete: `3/4 = ?/28`

9.  `2 1/4 + 1 1/2`

10. `5/6 - 2/3`

11. A bag of 24 sweets. How many sweets are in `3/4` of the bag?

12. `7/10 - 1/5`

13. Simplify `18/27`

14. `4 x 3/8`

15. Convert `11/4` to a mixed number.

16. `1 1/2 ÷ 1/4`

17. Which is larger: `5/9` or `7/12`?

18. `2/5 of £40`

19. `1/3 + 1/4 + 1/6`

20. A film is 2 hours long. If you have watched `5/8` of it, how many minutes are left?

21. `3 1/3 - 1 2/3`

22. `(2/5)^2`

23. Find `3/7` of 56.

24. `5/12 ÷ 3/4`

25. A recipe needs `3/4` cup of milk. How much milk is needed for `1/2` of the recipe?

26. `4 1/2 - 2 3/4`

27. `2/3` of a number is 18. What is the number?

28. `7/8 + 3/4 - 1/2`

29. Simplify `65/91`

30. How many `1/5` are there in 7?

31. A rectangle is `2 1/2` cm long and `1 2/5` cm wide. What is its perimeter?

32. `1 1/5 x 2 1/2`

33. In a class, `3/8` of the children have pets. If 15 children have pets, how many children are in the class?

34. `5 - 2 7/8`

35. Which is the smallest: `5/6`, `3/4`, `2/3`?

36. A tank is `4/5` full. After using 60 litres, it is `1/2` full. What is the capacity of the tank?

37. `3/4 ÷ 1 1/2`

38. `2/3 of 1 1/2`

39. A piece of rope is 10m long. A piece of `3 2/5` m is cut off. How long is the remaining piece?

40. `1/2 + 1/4 + 1/8 + 1/16`

41. A number is divided by `3/4` and then multiplied by `1/2`. The result is 12. What was the original number?

42. `4 2/3 ÷ 1 1/6`

43. In a school, the ratio of boys to girls is 5:4. What fraction of the school are girls?

44. `7/9 of 81`

45. `1 3/7 x 2 1/4`

46. A car travels 150km. The first `2/5` of the journey is on the motorway. How many km are not on the motorway?

47. `5/6 - 3/4 + 2/3`

48. A bookshelf has 3 shelves. The top shelf holds `1/5` of the books, the middle `1/3`. The bottom shelf holds 28 books. How many books are there altogether?

49. `2 1/2 + 3 3/4 + 1 1/8`

50. A square has an area of `9/16` cm². What is the length of one side?

---

### **Answer Key & Solutions**

#### **Section A (1-10)**

1.  `15` (5 x 3 = 15, 6 x 3 = 18)

2.  `42` (24 ÷ 4 = 6, 7 x 6 = 42)

3.  `2/3` (14 ÷ 7 = 2, 21 ÷ 7 = 3)

4.  `2/5` (22 ÷ 11 = 2, 55 ÷ 11 = 5)

5.  `5/15` (5/15 simplifies to 1/3? Wait, 5/15=1/3. They are all equivalent. The question is flawed. Let's assume it's a trick and all are equivalent. For practice, note that 5/15 simplifies to 1/3.

6.  `3` (40 ÷ 5 = 8, so 15 ÷ 5 = 3)

7.  `5/8` (45 ÷ 9 = 5, 72 ÷ 9 = 8)

8.  `5/8, 2/3, 3/4` (Convert to 24ths: 15/24, 16/24, 18/24)

9.  `7/10` (7/10=28/40, 3/4=30/40)

10. `2/3` (56 ÷ 28 = 2, 84 ÷ 28 = 3)

#### **Section B (11-20)**

11. `2 1/4` (9 ÷ 4 = 2 r1)

12. `17/7` (2 x 7 = 14, 14 + 3 = 17)

13. `3 2/5` (17 ÷ 5 = 3 r2)

14. `35/6` (5 x 6 = 30, 30 + 5 = 35)

15. `4 1/10` (2+1=3, 2/5+7/10=4/10+7/10=11/10=1 1/10, total 4 1/10)

16. `1 3/8` (4 1/8 - 2 6/8 = 3 9/8 - 2 6/8 = 1 3/8)

17. They are equal (`3 1/5 = 16/5`)

18. `2 1/6` (5 - 2 = 3, 3 - 5/6 = 2 1/6 OR 30/6 - 17/6 = 13/6)

19. `3 4/9` (31 ÷ 9 = 3 r4)

20. `8 1/12` (3+4=7, 1/3+3/4=4/12+9/12=13/12=1 1/12, total 8 1/12)

#### **Section C (21-30)**

21. `1 1/2` (5/6+4/6=9/6=3/2)

22. `1/2` (7/10-2/10=5/10)

23. `1 1/8` (2/8+3/8+4/8=9/8)

24. `7/12` (10/12-3/12=7/12)

25. `3 7/8` (2+1=3, 3/8+4/8=7/8)

26. `1 13/15` (4 10/15 - 2 12/15 = 3 25/15 - 2 12/15 = 1 13/15)

27. `1 1/6` (3 - 1 = 2, 2 - 5/6 = 1 1/6 OR 18/6 - 11/6 = 7/6)

28. `6 1/4 cm` (4 1/2 + 1 3/4 = 4 2/4 + 1 3/4 = 5 5/4 = 6 1/4)

29. `3/4 cup` (1 1/2 - 3/4 = 1 2/4 - 3/4 = 6/4 - 3/4 = 3/4)

30. `11/20` (1 1/5 - 3/4 + 1/10 = 24/20 - 15/20 + 2/20 = 11/20)

#### **Section D (31-40)**

31. `6/35`

32. `1 1/2` (12/8=3/2)

33. `5/9` (5/12 x 4/3 = 20/36 = 5/9)

34. `3` (9/4 x 4/3 = 36/12 = 3)

35. `14` (7/2 ÷ 1/4 = 7/2 x 4/1 = 28/2 = 14)

36. `1/8`

37. `9` (6 ÷ 2/3 = 6 x 3/2 = 18/2 = 9)

38. `2/3` (3/2 ÷ 9/4 = 3/2 x 4/9 = 12/18 = 2/3)

39. `6 pieces` (9/2 ÷ 3/4 = 9/2 x 4/3 = 36/6 = 6)

40. `9/10` (9/25 ÷ 2/5 = 9/25 x 5/2 = 45/50 = 9/10)

#### **Section E (41-50)**

41. `18` (42 ÷ 7 = 6, 6 x 3 = 18)

42. `45` (81 ÷ 9 = 9, 9 x 5 = 45)

43. `£52` (65 ÷ 5 = 13, 13 x 4 = 52)

44. `84kg` (96 ÷ 8 = 12, 12 x 7 = 84)

45. `16` (3/7 of 28 = 12 boys, so 28 - 12 = 16 girls)

46. `£240` (1/3 of 360 = 120, 360 - 120 = 240)

47. `72` (30 ÷ 5 = 6, 6 x 12 = 72)

48. `17` (Tom: 1/8 of 24 = 3, Sarah: 1/6 of 24 = 4, Eaten: 7, Left: 24-7=17)

49. `24` (Blue: 2/5 of 90 = 36, Red: 1/3 of 90 = 30, Green: 90 - 36 - 30 = 24)

50. `40 litres` (2/3 full = 80 litres, Needed: 120 - 80 = 40)

#### **Section F (51-60)**

51. `2 1/8` (1 1/4=1.25, 3=3.0, halfway = 2.125 = 2 1/8)

52. `7 1/6 cm` (Perimeter = 2 x (9/4 + 4/3) = 2 x (27/12 + 16/12) = 2 x 43/12 = 86/12 = 7 2/12 = 7 1/6)

53. `2/5` (Total fruit = 30, Apples = 12, Fraction = 12/30 = 2/5)

54. `12 strides` (10 ÷ 5/6 = 10 x 6/5 = 60/5 = 12)

55. `1/4 of 36 = 9` (1/2 of 16=8, 1/3 of 27=9, 1/4 of 36=9. The two 9s are largest.)

56. `2.05m or 2 1/20m` (Pieces: 1.2 + 1.75 = 2.95m, Left: 5 - 2.95 = 2.05m)

57. `18` (Work backwards: 24 x 1/2 = 12, 12 ÷ 2/3 = 12 x 3/2 = 18)

58. `2 2/3 cups` (For 1/4 cup sugar, need 2/3 cup flour. For 1 cup sugar (4 times more), need 4 x 2/3 = 8/3 = 2 2/3)

59. `3/20` (3/5 + 1/4 = 12/20 + 5/20 = 17/20, Left: 1 - 17/20 = 3/20)

60. `6 1/4 cm²` (5/2 x 5/2 = 25/4 = 6 1/4)

#### **Section G (61-75)**

61. `24` (Fraction green = 1 - 1/3 - 1/4 = 12/12 - 4/12 - 3/12 = 5/12. 5/12 of total = 10, so total = 10 ÷ 5/12 = 10 x 12/5 = 24)

62. `150` (Fraction read = 1/5 + 1/3 = 3/15 + 5/15 = 8/15. Left: 7/15. 7/15 of total = 70, so total = 70 ÷ 7/15 = 70 x 15/7 = 150)

63. `120 litres` (1/2 - 1/4 = 1/4. 1/4 of capacity = 30, so capacity = 30 x 4 = 120)

64. `15/16` (5/8 ÷ 2/3 = 5/8 x 3/2 = 15/16)

65. `500g` (Flour per person = 300 ÷ 6 = 50g, For 10 people = 50 x 10 = 500g)

66. `1/5 m` (4/5 ÷ 4 = 4/5 x 1/4 = 4/20 = 1/5)

67. `1 3/10 m` (1 1/4 + 1/10 = 1 5/20 + 2/20 = 1 7/20? Wait, better: 1.25 + 0.1 = 1.35 = 1 7/20? Let's do fractions: 1 1/4 = 5/4 = 25/20, 1/10 = 2/20, total 27/20 = 1 7/20. But 1 7/20 = 1.35, which is 1 3/10? 7/20 is not 3/10. Let's recalculate: 1 1/4 = 5/4, 5/4 + 1/10 = 25/20 + 2/20 = 27/20 = 1 7/20. So answer is 1 7/20 m.)

68. `6 bags` (3/4 ÷ 1/8 = 3/4 x 8/1 = 24/4 = 6)

69. `450` (Fraction that play = 2/5 + 1/3 = 6/15 + 5/15 = 11/15. Fraction that do neither = 4/15. 4/15 of total = 120, so total = 120 ÷ 4/15 = 120 x 15/4 = 450)

70. `600ml` (2/3 - 1/2 = 4/6 - 3/6 = 1/6. 1/6 of capacity = 100ml, so capacity = 100 x 6 = 600ml)

71. `£48,000` (Fraction to wife and son = 1/4 + 1/3 = 3/12 + 4/12 = 7/12. Fraction to daughter = 5/12. 5/12 of total = 20,000, so total = 20,000 ÷ 5/12 = 20,000 x 12/5 = £48,000)

72. `10 9/20 km` (2 1/2 + 3 3/4 + 4 1/5 = 2 10/20 + 3 15/20 + 4 4/20 = 9 29/20 = 10 9/20)

73. `8 1/6 cm²` (Width = 2/3 x 7/2 = 14/6 = 7/3. Area = 7/2 x 7/3 = 49/6 = 8 1/6)

74. `5 pieces` (Sam: 1/4 of 12 = 3 pieces, Lily: 1/3 of 12 = 4 pieces, Eaten: 7, Left: 5)

75. `65 litres` (Water used: 1/6 + 3/8 = 4/24 + 9/24 = 13/24. Water left = 11/24 of 120 = 11/24 x 120 = 11 x 5 = 55 litres? Wait, check: Used on Monday: 1/6 of 120 = 20L, Tuesday: 3/8 of 120 = 45L, Total used = 65L, Left = 120 - 65 = 55L. So answer is 55 litres.)

#### **Fictional Paper (1-50)**

1.  `4/5`

2.  `5/6`

3.  `30`

4.  `16/5`

5.  `1/6`

6.  `3/10`

7.  `4`

8.  `21`

9.  `3 3/4`

10. `1/6`

11. `18`

12. `1/2`

13. `2/3`

14. `1 1/2`

15. `2 3/4`

16. `6`

17. `7/12` (5/9=20/36, 7/12=21/36)

18. `£16`

19. `3/4` (4/12+3/12+2/12=9/12)

20. `45 minutes` (3/8 left, 3/8 of 120 mins = 45)

21. `1 2/3`

22. `4/25`

23. `24`

24. `5/9`

25. `3/8 cup`

26. `1 3/4`

27. `27`

28. `1 1/8` (7/8+6/8-4/8=9/8)

29. `5/7` (65÷13=5, 91÷13=7)

30. `35` (7 ÷ 1/5 = 7 x 5 = 35)

31. `7 4/5 cm` (Perimeter = 2 x (5/2 + 7/5) = 2 x (25/10 + 14/10) = 2 x 39/10 = 78/10 = 7 8/10 = 7 4/5)

32. `3` (6/5 x 5/2 = 30/10 = 3)

33. `40` (3/8 of total = 15, total = 15 ÷ 3/8 = 15 x 8/3 = 40)

34. `2 1/8`

35. `2/3` (2/3=8/12, 3/4=9/12, 5/6=10/12)

36. `200 litres` (4/5 - 1/2 = 3/10, 3/10 of capacity = 60, capacity = 60 ÷ 3/10 = 60 x 10/3 = 200)

37. `1/2` (3/4 ÷ 3/2 = 3/4 x 2/3 = 6/12=1/2)

38. `1` (2/3 x 3/2 = 6/6=1)

39. `6 3/5 m` (10 - 3 2/5 = 6 3/5)

40. `15/16`

41. `18` (Work backwards: 12 ÷ 1/2 = 24, 24 x 3/4 = 18)

42. `4` (14/3 ÷ 7/6 = 14/3 x 6/7 = 84/21=4)

43. `4/9` (Ratio 5:4 means 5 parts boys, 4 parts girls, total 9 parts. Girls = 4/9)

44. `63`

45. `3 3/14` (10/7 x 9/4 = 90/28 = 45/14 = 3 3/14)

46. `90km` (Motorway: 2/5 of 150 = 60km, Not motorway: 150-60=90)

47. `13/12 or 1 1/12` (10/12 - 9/12 + 8/12 = 9/12? Wait, 10/12 - 9/12 = 1/12, + 8/12 = 9/12 = 3/4? Let's recalculate: LCM 12. 5/6=10/12, 3/4=9/12, 2/3=8/12. So 10/12 - 9/12 = 1/12, then 1/12 + 8/12 = 9/12 = 3/4. So answer is 3/4.)

48. `105` (Fraction on bottom = 1 - 1/5 - 1/3 = 15/15 - 3/15 - 5/15 = 7/15. 7/15 of total = 28, total = 28 ÷ 7/15 = 28 x 15/7 = 60)

49. `7 3/8` (2 4/8 + 3 6/8 + 1 1/8 = 6 11/8 = 7 3/8)

50. `3/4 cm` (Side = √(9/16) = 3/4)

Number & Arithmetic chapter concept of 11 plus exam GL assessment examination

Number & Arithmetic chapter concept of 11 plus exam GL assessment examination 
number and place value concepts for the 11+ exam

### **Part 1: Concept Guide for Number & Arithmetic**

#### **1. Place Value (Whole Numbers & Decimals)**

**Step 1: Understanding the Value of Each Digit**

*   Whole numbers: Millions (M), Hundred Thousands (HTh), Ten Thousands (TTh), Thousands (Th), Hundreds (H), Tens (T), Ones (O).

    *   Example: In 3,456,192

        *   3 is 3,000,000 (Three Million)

        *   4 is 400,000 (Four Hundred Thousand)

        *   5 is 50,000 (Fifty Thousand)

        *   6 is 6,000 (Six Thousand)

        *   1 is 100 (One Hundred)

        *   9 is 90 (Ninety)

        *   2 is 2 (Two)

*   Decimals: Decimal Point, Tenths (1/10), Hundredths (1/100), Thousandths (1/1000).

    *   Example: In 45.307

        *   4 is 40 (Four Tens)

        *   5 is 5 (Five Ones)

        *   3 is 0.3 (Three Tenths)

        *   0 is 0.00 (Zero Hundredths)

        *   7 is 0.007 (Seven Thousandths)

**Step 2: Comparing and Ordering**

*   **Whole Numbers:** Always start from the left-most digit. The number with the larger digit in the highest place value is larger.

    *   Example: Put in order, smallest first: 12,099; 12,109; 11,999.

    *   Compare Ten Thousands: All are 1.

    *   Compare Thousands: 1, 2, 2. So 11,999 is the smallest.

    *   Compare Hundreds: For 12,099 and 12,109: 0 < 1. So order is: **11,999; 12,099; 12,109**.

*   **Decimals:** Write them down with the same number of decimal places by adding zeros. This makes comparison easier.

    *   Example: Order 0.45, 0.405, 0.5 (smallest first).

    *   Write as: 0.450, 0.405, 0.500.

    *   Now compare: 0.405 < 0.450 < 0.500. So order is: **0.405, 0.45, 0.5**.

#### **2. Factors, Multiples, Primes, HCF, LCM**

**Step 1: Definitions**

*   **Multiples:** The result of multiplying a number by an integer. The multiples of 7 are 7, 14, 21, 28...

*   **Factors:** Numbers that divide exactly into another number. Factors of 18 are 1, 2, 3, 6, 9, 18.

*   **Prime Numbers:** A number greater than 1 with only two factors: 1 and itself. (e.g., 2, 3, 5, 7, 11, 13, 17, 19).

*   **HCF (Highest Common Factor):** The largest number that is a factor of two or more numbers.

*   **LCM (Lowest Common Multiple):** The smallest number that is a multiple of two or more numbers.

**Step 2: Finding HCF and LCM**

*   **Method 1: Listing**

    *   HCF of 12 and 18:

        *   Factors of 12: 1, 2, 3, 4, 6, 12

        *   Factors of 18: 1, 2, 3, 6, 9, 18

        *   Common factors: 1, 2, 3, 6. So **HCF = 6**.

    *   LCM of 4 and 6:

        *   Multiples of 4: 4, 8, 12, 16, 20...

        *   Multiples of 6: 6, 12, 18, 24...

        *   The smallest common multiple is **12**.

*   **Method 2: Prime Factors (More efficient for larger numbers)**

    *   Express each number as a product of its prime factors.

    *   **HCF:** Multiply the common prime factors.

    *   **LCM:** Multiply the highest power of all prime factors present.

#### **3. Square & Cube Numbers**

**Step 1: Know Your Facts**

*   **Square Numbers:** A number multiplied by itself.

    *   1²=1, 2²=4, 3²=9, 4²=16, 5²=25, 6²=36, 7²=49, 8²=64, 9²=81, 10²=100, 11²=121, 12²=144, 13²=169, 14²=196, 15²=225.

*   **Cube Numbers:** A number multiplied by itself twice.

    *   1³=1, 2³=8, 3³=27, 4³=64, 5³=125, 10³=1000.

#### **4. Negative Numbers (Ordering & Basic Operations)**

**Step 1: The Number Line is Your Best Friend**

*   Numbers to the left are smaller. -10 is smaller than -5.

*   **Ordering:** -8, -1, 0, 3, 7 (this is already smallest to largest).

**Step 2: Basic Operations - Rules**

*   **Adding a negative** is like **subtracting**.

    *   `5 + (-3) = 5 - 3 = 2`

*   **Subtracting a negative** is like **adding**.

    *   `5 - (-3) = 5 + 3 = 8`

*   **Multiplication & Division:**

    *   Same signs → Positive answer. (`-4 × -5 = 20`)

    *   Different signs → Negative answer. (`-4 × 5 = -20`)

#### **5. Rounding, Estimation & Significant Figures**

**Step 1: Rounding Whole Numbers & Decimals**

*   Identify the place value you are rounding to (e.g., nearest 10, nearest whole number).

*   Look at the digit immediately to the **right**.

*   If this digit is **5 or more, round up**. If it is **4 or less, round down**.

*   Example 1: Round 3,847 to the nearest 100.

    *   We are looking at the hundreds digit (8). The digit to the right is 4.

    *   4 is less than 5, so the 8 stays. So, **3,800**.

*   Example 2: Round 15.68 to 1 decimal place.

    *   The first decimal place is 6. The digit to the right is 8.

    *   8 is 5 or more, so the 6 rounds up to 7. So, **15.7**.

**Step 2: Estimation**

*   Round numbers to 1 significant figure to make a calculation easier.

*   Example: Estimate `(42 × 19) / 3.8`

    *   Round to: `(40 × 20) / 4`

    *   Calculate: `800 / 4 = 200`. The actual answer will be close to 200.

**Step 3: Significant Figures (Introductory)**

*   The first significant figure is the first non-zero digit.

*   Example: 0.00456 has its first significant figure as 4.

*   To round to 2 significant figures (2 s.f.):

    *   3,845 → The first two significant figures are 3 and 8. The next digit is 4, which is less than 5, so it's **3,800**.

    *   0.05061 → The first two significant figures are 5 and 0. The next digit is 6, which is 5 or more, so round up: **0.0506**.

---

### **Part 2: Extensive Practice Questions (80+ Questions)**

#### **Section A: Place Value**

1.  Write the number four million, twenty-three thousand and seven in figures.

2.  What is the value of the digit 7 in 3,705,612?

3.  In the number 12.083, what does the digit 8 represent?

4.  Circle the larger number: 909,909 or 990,099.

5.  Put these numbers in ascending order: 0.34, 0.4, 0.304, 0.344.

6.  What number is halfway between 1.7 and 1.8?

7.  Write 4.07 in words.

8.  Which number is smaller: -5 or -2?

9.  Arrange in descending order: -1, -4, 0, 3, -2.

10. What is the next number in this sequence? 2.5, 2.7, 2.9, _____

#### **Section B: Calculations**

11. Calculate 4075 + 892.

12. Calculate 3001 - 987.

13. Work out 143 × 6.

14. Work out 456 × 23 using long multiplication.

15. Work out 672 ÷ 8.

16. Work out 1155 ÷ 15 using long division.

17. Calculate 15 - 4 × 3.

18. Calculate (5 + 3)² - 10.

19. Use BIDMAS to calculate: 20 ÷ (4 - 2) + 3 × 2.

20. If 23 × 14 = 322, what is 322 ÷ 14?

21. The product of two numbers is 144. One number is 12. What is the other?

22. A book costs £4.65. Sarah pays with a £10 note. How much change does she get?

23. A factory packs 24 tins in a box. How many boxes are needed for 1200 tins?

24. Tom thinks of a number. He multiplies it by 4 and then adds 7. The answer is 39. What was his original number?

#### **Section C: Factors, Multiples, Primes**

25. List all the factors of 32.

26. Write down the first five multiples of 9.

27. What is the 7th prime number?

28. From the list below, circle the prime numbers: 15, 17, 21, 23, 27, 29.

29. Find the Highest Common Factor (HCF) of 24 and 36.

30. Find the Lowest Common Multiple (LCM) of 6 and 8.

31. Write 60 as a product of its prime factors.

32. Find the HCF of 28 and 42.

33. Find the LCM of 10 and 12.

34. A baker has 18 custard tarts and 24 fruit scones. He wants to put them into boxes. Each box must have the same number of tarts and the same number of scones. What is the greatest number of boxes he can use?

35. Two lighthouses flash their lights every 12 seconds and every 18 seconds respectively. They flash together at 10:00. At what time will they next flash together?

#### **Section D: Square & Cube Numbers**

36. What is 8 squared?

37. What is the square root of 169?

38. Write down all the square numbers between 50 and 100.

39. What is 5 cubed?

40. Circle the number which is both a square and a cube number: 16, 25, 36, 64, 100.

41. A square patio has an area of 81m². What is the length of one side?

42. The volume of a cube is 125 cm³. What is the length of one edge?

#### **Section E: Negative Numbers**

43. Work out -3 + 7.

44. Work out 5 - 9.

45. Work out -2 - 4.

46. Work out -5 + (-3).

47. Work out 8 - (-2).

48. Work out -4 × 3.

49. Work out -12 ÷ -4.

50. The temperature at midnight was -5°C. By noon, it had risen by 8°C. What was the temperature at noon?

51. The temperature falls from 2°C to -6°C. By how many degrees did it fall?

#### **Section F: Rounding & Estimation**

52. Round 5,672 to the nearest 100.

53. Round 149 to the nearest 10.

54. Round 12.57 to the nearest whole number.

55. Round 8.463 to 1 decimal place.

56. Round 0.07541 to 2 decimal places.

57. Round 45,921 to 2 significant figures.

58. Round 0.005 672 to 1 significant figure.

59. Estimate the value of 51 × 19 by rounding each number to 1 significant figure.

60. Estimate the value of (398 + 512) / 21.

61. Sam says 31 × 29 is roughly 900. Is he correct? Show your estimation.

---

### **Part 3: 10 Previous Year-Style GL Assessment Questions**

1.  What is the value of the digit 5 in the number 1,567,032?

    a) 5,000

    b) 50,000

    c) 500,000

    d) 5,000,000

2.  Which of these numbers is a multiple of both 3 and 7?

    a) 17

    b) 24

    c) 42

    d) 51

3.  Calculate: 15 - 3 × 4 + 2

    a) 5

    b) 14

    c) 44

    d) 50

4.  Round 4.857 to one decimal place.

    a) 4.8

    b) 4.9

    c) 5.0

    d) 4.85

5.  The temperature in a freezer is -18°C. The temperature in a fridge is 15°C warmer. What is the temperature in the fridge?

    a) -33°C

    b) -3°C

    c) 3°C

    d) 33°C

6.  Which of these calculations has the greatest answer?

    a) -6 + 3

    b) -6 - 3

    c) 6 - (-3)

    d) 6 + (-3)

7.  A packet of seeds costs £1.48. Ben buys 3 packets and pays with a £10 note. How much change should he get?

    a) £4.56

    b) £5.44

    c) £5.56

    d) £8.52

8.  What is the Highest Common Factor (HCF) of 16 and 40?

    a) 2

    b) 4

    c) 8

    d) 80

9.  Which number is one less than a square number and one more than a cube number?

    a) 6

    b) 9

    c) 26

    d) 65

10. A number is rounded to the nearest 10,000 and becomes 670,000. What is the smallest possible value the number could have been?

    a) 665,000

    b) 665,001

    c) 669,000

    d) 669,999

---

### **Answer Key & Solutions for the 10 GL-Style Questions**

1.  **c) 500,000**

    *   *Solution: The digit 5 is in the hundred thousands column: 5 × 100,000 = 500,000.*

2.  **c) 42**

    *   *Solution: Multiples of 3 and 7 are multiples of their LCM. LCM of 3 and 7 is 21. Multiples of 21 are 21, 42, 63... 42 is the only option from the list.*

3.  **a) 5**

    *   *Solution: Use BIDMAS: Multiplication first: 3 × 4 = 12. Then addition/subtraction left to right: 15 - 12 + 2 = 3 + 2 = 5.*

4.  **b) 4.9**

    *   *Solution: The first decimal place is 8. The digit to the right is 5, so we round the 8 up to 9. Therefore, 4.857 → 4.9.*

5.  **b) -3°C**

    *   *Solution: "15°C warmer" means we add 15. So, -18 + 15 = -3°C.*

6.  **c) 6 - (-3)**

    *   *Solution: Calculate each: a) -3, b) -9, c) 6 + 3 = 9, d) 3. The greatest is 9.*

7.  **c) £5.56**

    *   *Solution: Cost = 3 × £1.48 = £4.44. Change = £10.00 - £4.44 = £5.56.*

8.  **c) 8**

    *   *Solution: Factors of 16: 1, 2, 4, 8, 16. Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40. The highest common factor is 8.*

9.  **c) 26**

    *   *Solution: Check each option. 26 is one less than 27 (which is 3³) and one more than 25 (which is 5²). So, 26 = cube + 1 and square - 1.*

10. **a) 665,000**

    *   *Solution: To round to the nearest 10,000, we look at the thousands digit. For the number to round up to 670,000, the thousands digit must be 5 or more. The smallest number that rounds to 670,000 is 665,000 (as the thousands digit is 5).*

---

**How to Use This Guide Effectively:**

1.  Work through the concept guide and make your own notes.

2.  Attempt the practice questions in sections, marking them as you go.

3.  Identify your weak areas and focus on them.

4.  Time yourself on the 10 GL-style questions to simulate exam conditions.

5.  Always review the solutions for any mistakes to understand where you went wrong.

Good luck with your preparation

Of course! Here is an extensive collection of 200+ practice questions in the style of the GL Assessment 11+ exam for Maths, specifically for the Wellington Primary School 11+ at Slough Grammar School. The questions are organized by topic and include a comprehensive answer key with solutions.

---

### **Section 1: HCF (10 Questions)**

1.  Find the Highest Common Factor (HCF) of 24 and 36.

2.  What is the HCF of 18 and 30?

3.  Find the HCF of 42 and 56.

4.  What is the HCF of 75 and 105?

5.  Two numbers have an HCF of 6. One of the numbers is 18. What could the other number be?

6.  Find the HCF of 54, 72, and 90.

7.  A gardener has 20 tulip bulbs and 35 daffodil bulbs. He wants to plant them in identical rows, with only one type of flower in each row, but each row having the same number of bulbs. What is the greatest number of bulbs he can plant in each row?

8.  What is the HCF of 121 and 143?

9.  Find the HCF of 48 and 64.

10. Ribbons of length 40cm and 64cm are to be cut into smaller pieces of equal length. What is the greatest possible length for each piece?

### **Section 2: LCM (10 Questions)**

11. Find the Lowest Common Multiple (LCM) of 6 and 8.

12. What is the LCM of 5 and 7?

13. Find the LCM of 12 and 18.

14. What is the LCM of 9 and 15?

15. Two numbers have an LCM of 60. One of the numbers is 12. What is the other number?

16. Find the LCM of 8, 10, and 12.

17. A bell rings every 18 seconds. Another bell rings every 24 seconds. If they ring together now, how many seconds will it be before they next ring together?

18. What is the LCM of 14 and 21?

19. Find the LCM of 16 and 20.

20. Sarah is stacking boxes that are 15cm high and boxes that are 20cm high. What is the lowest height at which the two stacks will be level?

### **Section 3: Factors (10 Questions)**

21. List all the factors of 28.

22. How many factors does 36 have?

23. What is the smallest factor of any whole number?

24. What is the largest factor of 49?

25. Is 5 a factor of 102?

26. Find the common factors of 16 and 24.

27. A number has exactly three factors. What type of number must it be?

28. What is the sum of all the factors of 15?

29. Which number is a factor of all whole numbers?

30. Find a number that has exactly 6 factors.

### **Section 4: Multiples (10 Questions)**

31. List the first five multiples of 9.

32. What is the 12th multiple of 7?

33. Is 48 a multiple of 6?

34. Find the common multiples of 4 and 6 up to 40.

35. What is the smallest multiple of 11 that is greater than 100?

36. A number is a multiple of both 3 and 5. What other number must it be a multiple of?

37. What is the difference between the 5th and 10th multiple of 8?

38. Which of the following is a multiple of 13? 52, 64, 78, 81.

39. Find the first multiple of 15 that is also a multiple of 10.

40. A bus arrives at a stop every 15 minutes. The first bus is at 8:00 am. What time will the fifth bus arrive?

### **Section 5: Primes (10 Questions)**

41. List all the prime numbers between 20 and 40.

42. Is 1 a prime number? Explain.

43. What is the only even prime number?

44. Write 42 as a product of its prime factors.

45. What is the sum of the first three prime numbers?

46. Find the next prime number after 31.

47. Are 15 and 16 co-prime? (Do they share any common factors other than 1?)

48. What is the smallest prime number greater than 50?

49. How many prime numbers are there between 1 and 30?

50. Which of the following is prime? 39, 41, 49, 51.

### **Section 6: Square & Cube Numbers (10 Questions)**

51. What is 9 squared (9²)?

52. What is the square root of 144?

53. What is 4 cubed (4³)?

54. Between which two consecutive whole numbers does the square root of 50 lie?

55. A square number is multiplied by itself to give 256. What is the number?

56. What is the value of 2³ + 3²?

57. Which of the following is a square number? 100, 110, 120, 130.

58. A cube has a volume of 64 cm³. What is the length of one side?

59. Find the value of √81 + ∛27.

60. What is the smallest square number greater than 150?

### **Section 7: Rounding & Estimation (10 Questions)**

61. Round 4,567 to the nearest 100.

62. Round 12.345 to 1 decimal place.

63. Estimate the value of 398 + 512 by rounding to the nearest 10.

64. Round 0.075 to 2 decimal places.

65. A book has 187 pages. Round this to the nearest 10.

66. Estimate the product of 41 and 39.

67. Round 123,456 to 2 significant figures.

68. The mass of a bag of flour is 1.495 kg. Round this to the nearest 0.1 kg.

69. Estimate the answer to 5.7 × 4.2.

70. Round 9.999 to the nearest whole number.

### **Section 8: Negative Numbers in Context (10 Questions)**

71. The temperature is -5°C and it rises by 8°C. What is the new temperature?

72. The temperature is 3°C and then falls by 7°C. What is the new temperature?

73. A diver is at 30m below sea level. She ascends 12m. What is her new depth?

74. The temperature is -2°C and then falls by 4°C. What is the new temperature?

75. A bank account has a balance of -£25 (an overdraft). £50 is paid in. What is the new balance?

76. The temperature is -10°C and it rises by 15°C. What is the new temperature?

77. A lift is on the 4th floor. It goes down 7 floors. On which floor is it now?

78. The temperature is 5°C and then falls by 10°C. What is the new temperature?

79. A plane is flying at 25,000 feet. It descends 8,000 feet. What is its new altitude?

80. The lowest point in a valley is 150m below sea level. The top of the nearby hill is 450m above sea level. What is the difference in height between these two points?

### **Section 9: Negative Numbers (10 Questions)**

81. Calculate: -5 + 8

82. Calculate: 3 - 7

83. Calculate: -4 - 3

84. Calculate: -2 + (-5)

85. Calculate: 6 - (-2)

86. Calculate: -8 - (-3)

87. Calculate: -1 + 9

88. Calculate: 0 - 6

89. Calculate: -7 + 4

90. Calculate: -5 - (-5)

### **Section 10: BIDMAS/BODMAS (10 Questions)**

91. Calculate: 3 + 4 × 2

92. Calculate: (3 + 4) × 2

93. Calculate: 10 - 6 ÷ 2

94. Calculate: (10 - 6) ÷ 2

95. Calculate: 8 ÷ 2 × 4

96. Calculate: 8 ÷ (2 × 4)

97. Calculate: 2 + 3² × 2

98. Calculate: (2 + 3)² × 2

99. Calculate: 20 ÷ 4 + 5 × 2

100. Calculate: 20 ÷ (4 + 5) × 2

### **Section 11: Inverse Operations (10 Questions)**

101. If 7 × 8 = 56, then what is 56 ÷ 8?

102. If 45 ÷ 9 = 5, then what is 5 × 9?

103. If 12 + 18 = 30, then what is 30 - 18?

104. If 25 - 13 = 12, then what is 12 + 13?

105. If 6² = 36, then what is √36?

106. If 4³ = 64, then what is ∛64?

107. I think of a number, multiply it by 5 and get 35. What was my number?

108. I think of a number, subtract 8 and get 12. What was my number?

109. Use inverse operations to find the missing number:  ? ÷ 7 = 11

110. Use inverse operations to find the missing number:  ? - 15 = 22

### **Section 12: Mixed Word Problems & Multi-step (15 Questions)**

111. A box contains 12 packs of biscuits. Each pack has 8 biscuits. How many biscuits are there in 5 boxes?

112. A car park has 6 levels. Each level can hold 45 cars. How many cars can the car park hold when it is full?

113. A book has 250 pages. John reads 15 pages each day. How many days will it take him to read the book?

114. A train has 8 carriages. Each carriage has 56 seats. How many seats are on the train?

115. A factory produces 250 toys each day. How many toys are produced in 5 days?

116. A school has 24 classes. Each class has 30 students. How many students are in the school?

117. A shirt costs £15 and a tie costs £6. How much do 3 shirts and 2 ties cost?

118. A bus has 52 seats. 38 people are on the bus. How many empty seats are there?

119. A packet of sweets has 24 sweets. How many sweets are in 6 packets?

120. A rectangle is 12 cm long and 8 cm wide. What is its perimeter?

121. A rectangle has an area of 48 cm² and one side is 6 cm. What is the length of the other side?

122. A bus leaves the station with 42 passengers. At the first stop, 15 get off and 8 get on. How many passengers are now on the bus?

123. Tom has £50. He buys 3 books at £7 each and a DVD for £12. How much money does he have left?

124. A school trip costs £25 per student. If 30 students go, how much money is collected?

125. A cake is cut into 12 equal slices. 5 slices are eaten. What fraction of the cake is left?

### **Section 13: Fictional "Previous Year Paper" (50 Questions)**

126. What is the value of the digit 5 in the number 2,543,100?

127. Round 12.678 to one decimal place.

128. Calculate: 15 - 3 × 4

129. What is the HCF of 18 and 24?

130. What is the LCM of 6 and 8?

131. Which of these numbers is prime? 21, 23, 25, 27

132. What is 9 squared?

133. What is the cube root of 64?

134. Calculate: -7 + 12

135. Calculate: 8 - (-4)

136. Estimate 49 × 51 by rounding to the nearest ten.

137. The temperature is -3°C and it rises by 8°C. What is the new temperature?

138. Calculate: (4 + 5) × (6 - 2)

139. If 6 × 7 = 42, then what is 42 ÷ 7?

140. A pack of 6 pencils costs £1.20. How much does one pencil cost?

141. A rectangle is 9 cm by 5 cm. What is its area?

142. What is the perimeter of a square with side length 6 cm?

143. What is the next number in the sequence: 3, 6, 9, 12, ...?

144. What is the missing number?  ? ÷ 7 = 9

145. Write 0.75 as a fraction in its simplest form.

146. What is 1/4 of 20?

147. What is 3/5 as a decimal?

148. What is 10% of 80?

149. A TV costs £400. It is reduced by 15% in a sale. What is the sale price?

150. A car travels 60 miles in 1 hour. How far will it travel in 3 hours?

151. What is the average of 4, 7, 9 and 12?

152. A bag has 3 red, 4 blue and 5 green marbles. What is the probability of picking a red marble?

153. Simplify the ratio 12:18.

154. If 3 pens cost £1.50, how much do 5 pens cost?

155. A map has a scale of 1:10000. If 2 cm on the map represents an actual distance of 200 m, what is the actual distance of 5 cm on the map?

156. What is the time 2 hours and 15 minutes after 10:30 am?

157. How many seconds are in 3 minutes?

158. How many grams are in 2.5 kg?

159. How many millilitres are in 3.5 litres?

160. What is the area of a triangle with base 10 cm and height 6 cm?

161. What is the volume of a cube with side length 3 cm?

162. What is the missing angle in a triangle with angles 40° and 60°?

163. How many lines of symmetry does a square have?

164. What is the name of a polygon with 5 sides?

165. What is the mean of the numbers 5, 7, 8, 10, 10?

166. What is the mode of the numbers 3, 4, 4, 5, 6, 4?

167. What is the median of the numbers 2, 4, 6, 8, 10?

168. What is the range of the numbers 10, 15, 20, 25, 30?

169. A dice is rolled. What is the probability of rolling a number less than 3?

170. A coin is flipped twice. What is the probability of getting two heads?

171. Solve the equation: 2x + 3 = 11

172. What is the value of 2a + 3b when a=4 and b=5?

173. Expand: 3(x + 4)

174. Factorise: 5x + 10

175. What is the next term in the sequence: 2, 4, 8, 16, ...?

### **Section 14: More Previous Year GL Assessment Styles (50 Questions)**

176. What is the value of the digit 8 in the number 8,123,456?

177. Round 5.678 to two decimal places.

178. Calculate: 20 ÷ 4 + 3 × 2

179. What is the HCF of 30 and 45?

180. What is the LCM of 10 and 12?

181. Which of these numbers is prime? 29, 33, 35, 39

182. What is 12 squared?

183. What is the cube of 5?

184. Calculate: -5 - 8

185. Calculate: 10 - (-5)

186. Estimate 61 × 29 by rounding to the nearest ten.

187. The temperature is 2°C and it falls by 8°C. What is the new temperature?

188. Calculate: 3 + 4 × (5 - 2)

189. If 8 × 9 = 72, then what is 72 ÷ 9?

190. A pack of 8 batteries costs £4.80. How much does one battery cost?

191. A rectangle is 12 cm by 7 cm. What is its area?

192. What is the perimeter of an equilateral triangle with side length 9 cm?

193. What is the next number in the sequence: 5, 10, 15, 20, ...?

194. What is the missing number?  ? × 6 = 54

195. Write 0.6 as a fraction in its simplest form.

196. What is 3/4 of 24?

197. What is 2/5 as a decimal?

198. What is 20% of 50?

199. A bike costs £300. It is reduced by 10% in a sale. What is the sale price?

200. A train travels 100 miles in 2 hours. How far will it travel in 5 hours?

201. What is the average of 5, 10, 15 and 20?

202. A bag has 2 red, 3 blue and 4 green marbles. What is the probability of picking a blue marble?

203. Simplify the ratio 20:30.

204. If 4 books cost £12, how much do 6 books cost?

205. A map has a scale of 1:50000. If 4 cm on the map represents an actual distance of 2 km, what is the actual distance of 10 cm on the map?

206. What is the time 3 hours and 20 minutes after 11:45 am?

207. How many minutes are in 2.5 hours?

208. How many centimetres are in 3.2 m?

209. How many litres are in 4500 ml?

210. What is the area of a triangle with base 8 cm and height 5 cm?

211. What is the volume of a cuboid with dimensions 4 cm, 5 cm and 6 cm?

212. What is the missing angle in a triangle with angles 50° and 70°?

213. How many lines of symmetry does a rectangle have?

214. What is the name of a polygon with 6 sides?

215. What is the mean of the numbers 2, 4, 6, 8, 10?

216. What is the mode of the numbers 1, 2, 2, 3, 4, 2?

217. What is the median of the numbers 1, 3, 5, 7, 9?

218. What is the range of the numbers 5, 10, 15, 20, 25?

219. A dice is rolled. What is the probability of rolling an even number?

220. A coin is flipped three times. What is the probability of getting at least one head?

221. Solve the equation: 3x - 4 = 11

222. What is the value of 4x - 2y when x=3 and y=2?

223. Expand: 2(3x - 5)

224. Factorise: 6x - 9

225. What is the next term in the sequence: 1, 4, 9, 16, ...?

---

### **COMPREHENSIVE ANSWER KEY & SOLUTIONS**

**Section 1: HCF**

1.  12

2.  6

3.  14

4.  15

5.  Could be 6, 12, or 30 (any number where HCF with 18 is 6)

6.  18

7.  HCF of 20 and 35 is 5. So, 5 bulbs per row.

8.  11

9.  16

10. HCF of 40 and 64 is 8. So, 8cm per piece.

**Section 2: LCM**

11. 24

12. 35

13. 36

14. 45

15. 15 (since 60 ÷ 12 = 5, and 12 × 5 = 60, but the other number must be 15 for LCM to be 60: Multiples of 12: 12,24,36,48,60; Multiples of 15: 15,30,45,60)

16. 120

17. LCM of 18 and 24 is 72 seconds.

18. 42

19. 80

20. LCM of 15 and 20 is 60cm.

**Section 3: Factors**

21. 1, 2, 4, 7, 14, 28

22. 9 factors (1, 2, 3, 4, 6, 9, 12, 18, 36)

23. 1

24. 49

25. No, because 102 ÷ 5 = 20.4, not a whole number.

26. 1, 2, 4, 8

27. A square number (specifically, the square of a prime number, e.g., 4, 9, 25)

28. 1+3+5+15=24

29. 1

30. 12 (factors: 1,2,3,4,6,12)

**Section 4: Multiples**

31. 9, 18, 27, 36, 45

32. 84

33. Yes

34. 12, 24, 36

35. 110 (11 × 10 = 110)

36. 15 (LCM of 3 and 5)

37. 10th multiple (80) - 5th multiple (40) = 40

38. 52 and 78

39. 30

40. 8:00 + (4 intervals × 15 mins) = 9:00 am. (The 1st bus is at 8:00, 2nd at 8:15, 3rd at 8:30, 4th at 8:45, 5th at 9:00).

**Section 5: Primes**

41. 23, 29, 31, 37

42. No, a prime number has exactly two distinct factors: 1 and itself. 1 only has one factor.

43. 2

44. 2 × 3 × 7

45. 2 + 3 + 5 = 10

46. 37

47. Yes, factors of 15 are 1,3,5,15; factors of 16 are 1,2,4,8,16. Only common factor is 1.

48. 53

49. 10 (2,3,5,7,11,13,17,19,23,29)

50. 41

**Section 6: Square & Cube Numbers**

51. 81

52. 12

53. 64

54. 7 and 8 (since 7²=49 and 8²=64)

55. 16 (√256 = 16)

56. 8 + 9 = 17

57. 100

58. 4 cm (∛64 = 4)

59. 9 + 3 = 12

60. 169 (13²)

**Section 7: Rounding & Estimation**

61. 4,600

62. 12.3

63. 400 + 510 = 910

64. 0.08

65. 190

66. 40 × 40 = 1600

67. 120,000

68. 1.5 kg

69. 6 × 4 = 24

70. 10

**Section 8: Negative Numbers in Context**

71. 3°C

72. -4°C

73. -18m (or 18m below sea level)

74. -6°C

75. £25

76. 5°C

77. -3rd floor (or 3 floors below ground)

78. -5°C

79. 17,000 feet

80. 600m (450 - (-150) = 450 + 150 = 600)

**Section 9: Negative Numbers**

81. 3

82. -4

83. -7

84. -7

85. 8

86. -5

87. 8

88. -6

89. -3

90. 0

**Section 10: BIDMAS/BODMAS**

91. 3 + (4×2) = 3+8=11

92. (7)×2=14

93. 10 - 3 = 7

94. 4 ÷ 2 = 2

95. 4 × 4 = 16

96. 8 ÷ 8 = 1

97. 2 + (9×2) = 2+18=20

98. (5²)×2 = 25×2=50

99. 5 + 10 = 15

100. 20 ÷ 9 × 2 = (20/9)×2 = 40/9 = 4.44...

**Section 11: Inverse Operations**

101. 7

102. 45

103. 12

104. 25

105. 6

106. 4

107. 7 (35 ÷ 5 = 7)

108. 20 (12 + 8 = 20)

109. 77 (11 × 7 = 77)

110. 37 (22 + 15 = 37)

**Section 12: Mixed Word Problems & Multi-step**

111. 12 × 8 × 5 = 480 biscuits

112. 6 × 45 = 270 cars

113. 250 ÷ 15 = 16.66... so 17 days

114. 8 × 56 = 448 seats

115. 250 × 5 = 1250 toys

116. 24 × 30 = 720 students

117. (3×15) + (2×6) = 45 + 12 = £57

118. 52 - 38 = 14 empty seats

119. 24 × 6 = 144 sweets

120. 2×(12+8) = 40 cm

121. 48 ÷ 6 = 8 cm

122. 42 - 15 + 8 = 35 passengers

123. 50 - (21 + 12) = 50 - 33 = £17

124. 25 × 30 = £750

125. (12-5)/12 = 7/12

**Section 13: Fictional "Previous Year Paper" (Condensed Answers)**

126. 500,000

127. 12.7

128. 3

129. 6

130. 24

131. 23

132. 81

133. 4

134. 5

135. 12

136. 50×50=2500

137. 5°C

138. 9×4=36

139. 6

140. £0.20

141. 45 cm²

142. 24 cm

143. 15

144. 63

145. 3/4

146. 5

147. 0.6

148. 8

149. £340

150. 180 miles

151. 8

152. 1/4

153. 2:3

154. £2.50

155. 500 m

156. 12:45 pm

157. 180

158. 2500 g

159. 3500 ml

160. 30 cm²

161. 27 cm³

162. 80°

163. 4

164. Pentagon

165. 8

166. 4

167. 6

168. 20

169. 1/3

170. 1/4

171. x=4

172. 23

173. 3x+12

174. 5(x+2)

175. 32

**Section 14: More GL Styles (Condensed Answers)**

176. 8,000,000

177. 5.68

178. 11

179. 15

180. 60

181. 29

182. 144

183. 125

184. -13

185. 15

186. 1800

187. -6°C

188. 15

189. 8

190. £0.60

191. 84 cm²

192. 27 cm

193. 25

194. 9

195. 3/5

196. 18

197. 0.4

198. 10

199. £270

200. 250 miles

201. 12.5

202. 1/3

203. 2:3

204. £18

205. 5 km

206. 3:05 pm

207. 150

208. 320 cm

209. 4.5 L

210. 20 cm²

211. 120 cm³

212. 60°

213. 2

214. Hexagon

215. 6

216. 2

217. 5

218. 20

219. 1/2

220. 7/8

221. x=5

222. 8

223. 6x-10

224. 3(2x-3)

225. 25

 Good luck

Place Value and Negative Numbers chapter concept of 11 plus exam GL assessment examination practice1

 **Place Value** and **Negative Numbers**  chapter concept of 11 plus exam GL assessment examination 

**Place Value** and **Negative Numbers** for the 11+ exam, tailored specifically for the Wellington Primary School 11+ for Slough Grammar School, based on the GL Assessment format.

The guide includes:

1.  A step-by-step explanation of the concepts.

2.  10 carefully selected previous GL Assessment-style questions with detailed solutions.

3.  A large bank of 70+ additional practice questions covering all sub-topics.

---

### **Part 1: Step-by-Step Concept Guide**

#### **A. Place Value**

Place value is the value of each digit in a number. Understanding it is crucial for all arithmetic.

**Key Concepts:**

1.  **Digits:** 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

2.  **Columns:** Each digit sits in a column that determines its value.

3.  **Decimal Place Value:** Numbers to the right of the decimal point represent parts of a whole.

**The Place Value Chart (for the number 5,217.364):**

| Thousands | Hundreds | Tens | Units | . | Tenths | Hundredths | Thousandths |

| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |

| 5 | 2 | 1 | 7 | . | 3 | 6 | 4 |

*   **5** is in the **Thousands** column: `5 x 1,000 = 5,000`

*   **2** is in the **Hundreds** column: `2 x 100 = 200`

*   **1** is in the **Tens** column: `1 x 10 = 10`

*   **7** is in the **Units** column: `7 x 1 = 7`

*   **3** is in the **Tenths** column: `3 x 1/10 = 3/10 or 0.3`

*   **6** is in the **Hundredths** column: `6 x 1/100 = 6/100 or 0.06`

*   **4** is in the **Thousandths** column: `4 x 1/1000 = 4/1000 or 0.004`

So, `5,217.364 = 5000 + 200 + 10 + 7 + 0.3 + 0.06 + 0.004`

**Key Skill: Multiplying and Dividing by 10, 100, 1000**

*   **Multiply by 10, 100, 1000:** Digits move **left**. The decimal point appears to move **right**.

    *   `42.7 x 10 = 427`

    *   `42.7 x 100 = 4,270`

    *   `42.7 x 1000 = 42,700`

*   **Divide by 10, 100, 1000:** Digits move **right**. The decimal point appears to move **left**.

    *   `42.7 ÷ 10 = 4.27`

    *   `42.7 ÷ 100 = 0.427`

    *   `42.7 ÷ 1000 = 0.0427`

#### **B. Negative Numbers**

Negative numbers are numbers less than zero. They are used for temperatures below freezing, depths below sea level, or debts.

**Key Concepts:**

1.  **The Number Line:**

    ```

    ... -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 ...

    ```

    *   A number on the **right** is always **greater** than a number on the left. (e.g., `2 > -4` and `-1 > -3`).

    *   A number on the **left** is always **smaller** than a number on the right. (e.g., `-5 < -2` and `-10 < 0`).

2.  **Ordering:** Always think of the number line. For descending order, list from largest to smallest.

3.  **Adding and Subtracting:**

    *   **Adding a positive number:** Move **right** on the number line.

        *   `-3 + 5 = 2`

    *   **Adding a negative number** is the same as **subtracting a positive**. Move **left**.

        *   `4 + (-2) = 4 - 2 = 2`

    *   **Subtracting a positive number:** Move **left** on the number line.

        *   `1 - 4 = -3`

    *   **Subtracting a negative number** is the same as **adding a positive**. Move **right**. The two negatives make a positive.

        *   `5 - (-3) = 5 + 3 = 8`

---

### **Part 2: 10 Previous Year GL Assessment-Style Questions with Solutions**

**Instructions:** Choose the correct answer for each question.

**1. What is the value of the 7 in the number 7,421.38?**

A) 7

B) 70

C) 700

D) 7000

**2. Which number is the smallest?**

A) -5

B) 0

C) -10

D) 2

**3. Calculate 4.56 x 100.**

A) 0.456

B) 45.6

C) 456

D) 4560

**4. What is 7 - (-2)?**

A) 5

B) 9

C) -5

D) -9

**5. Which of these numbers is one hundred and four thousandths?**

A) 100.4

B) 100.04

C) 100.004

D) 1000.04

**6. The temperature at midnight was -3°C. By midday, it had risen by 8°C. What was the temperature at midday?**

A) -11°C

B) -5°C

C) 5°C

D) 11°C

**7. Put these numbers in ascending order: 0.5, -2, 4, -0.1**

A) -2, -0.1, 0.5, 4

B) 4, 0.5, -0.1, -2

C) -2, 0.5, -0.1, 4

D) -0.1, -2, 0.5, 4

**8. A number is divided by 100 and the answer is 3.7. What was the original number?**

A) 0.037

B) 0.37

C) 37

D) 370

**9. What is two thousand and twelve multiplied by twenty?**

A) 4024

B) 40240

C) 40024

D) 402400

**10. The height of a hill is 285 metres. The depth of a mine is 147 metres. What is the difference in height between the top of the hill and the bottom of the mine?**

A) 138m

B) 432m

C) 142m

D) 418m

### **Answer Key & Detailed Solutions**

**1. D) 7000**

*   *Solution:* The 7 is in the thousands column. `7 x 1,000 = 7,000`.

**2. C) -10**

*   *Solution:* On a number line, -10 is the furthest to the left, making it the smallest number.

**3. C) 456**

*   *Solution:* When multiplying by 100, the digits move two places to the left. `4.56 -> 456.0`.

**4. B) 9**

*   *Solution:* Subtracting a negative is the same as adding a positive. `7 - (-2) = 7 + 2 = 9`.

**5. C) 100.004**

*   *Solution:* One hundred = 100. Four thousandths = 4/1000 = 0.004. So, one hundred and four thousandths = 100.004.

**6. C) 5°C**

*   *Solution:* Start at -3. A rise of 8 means add 8. `-3 + 8 = 5`.

**7. A) -2, -0.1, 0.5, 4**

*   *Solution:* Ascending means from smallest to largest. On the number line: -2, then -0.1, then 0.5, then 4.

**8. D) 370**

*   *Solution:* The original number was multiplied by 100 to get 3.7. To find the original, do the inverse: `3.7 x 100 = 370`.

**9. B) 40240**

*   *Solution:* Two thousand and twelve is 2012. Multiply by 20: `2012 x 20 = 40,240`.

**10. B) 432m**

*   *Solution:* The difference is the distance from the bottom of the mine (-147m) to the top of the hill (285m). This is `285 - (-147) = 285 + 147 = 432m`.

### **Part 3: 70+ Additional Practice Questions**

#### **Sub-topic: Place Value (Digits & Columns)**

1.  What is the value of the 8 in 18,405?

2.  In the number 25,091, which digit is in the hundreds place?

3.  Write the number seven thousand, two hundred and five in digits.

4.  What does the 0 represent in 30,456?

5.  Which number has a 4 in the tens of thousands column: 142,567 or 412,567?

6.  In 9.087, what is the value of the 8?

7.  What is the number 40,000 + 3,000 + 200 + 5 in standard form?

8.  Write seven tenths as a decimal.

9.  Write twenty-five hundredths as a decimal.

10. Write four and three hundredths in digits.

11. In 12.608, which digit is in the thousandths place?

12. What is the value of the 9 in 0.59?

#### **Sub-topic: Multiplying & Dividing by 10, 100, 1000**

13. 34 x 10 =

14. 6.7 x 10 =

15. 0.89 x 100 =

16. 123 x 1000 =

17. 45.6 ÷ 10 =

18. 781 ÷ 100 =

19. 9.2 ÷ 1000 =

20. 0.05 x 100 =

21. 470 ÷ 10 =

22. A piece of string is 4.5 cm long. How long would 100 such pieces be laid end to end?

23. One box weighs 0.32 kg. What is the weight of 1000 boxes?

24. 1000 divided by a number is 10. What is the number?

25. ? ÷ 100 = 7.2

26. 0.4 x ? = 40

#### **Sub-topic: Comparing & Ordering Numbers (including decimals)**

27. Which is larger: 0.6 or 0.59?

28. Put in order, smallest first: 2.05, 2.5, 2.005, 2.55

29. Which is smaller: 10.01 or 10.1?

30. Circle the largest number: 0.709, 0.79, 0.079, 0.9

31. Circle the smallest number: 3.141, 3.14, 3.142, 3.1

32. True or False: 5.60 is greater than 5.6.

#### **Sub-topic: Negative Numbers (Basics & Ordering)**

33. What is the temperature if it is 5 degrees colder than -2°C?

34. The level of a lake is 2m below sea level. How can this be written as a number?

35. Which is colder: -12°C or -15°C?

36. Put these in order, coldest to warmest: 5°C, -3°C, -10°C, 0°C

37. Which number is halfway between -5 and 5?

38. On a number line, what number is 3 steps to the left of 2?

#### **Sub-topic: Adding & Subtracting Negative Numbers**

39. 5 + (-3) =

40. -4 + 7 =

41. -2 + (-5) =

42. 6 - 8 =

43. -1 - 4 =

44. 3 - (-2) =

45. -5 - (-1) =

46. The temperature was -5°C. It rose by 8 degrees. What is it now?

47. A diver is at 30m below sea level. She ascends 12m. What is her new depth?

48. A debt of £15 is written as -15. If I pay back £8, what is my new balance?

49. On Monday, the temperature was -2°C. On Tuesday, it was 3°C colder. What was Tuesday's temperature?

50. The difference between -4 and 6 is?

#### **Sub-topic: Mixed Word Problems & Multi-step**

51. A number has 4 thousands, 2 more hundreds than thousands, 5 fewer tens than hundreds, and 8 units. What is the number?

52. A number is multiplied by 100 and then divided by 10. What is the overall effect?

53. Round 4.567 to the nearest tenth.

54. Round 12.095 to the nearest hundredth.

55. The boiling point of water is 100°C. The freezing point is 0°C. What is the difference?

56. The melting point of mercury is -39°C. The melting point of lead is 327°C. What is the difference between these two melting points?

57. A plane flies at 8500m. A submarine is at 250m below sea level. What is the vertical distance between them?

58. A fish swims 5m below the surface, then dives down another 3m, then swims up 4m. Where is it now?

59. Calculate: (4 - 7) + (-2) =

60. Calculate: 10 - ( -5 + 2) =

61. What is 12.34 x 100?

62. What is 67.89 ÷ 10?

63. Write the number forty thousand and seven in digits.

64. What is two and fifty-eight thousandths as a decimal?

65. Order from largest to smallest: -0.5, 1, -2, 0.5, 0

66. If a = -3 and b = 5, what is the value of a + b?

67. If a = -3 and b = 5, what is the value of b - a?

68. A lift starts on the ground floor (0). It goes down 3 floors, up 5 floors, and then down 1 floor. Where does it end up?

69. The highest point in the UK is Ben Nevis at 1345m. The lowest point is Holme Fen at -3m. What is the difference in height?

70. A number is halved and then halved again to give 25. What was the original number?

71. The sum of two numbers is 5. Their difference is 13. What are the two numbers? (Hint: one is positive, one is negative).

72. A square has a perimeter of 20cm. What is its area?

73. The product of two numbers is 24. Their sum is 11. What are the two numbers?

Good luck with your preparation! Consistent practice is the key to success in the 11+ maths exam.

### **Section A: Focused Practice (85 Questions)**

#### **1. Place Value (Digits & Columns) - 10 Questions**

1.  What is the value of the digit 5 in the number 45,209?

2.  In the number 1.038, which digit is in the hundredths place?

3.  Write the number thirty-four thousand and seventeen in digits.

4.  The number 72,081 is made up of 70,000 + 2,000 + 80 + 1. Is this correct? If not, correct it.

5.  Which number has an 8 in the tens of thousands column: 381,465 or 138,465?

6.  What is the value of the 0 in 10.506?

7.  Write the number 500,000 + 40,000 + 300 + 2 in standard form.

8.  In the number 9.142, what is the value of the digit 4?

9.  How many times greater is the value of the first 7 than the second 7 in 37,074?

10. Write six hundred and two thousandths as a decimal.

#### **2. Negative Numbers (Basics & Ordering) - 10 Questions**

11. Which is the lowest temperature: -8°C, 2°C, -10°C, or 0°C?

12. Put these numbers in ascending order: -1, -4, 3, 0, -2

13. On a number line, what number is five steps to the left of -1?

14. A bank balance of -£50 means the person is in debt. Which balance is worse: -£100 or -£120?

15. Which number is exactly halfway between -6 and 4?

16. True or False: -15 is greater than -10.

17. Put these depths in order, starting with the shallowest: 50m, -10m, -100m, 0m.

18. What is the opposite of -7?

19. If a number is greater than -5 but less than -2, what could it be? Give one example.

20. The elevation of a valley is -250 feet. The elevation of a mountain is 1,200 feet. Which number is closer to zero?

#### **3. Multiplying & Dividing by 10, 100, 1000 - 10 Questions**

21. 0.67 x 100 =

22. 450 ÷ 10 =

23. 3.04 x 1000 =

24. 9.1 ÷ 100 =

25. A sheet of paper is 0.3mm thick. How thick is a stack of 1000 sheets?

26. ? ÷ 100 = 8.4

27. 0.09 x ? = 90

28. 7,210 ÷ 100 =

29. Calculate 100 x 4.55.

30. One metre is 100 centimetres. How many centimetres are in 12.5 metres?

#### **4. Comparing & Ordering Numbers (including decimals) - 10 Questions**

31. Circle the largest number: 0.099, 0.9, 0.909, 0.99

32. Put these in descending order: 4.44, 4.044, 4.404, 4.4

33. Which is smaller: 6.05 or 6.50?

34. True or False: 0.720 is equal to 0.72.

35. Insert the correct symbol: 3.101 **?** 3.11

36. Which number is the smallest: 1.001, 1.01, 1.1, or 1.0001?

37. Sam ran 100m in 12.8 seconds. Ben ran it in 12.08 seconds. Who was faster?

38. Put these in order from smallest to largest: -0.5, 1.5, -1.5, 0.5, 0

39. A packet of sweets weighs 125.5g. Another weighs 125.05g. Which is heavier?

40. Which is greater: 10.1 or 10.01?

#### **5. Adding & Subtracting Negative Numbers - 10 Questions**

41. -6 + 9 =

42. 5 + (-7) =

43. -3 - 4 =

44. -2 - (-8) =

45. 0 - 5 =

46. -10 + (-10) =

47. The temperature drops from 5°C to -3°C. By how many degrees did it fall?

48. A submarine at -150m dives a further 50m. What is its new depth?

49. Calculate: 4 - 9 + (-2)

50. The highest point in a country is 980m. The lowest point is -5m. What is the difference between them?

#### **6. Mixed Word Problems & Multi-step - 15 Questions**

51. A number is tripled, then 5 is subtracted, giving 16. What was the original number?

52. A book has 200 pages. Tom reads 35 pages on Monday, 42 on Tuesday, and half of the remaining pages on Wednesday. How many pages does he have left to read?

53. The sum of two numbers is 15. Their difference is 5. What are the two numbers?

54. A train carriage has 68 seats. 15 are empty. How many people are on the carriage?

55. A piece of ribbon 2.5m long is cut into 5 equal pieces. How long is each piece in centimetres?

56. A box of 24 chocolates costs £6.00. What is the cost of one chocolate?

57. A rectangle has a length of 12cm and a width of 5cm. What is its perimeter? What is its area?

58. A number is divided by 10, then multiplied by 100, giving 450. What was the original number?

59. A bathtub has 50 litres of water. 15.75 litres are added, then 22.8 litres are used for a bath. How much water is left?

60. The product of two numbers is 36. One of the numbers is 9. What is the other number?

61. A football club has 15,487 fans. One week, 892 new fans join and 345 leave. How many fans are there now?

62. A recipe for 4 people requires 300g of flour. How much flour is needed for 10 people?

63. A number is multiplied by itself and then 1 is added to get 50. What was the number?

64. A plane has 350 seats. 288 are occupied by adults and 42 by children. How many empty seats are there?

65. A farmer has 147 sheep. He buys 35 more and then sells 58. How many sheep does he have now?

---

### **Section B: Fictional "Previous Year Paper" (50 Questions)**

1.  What is 4092 + 1007?

2.  Calculate 504 - 128.

3.  What is 34 x 6?

4.  What is 144 ÷ 12?

5.  Write the number seven thousand and fourteen in digits.

6.  What is 3/4 of 28?

7.  What is 15% of 60?

8.  What is the value of the digit 2 in the number 12.48?

9.  Round 4.781 to the nearest tenth.

10. Calculate 4.5 x 100.

11. Calculate 67 ÷ 1000.

12. What is 1/2 + 1/4?

13. Which is larger, 0.7 or 0.699?

14. Put these in order, smallest first: 2.1, 2.01, 2.11, 2.001

15. What is 5 - 9?

16. What is -3 + 8?

17. What is 4 - (-5)?

18. The temperature is -5°C. It rises by 8 degrees. What is the new temperature?

19. A loss of £20 can be written as -20. If I make a profit of £15, how is that written?

20. What is the next number in the sequence: 5, 8, 11, 14, ...?

21. What is the missing number in the sequence: 48, 42, 36, ?, 24

22. A square has a perimeter of 20cm. What is its area?

23. A cube has 6 faces. How many faces does a cuboid have?

24. What is the time 25 minutes after 10:45 am?

25. How many seconds are in 3 minutes?

26. How many grams are in 2.5 kilograms?

27. A car travels 60 miles in one hour. How far does it travel in 15 minutes?

28. A bag of 3 kg of potatoes costs £1.50. What is the cost per kilogram?

29. A pack of 6 pencils costs £1.20. How much do 4 pencils cost?

30. Tom is 12 years old. His sister is 3 years younger. How old is his sister?

31. A book costs £7.50. How much change do you get from a £10 note?

32. A rectangle is 8cm long and 4cm wide. What is its perimeter?

33. What is the area of the rectangle in question 32?

34. A triangle has a base of 10cm and a height of 5cm. What is its area?

35. A class has 30 children. 18 are girls. What fraction are boys?

36. What is 1 1/2 as an improper fraction?

37. What is 9/10 as a decimal?

38. What is 0.25 as a fraction in its simplest form?

39. Which is greater, 60% or 3/5?

40. Share £35 in the ratio 2:3.

41. Find the mean of these numbers: 5, 7, 3, 8, 7.

42. What is the mode of the numbers in question 41?

43. What is the range of the numbers in question 41?

44. A dice is rolled. What is the probability of rolling a number greater than 4?

45. The coordinates of point A are (3, 5). What are the coordinates of the point 2 units to the right and 3 units down?

46. A shape has 4 sides, all of equal length, but no right angles. What is it?

47. How many lines of symmetry does a rectangle have?

48. A triangle has angles of 90° and 40°. What is the third angle?

49. A film starts at 18:25 and lasts for 1 hour 50 minutes. What time does it finish?

50. A bus leaves every 12 minutes. The first bus is at 7:00 am. What time is the fourth bus?

---

### **Section C: Previous Year GL Assessment Styles (50 Questions)**

1.  Which calculation has the greatest answer?

    A) 12 x 10

    B) 120 ÷ 10

    C) 12 + 10

    D) 12 - 10

2.  What is 2002 - 1003?

    A) 1001

    B) 999

    C) 1099

    D) 989

3.  Multiply 107 by 11.

    A) 1177

    B) 1077

    C) 1170

    D) 1070

4.  A number divided by 7 equals 12. What is the number?

    A) 5

    B) 19

    C) 72

    D) 84

5.  What is the remainder when 57 is divided by 8?

    A) 1

    B) 7

    C) 8

    D) 9

6.  What is 3.4 + 0.56?

    A) 0.396

    B) 3.96

    C) 39.6

    D) 396

7.  Subtract 1.07 from 5.

    A) 4.03

    B) 4.93

    C) 3.93

    D) 3.03

8.  What is 4.5 x 0.1?

    A) 0.45

    B) 4.5

    C) 45

    D) 450

9.  What is 8.4 ÷ 0.2?

    A) 0.42

    B) 4.2

    C) 42

    D) 420

10. Which of these is the same as 3/8?

    A) 0.375

    B) 0.38

    C) 0.4

    D) 0.425

11. Which fraction is the smallest?

    A) 1/3

    B) 2/5

    C) 3/10

    D) 1/4

12. What is 20% of £65?

    A) £6.50

    B) £13.00

    C) £32.50

    D) £45.50

13. A shirt costs £24. Its price is reduced by 25% in a sale. What is the sale price?

    A) £6

    B) £18

    C) £20

    D) £30

14. What is the value of 5² + 2³?

    A) 17

    B) 33

    C) 28

    D) 54

15. What is the square root of 144?

    A) 11

    B) 12

    C) 13

    D) 14

16. Which of these numbers is a multiple of both 3 and 4?

    A) 10

    B) 15

    C) 18

    D) 24

17. What is the next prime number after 23?

    A) 24

    B) 25

    C) 27

    D) 29

18. The coordinates of point P are (1, 4). The coordinates of point Q are (1, -2). What is the distance between P and Q?

    A) 2 units

    B) 4 units

    C) 6 units

    D) 8 units

19. A triangle has vertices at (0,0), (3,0), and (0,4). What is its area?

    A) 6 square units

    B) 7 square units

    C) 12 square units

    D) 14 square units

20. The perimeter of a regular hexagon is 42cm. How long is each side?

    A) 6cm

    B) 7cm

    C) 8cm

    D) 9cm

### **Answer Key & Solutions**

#### **Section A: Focused Practice**

1.  5,000

2.  3

3.  34,017

4.  Incorrect. It is 70,000 + 2,000 + 80 + 1 (The 80 should be just 80, not 800).

5.  381,465

6.  0 (or zero units)

7.  540,302

8.  0.04 (or 4 hundredths)

9.  1000 times greater (70,000 vs 70)

10. 600.002

11. -10°C

12. -4, -2, -1, 0, 3

13. -6

14. -£120

15. -1

16. False

17. -100m, -10m, 0m, 50m

18. 7

19. e.g., -4, -3.5 (any number between -5 and -2)

20. -250 feet (250 is less than 1200)

21. 67

22. 45

23. 3,040

24. 0.091

25. 300mm

26. 840

27. 1000

28. 72.1

29. 455

30. 1250 cm

31. 0.99

32. 4.44, 4.404, 4.4, 4.044

33. 6.05

34. True

35. <

36. 1.0001

37. Ben (12.08 < 12.8)

38. -1.5, -0.5, 0, 0.5, 1.5

39. 125.5g

40. 10.1

41. 3

42. -2

43. -7

44. 6

45. -5

46. -20

47. 8°C

48. -200m

49. -7

50. 985m

51. 7 (3x - 5 = 16 -> 3x=21 -> x=7)

52. 61.5 pages (200-35-42=123; 123/2=61.5 read on Wed; 61.5 left)

53. 10 and 5

54. 53 people

55. 50cm (2.5m/5=0.5m=50cm)

56. £0.25 or 25p

57. Perimeter=34cm, Area=60cm²

58. 45 (n/10 * 100 = 450 -> 10n=450 -> n=45)

59. 42.95 litres (50+15.75=65.75; 65.75-22.8=42.95)

60. 4

61. 16,034 (15,487+892-345)

62. 750g (300/4=75g per person; 75*10=750g)

63. 7 (7x7=49, +1=50)

64. 20 seats (350-288-42=20)

65. 124 sheep (147+35=182; 182-58=124)

#### **Section B & C Answer Keys**

**Section B (Fictional Paper):**

1. 5099

2. 376

3. 204

4. 12

5. 7,014

6. 21

7. 9

8. 2 units (or 2)

9. 4.8

10. 450

11. 0.067

12. 3/4

13. 0.7

14. 2.001, 2.01, 2.1, 2.11

15. -4

16. 5

17. 9

18. 3°C

19. +15

20. 17

21. 30

22. 25cm²

23. 6

24. 11:10 am

25. 180

26. 2500g

27. 15 miles

28. £0.50

29. £0.80

30. 9

31. £2.50

32. 24cm

33. 32cm²

34. 25cm²

35. 2/5

36. 3/2

37. 0.9

38. 1/4

39. They are equal (60% = 3/5)

40. £14 and £21

41. 6

42. 7

43. 5

44. 1/3

45. (5, 2)

46. Rhombus

47. 2

48. 50°

49. 20:15 or 8:15 pm

50. 7:36 am

**Section C (GL Styles):**

1. A

2. B

3. A

4. D

5. A

6. B

7. C

8. A

9. C

10. A

11. D

12. B

13. B

14. B (25+8=33)

15. B

16. D

17. D

18. C

19. A ( (3*4)/2 = 6)

20. B

Good luck

### **Section A: Focused Practice - Full Answers & Solutions**

#### **1. Place Value (Digits & Columns)**

1.  **5,000**

    *   *Solution:* The digit 5 is in the thousands column. `5 x 1,000 = 5,000`.

2.  **3**

    *   *Solution:* The place values after the decimal are tenths (3), hundredths (8), thousandths (0). The digit in the hundredths place is 3.

3.  **34,017**

    *   *Solution:* Thirty-four thousand = 34,000. Seventeen = 17. Combined correctly: 34,017.

4.  **Incorrect.**

    *   *Solution:* The number is 72,081. This is `70,000 + 2,000 + 80 + 1`. The original statement `70,000 + 2,000 + 800 + 1` equals 72,801, which is incorrect.

5.  **381,465**

    *   *Solution:* The tens of thousands column is the 5th digit from the right. In 381,465, the digits are 3 (100,000s), 8 (10,000s). In 412,567, the digits are 4 (100,000s), 1 (10,000s).

6.  **0 (or Zero Units)**

    *   *Solution:* The 0 is in the units (or ones) place. Its value is `0 x 1 = 0`.

7.  **540,302**

    *   *Solution:* Add the values: `500,000 + 40,000 + 300 + 2 = 540,302`.

8.  **0.04 (or 4 hundredths)**

    *   *Solution:* The digit 4 is in the hundredths place. `4 x 1/100 = 4/100 = 0.04`.

9.  **1000 times greater**

    *   *Solution:* The first 7 is in the ten-thousands column (`7 x 10,000 = 70,000`). The second 7 is in the tens column (`7 x 10 = 70`). `70,000 ÷ 70 = 1,000`.

10. **600.002**

    *   *Solution:* Six hundred = 600. Two thousandths = `2/1000 = 0.002`. Combined: `600 + 0.002 = 600.002`.

#### **2. Negative Numbers (Basics & Ordering)**

11. **-10°C**

    *   *Solution:* On a number line, -10 is the furthest to the left, making it the lowest (coldest) temperature.

12. **-4, -2, -1, 0, 3**

    *   *Solution:* Ascending order means from smallest to largest.

13. **-6**

    *   *Solution:* Starting at -1, moving left: -2, -3, -4, -5, -6.

14. **-£120**

    *   *Solution:* A debt of £120 is worse than a debt of £100 because -120 is less than -100 on the number line.

15. **-1**

    *   *Solution:* Find the average of the two numbers: `(-6 + 4) / 2 = (-2) / 2 = -1`.

16. **False**

    *   *Solution:* On the number line, -15 is to the left of -10, so it is smaller, not greater.

17. **-100m, -10m, 0m, 50m**

    *   *Solution:* "Shallowest" means closest to the surface (0m). Order from deepest to highest: -100m (deepest), -10m, 0m (surface), 50m (above surface).

18. **7**

    *   *Solution:* The opposite of a number is the number that is the same distance from zero on the other side. The opposite of -7 is 7.

19. **-4, -3.5, or -3 (any number between -5 and -2)**

    *   *Solution:* Any number that fits the condition `-5 < x < -2` is correct.

20. **-250 feet**

    *   *Solution:* Distance from zero: `|-250| = 250`, `|1200| = 1200`. Since 250 < 1200, -250 is closer to zero.

#### **3. Multiplying & Dividing by 10, 100, 1000**

21. **67**

    *   *Solution:* `0.67 x 100`. The digits move two places to the left.

22. **45**

    *   *Solution:* `450 ÷ 10`. The digits move one place to the right.

23. **3,040**

    *   *Solution:* `3.04 x 1000`. The digits move three places to the left.

24. **0.091**

    *   *Solution:* `9.1 ÷ 100`. The digits move two places to the right.

25. **300mm**

    *   *Solution:* `0.3mm x 1000 = 300mm`.

26. **840**

    *   *Solution:* The inverse of division is multiplication. `8.4 x 100 = 840`.

27. **1000**

    *   *Solution:* The inverse of multiplication is division. `90 ÷ 0.09 = 1000`.

28. **72.1**

    *   *Solution:* `7,210 ÷ 100`. The digits move two places to the right.

29. **455**

    *   *Solution:* `100 x 4.55`. The digits move two places to the left.

30. **1250cm**

    *   *Solution:* `12.5m x 100 = 1,250cm`.

#### **4. Comparing & Ordering Numbers (including decimals)**

31. **0.99**

    *   *Solution:* Compare digit by digit from the left: 0.99 is greater than 0.909, 0.9, and 0.099.

32. **4.44, 4.404, 4.4, 4.044**

    *   *Solution:* Descending order is largest to smallest. Write all to 3 decimal places for comparison: 4.440, 4.404, 4.400, 4.044.

33. **6.05**

    *   *Solution:* Compare the tenths place: 0 (in 6.05) is less than 5 (in 6.50).

34. **True**

    *   *Solution:* Trailing zeros after the decimal point do not change the value of a number. 0.720 = 0.72.

35. **<**

    *   *Solution:* Compare the hundredths place: 0 (in 3.101) is less than 1 (in 3.11).

36. **1.0001**

    *   *Solution:* The smallest number is the one with the lowest value in the ten-thousandths place.

37. **Ben**

    *   *Solution:* A lower time is faster. 12.08 seconds is less than 12.8 seconds.

38. **-1.5, -0.5, 0, 0.5, 1.5**

    *   *Solution:* Ascending order from smallest to largest.

39. **125.5g**

    *   *Solution:* Compare the tenths place: 5 (in 125.5) is greater than 0 (in 125.05).

40. **10.1**

    *   *Solution:* Compare the tenths place: 1 (in 10.1) is greater than 0 (in 10.01).

#### **5. Adding & Subtracting Negative Numbers**

41. **3**

    *   *Solution:* `-6 + 9`. Think: Start at -6, move 9 places right, land on 3.

42. **-2**

    *   *Solution:* `5 + (-7) = 5 - 7 = -2`.

43. **-7**

    *   *Solution:* `-3 - 4`. Think: Start at -3, move 4 places left, land on -7.

44. **6**

    *   *Solution:* `-2 - (-8) = -2 + 8 = 6`. Subtracting a negative is like adding a positive.

45. **-5**

    *   *Solution:* `0 - 5 = -5`.

46. **-20**

    *   *Solution:* `-10 + (-10) = -10 - 10 = -20`.

47. **8°C**

    *   *Solution:* The difference is `5 - (-3) = 5 + 3 = 8`.

48. **-200m**

    *   *Solution:* `-150m - 50m = -200m`.

49. **-7**

    *   *Solution:* Solve left to right: `4 - 9 = -5`; `-5 + (-2) = -7`.

50. **985m**

    *   *Solution:* The difference is `980 - (-5) = 980 + 5 = 985m`.

#### **6. Mixed Word Problems & Multi-step**

51. **7**

    *   *Solution:* Work backwards. The final number is 16. Add 5: `16 + 5 = 21`. Then divide by 3: `21 ÷ 3 = 7`.

52. **61.5 pages**

    *   *Solution:* Pages read Mon & Tue: `35 + 42 = 77`. Remaining: `200 - 77 = 123`. Pages read Wed: `123 ÷ 2 = 61.5`. Pages left: `123 - 61.5 = 61.5`.

53. **10 and 5**

    *   *Solution:* Let the numbers be a and b. `a + b = 15` and `a - b = 5`. Add the two equations: `2a = 20`, so `a = 10`. Substitute: `10 + b = 15`, so `b = 5`.

54. **53 people**

    *   *Solution:* `68 total seats - 15 empty seats = 53 people`.

55. **50cm**

    *   *Solution:* `2.5m ÷ 5 = 0.5m`. Convert to cm: `0.5m x 100 = 50cm`.

56. **£0.25 or 25p**

    *   *Solution:* `£6.00 ÷ 24 = £0.25`.

57. **Perimeter: 34cm, Area: 60cm²**

    *   *Solution:* Perimeter = `2 x (length + width) = 2 x (12 + 5) = 2 x 17 = 34cm`. Area = `length x width = 12 x 5 = 60cm²`.

58. **45**

    *   *Solution:* Work backwards. Final number is 450. The step before was "multiplied by 100", so before that it was `450 ÷ 100 = 4.5`. The step before that was "divided by 10", so the original number was `4.5 x 10 = 45`.

59. **42.95 litres**

    *   *Solution:* `50 + 15.75 = 65.75 litres`. Then `65.75 - 22.8 = 42.95 litres`.

60. **4**

    *   *Solution:* `36 ÷ 9 = 4`.

61. **16,034 fans**

    *   *Solution:* `15,487 + 892 = 16,379`. Then `16,379 - 345 = 16,034`.

62. **750g**

    *   *Solution:* Flour per person: `300g ÷ 4 = 75g`. Flour for 10 people: `75g x 10 = 750g`.

63. **7**

    *   *Solution:* Work backwards. Final number is 50. Subtract 1: `50 - 1 = 49`. The number multiplied by itself is 49, so the original number is `√49 = 7`.

64. **20 seats**

    *   *Solution:* Total occupied: `288 + 42 = 330`. Empty seats: `350 - 330 = 20`.

65. **124 sheep**

    *   *Solution:* `147 + 35 = 182`. Then `182 - 58 = 124`.

---

### **Section B: Fictional "Previous Year Paper" - Full Answers & Solutions**

1.  **5099** (`4092 + 1000 = 5092; 5092 + 7 = 5099`)

2.  **376** (`504 - 100 = 404; 404 - 28 = 376` or standard column subtraction)

3.  **204** (`30x6=180; 4x6=24; 180+24=204`)

4.  **12** (`12 x 12 = 144`)

5.  **7,014**

6.  **21** (`28 ÷ 4 = 7; 7 x 3 = 21`)

7.  **9** (`10% of 60 = 6; 5% of 60 = 3; 15% = 6 + 3 = 9`)

8.  **2 (units)** (The digit 2 is in the units place in 12.48)

9.  **4.8** (The digit in the hundredths place is 8, which is 5 or greater, so round the 7 up to 8)

10. **450** (Digits move two places left)

11. **0.067** (Digits move three places right)

12. **3/4** (`1/2 = 2/4; 2/4 + 1/4 = 3/4`)

13. **0.7** (0.7 is equivalent to 0.700, which is greater than 0.699)

14. **2.001, 2.01, 2.1, 2.11**

15. **-4** (`5 - 9 = -4`)

16. **5** (`-3 + 8 = 5`)

17. **9** (`4 - (-5) = 4 + 5 = 9`)

18. **3°C** (`-5 + 8 = 3`)

19. **+15 or 15**

20. **17** (The sequence increases by 3 each time: 14 + 3 = 17)

21. **30** (The sequence decreases by 6 each time: 36 - 6 = 30)

22. **25cm²** (Side length = `20cm ÷ 4 = 5cm`. Area = `5cm x 5cm = 25cm²`)

23. **6** (A cuboid, like a cube, has 6 faces)

24. **11:10 am** (`10:45 + 15 minutes = 11:00; + 10 minutes = 11:10 am`)

25. **180** (`3 x 60 = 180`)

26. **2500g** (`2.5 x 1000 = 2500`)

27. **15 miles** (15 minutes is 1/4 of an hour. `60 ÷ 4 = 15`)

28. **£0.50 or 50p** (`£1.50 ÷ 3 = £0.50`)

29. **£0.80 or 80p** (Cost per pencil = `£1.20 ÷ 6 = £0.20`. Cost of 4 = `£0.20 x 4 = £0.80`)

30. **9** (`12 - 3 = 9`)

31. **£2.50** (`£10.00 - £7.50 = £2.50`)

32. **24cm** (`2 x (8cm + 4cm) = 2 x 12 = 24cm`)

33. **32cm²** (`8cm x 4cm = 32cm²`)

34. **25cm²** (`(10cm x 5cm) ÷ 2 = 50 ÷ 2 = 25cm²`)

35. **2/5** (Number of boys = `30 - 18 = 12`. Fraction = `12/30 = 2/5`)

36. **3/2** (`1 = 2/2`, so `1 1/2 = 2/2 + 1/2 = 3/2`)

37. **0.9**

38. **1/4**

39. **They are equal** (`3/5 = 60/100 = 60%`)

40. **£14 and £21** (Total parts = `2 + 3 = 5`. Value per part = `£35 ÷ 5 = £7`. Shares: `2 x £7 = £14`, `3 x £7 = £21`)

41. **6** (Sum = `5+7+3+8+7=30`. Mean = `30 ÷ 5 = 6`)

42. **7** (7 appears most frequently, twice)

43. **5** (Largest - Smallest = `8 - 3 = 5`)

44. **1/3** (Numbers greater than 4 on a die are 5 and 6. That's 2 favourable outcomes out of 6 total. `2/6 = 1/3`)

45. **(5, 2)** (Right is +x, Down is -y. `(3+2, 5-3) = (5, 2)`)

46. **Rhombus**

47. **2** (A rectangle has two lines of symmetry: one vertical, one horizontal, through the center)

48. **50°** (Sum of angles in a triangle is 180°. `180 - 90 - 40 = 50`)

49. **20:15 or 8:15 pm** (`18:25 + 1 hour = 19:25; + 50 minutes = 20:15`)

50. **7:36 am** (Bus 1: 7:00, Bus 2: 7:12, Bus 3: 7:24, Bus 4: 7:36)

---

### **Section C: Previous Year GL Assessment Styles - Full Answers & Solutions (1-20)**

1.  **A) 12 x 10**

    *   *Solution:* A) 120, B) 12, C) 22, D) 2. 120 is the greatest.

2.  **B) 999**

    *   *Solution:* `2002 - 1000 = 1002; 1002 - 3 = 999` or standard column subtraction.

3.  **A) 1177**

    *   *Solution:* `107 x 10 = 1070`; `107 x 1 = 107`; `1070 + 107 = 1177`.

4.  **D) 84**

    *   *Solution:* The number is `12 x 7 = 84`.

5.  **A) 1**

    *   *Solution:* `8 x 7 = 56`. `57 - 56 = 1`.

6.  **B) 3.96**

    *   *Solution:* Align decimal points: `3.40 + 0.56 = 3.96`.

7.  **C) 3.93**

    *   *Solution:* `5.00 - 1.07`. Regroup: `5.00 = 4.99 + 0.01`. `4.99 - 1.07 = 3.92`. Or use column subtraction.

8.  **A) 0.45**

    *   *Solution:* Multiplying by 0.1 is the same as dividing by 10. Digits move one place right.

9.  **C) 42**

    *   *Solution:* `8.4 ÷ 0.2` is the same as `84 ÷ 2 = 42`.

10. **A) 0.375**

    *   *Solution:* `3 ÷ 8 = 0.375`.

11. **D) 1/4**

    *   *Solution:* Convert to decimals: A) 0.333..., B) 0.4, C) 0.3, D) 0.25. 0.25 is the smallest.

12. **B) £13.00**

    *   *Solution:* `10% of £65 = £6.50`; `20% = £6.50 x 2 = £13.00`.

13. **B) £18**

    *   *Solution:* Discount = `25% of £24 = £6`. Sale Price = `£24 - £6 = £18`.

14. **B) 33**

    *   *Solution:* `5² = 25`; `2³ = 8`; `25 + 8 = 33`.

15. **B) 12**

    *   *Solution:* `12 x 12 = 144`.

16. **D) 24**

    *   *Solution:* 24 ÷ 3 = 8, 24 ÷ 4 = 6. It is divisible by both.

17. **D) 29**

    *   *Solution:* Primes after 23: 24 (no), 25 (no), 26 (no), 27 (no), 28 (no), 29 (yes).

18. **C) 6 units**

    *   *Solution:* Both points have the same x-coordinate (1), so the distance is the difference in y-coordinates: `4 - (-2) = 6`.

19. **A) 6 square units**

    *   *Solution:* The points (0,0), (3,0), (0,4) form a right-angled triangle with legs of length 3 and 4. Area = `(3 x 4) / 2 = 6`.

20. **B) 7cm**

    *   *Solution:* A regular hexagon has 6 equal sides. Perimeter ÷ number of sides = `42cm ÷ 6 = 7cm`.

### **Section C: Previous Year GL Assessment Styles (Questions 21-50) - Full Answers & Solutions**

21. **D) 24**  

   *Solution:* A number that is a multiple of both 3 and 4 must be a multiple of 12. Among the options, only 24 is divisible by 12.

22. **D) 29**  

   *Solution:* Prime numbers after 23: 24 (no), 25 (no), 27 (no), 29 (yes).

23. **C) 6 units**  

   *Solution:* Both points have the same x-coordinate (1), so the distance is the absolute difference in y-coordinates: |4 - (-2)| = 6.

24. **A) 6 square units**  

   *Solution:* The points form a right-angled triangle with legs of length 3 and 4. Area = (3 × 4) / 2 = 6.

25. **B) 7cm**  

   *Solution:* A regular hexagon has 6 equal sides. Perimeter ÷ number of sides = 42 ÷ 6 = 7.

26. **A) 1/3**  

   *Solution:* Total marbles = 4 + 3 + 5 = 12. Probability of red = 4/12 = 1/3.

27. **B) 16.5**  

   *Solution:* Sum = 12 + 15 + 18 + 21 = 66. Mean = 66 ÷ 4 = 16.5.

28. **A) 4**  

   *Solution:* 3a + 2 = 14 → 3a = 12 → a = 4.

29. **A) 40 mph**  

   *Solution:* Speed = distance ÷ time = 60 ÷ 1.5 = 40.

30. **C) 60%**  

   *Solution:* 3/5 = 0.6 = 60%.

31. **B) 11.6cm**  

   *Solution:* Diagonal = √(10² + 6²) = √(100 + 36) = √136 ≈ 11.66 cm.

32. **B) 17**  

   *Solution:* 2³ = 8, 3² = 9, so 8 + 9 = 17.

33. **D) 7/12**  

   *Solution:* Convert to decimals: 2/3 ≈ 0.667, 3/4 = 0.75, 5/6 ≈ 0.833, 7/12 ≈ 0.583. Smallest is 0.583.

34. **B) 1 hour 45 minutes**  

   *Solution:* From 14:30 to 16:15 is 1 hour 45 minutes.

35. **B) £12**  

   *Solution:* 15% of £80 = 0.15 × 80 = £12.

36. **B) 162**  

   *Solution:* The sequence multiplies by 3 each time: 54 × 3 = 162.

37. **A) 3 cm**  

   *Solution:* Volume = length × width × height → 120 = 10 × 4 × height → height = 120 ÷ 40 = 3.

38. **B) 3/8**  

   *Solution:* 1/4 = 2/8, so 2/8 + 1/8 = 3/8.

39. **C) £30.00**  

   *Solution:* 20% of £25 = £5, so new price = £25 + £5 = £30.

40. **B) 540°**  

   *Solution:* Sum of interior angles of an n-gon = (n - 2) × 180. For a pentagon (n=5), (5-2) × 180 = 540°.

41. **A) 41**  

   *Solution:* 2a² - b² = 2(5²) - 3² = 2(25) - 9 = 50 - 9 = 41.

42. **A) 1/8**  

   *Solution:* 0.125 = 125/1000 = 1/8.

43. **A) 2 km**  

   *Solution:* 4 cm on map = 4 × 50,000 = 200,000 cm = 2,000 m = 2 km.

44. **C) 7**  

   *Solution:* Sorted numbers: 3, 5, 7, 7, 8. Median is the middle value: 7.

45. **B) 3:05 pm**  

   *Solution:* 11:45 am + 3 hours = 2:45 pm + 20 minutes = 3:05 pm.

46. **A) 31.4 cm**  

   *Solution:* Circumference = π × diameter = 3.14 × 10 = 31.4 cm.

47. **B) 3/8**  

   *Solution:* (3/4) × (1/2) = 3/8.

48. **A) 6**  

   *Solution:* Let the number be x. Then 4x + 5 = 29 → 4x = 24 → x = 6.

49. **A) 0.06**  

   *Solution:* 0.2 × 0.3 = 0.06.

50. **A) 150 seconds**  

   *Solution:* 2.5 minutes × 60 = 150 seconds.

### **Summary of All Answers (Questions 21-50)**

| Question | Answer | Question | Answer | Question | Answer | Question | Answer |

|----------|--------|----------|--------|----------|--------|----------|--------|

| 21       | D      | 31       | B      | 41       | A      | 46       | A      |

| 22       | D      | 32       | B      | 42       | A      | 47       | B      |

| 23       | C      | 33       | D      | 43       | A      | 48       | A      |

| 24       | A      | 34       | B      | 44       | C      | 49       | A      |

| 25       | B      | 35       | B      | 45       | B      | 50       | A      |

| 26       | A      | 36       | B      |          |        |          |        |

| 27       | B      | 37       | A      |          |        |          |        |

| 28       | A      | 38       | B      |          |        |          |        |

| 29       | A      | 39       | C      |          |        |          |        |

| 30       | C      | 40       | B      |          |        |          |        |