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.

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### **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?

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### **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)