**11+ Algebra Concepts**
**Concept 1: Understanding the Basics - What is a Letter in Maths?**
* A letter (like \( n \), \( x \), \( a \)) represents an unknown number. It's a placeholder.
* If you see \( n + 5 \), it means "a number plus five".
**Concept 2: Simplifying Expressions (Collecting Like Terms)**
* You can only add or subtract terms that have the exact same letter combination.
* **Example:** \( 3a + 2a = 5a \) but \( 3a + 2b \) cannot be simplified.
* **Example with numbers:** \( 5 + 3n + 2 - n = (5 + 2) + (3n - n) = 7 + 2n \)
**Concept 3: Substitution**
* Replace the letter with the number you are given.
* **Example:** If \( a = 3 \), what is \( 4a + 7 \)?
* \( 4 \times 3 + 7 = 12 + 7 = 19 \)
**Concept 4: Solving Simple Equations (The Balancing Act)**
* An equation has an equals sign. Your goal is to find the value of the letter that makes it true.
* Whatever you do to one side of the equation, you MUST do to the other side to keep it balanced.
* **Example:** Solve \( n + 7 = 15 \)
* Subtract 7 from both sides: \( n + 7 - 7 = 15 - 7 \)
* So, \( n = 8 \)
* **Example:** Solve \( 3p = 18 \)
* Divide both sides by 3: \( 3p \div 3 = 18 \div 3 \)
* So, \( p = 6 \)
* **Two-Step Example:** Solve \( 2y - 5 = 11 \)
* Step 1: Add 5 to both sides: \( 2y - 5 + 5 = 11 + 5 \) → \( 2y = 16 \)
* Step 2: Divide both sides by 2: \( 2y \div 2 = 16 \div 2 \) → \( y = 8 \)
**Concept 5: Forming Simple Expressions**
* Translate a word problem into a mathematical expression.
* **Example:** "I think of a number, multiply it by 4 and then add 6."
* Let the number be \( x \).
* The expression is \( 4x + 6 \).
---
### **Practice Questions (50+) Styled on GL Assessment**
#### **Section A: Simplifying Expressions**
1. Simplify \( 5m + 3m - m \)
2. Simplify \( 7p + 3q + 2p - q \)
3. Simplify \( 2a + 3 + 4a + 7 \)
4. Simplify \( 10x - 3y - x + 5y \)
5. Simplify \( n \times n \times 3 \)
6. Simplify \( 4t \times 2s \)
7. Simplify \( 8d - 3d + 2d \)
8. Simplify \( 5 + 2k - 1 + 4k \)
9. Simplify \( 9g - 4h - 2g + h \)
10. Simplify \( 3 \times a \times b \times 2 \)
#### **Section B: Substitution**
11. If \( a = 5 \), work out \( a + 12 \)
12. If \( b = 4 \), work out \( 3b \)
13. If \( p = 7 \) and \( q = 2 \), work out \( p - q \)
14. If \( x = 3 \), work out \( 4x + 5 \)
15. If \( m = 6 \) and \( n = 1 \), work out \( 2m + 3n \)
16. If \( y = 10 \), work out \( y^2 \) (this means \( y \times y \))
17. If \( a = 8 \) and \( b = 5 \), work out \( 2a - b \)
18. If \( h = 3 \), work out \( 30 - 4h \)
19. If \( r = 4 \) and \( s = 6 \), work out \( \frac{r + s}{2} \)
20. The formula for the perimeter of a rectangle is \( P = 2(l + w) \). If \( l = 9 \) and \( w = 5 \), work out the perimeter \( P \).
#### **Section C: Solving Equations**
21. Solve \( x + 9 = 15 \)
22. Solve \( y - 4 = 11 \)
23. Solve \( 6p = 42 \)
24. Solve \( \frac{t}{3} = 7 \)
25. Solve \( 2n + 1 = 13 \)
26. Solve \( 5k - 7 = 28 \)
27. Solve \( 3m + 8 = 35 \)
28. Solve \( \frac{a}{4} - 2 = 3 \)
29. Solve \( 20 = 4(x + 1) \)
30. Solve \( 3(2y - 5) = 9 \)
#### **Section D: Forming Expressions & Equations**
31. Write an expression for "7 more than \( x \)".
32. Write an expression for "\( y \) multiplied by 5".
33. Write an expression for "12 less than \( m \)".
34. I think of a number \( n \), double it and add 3. Write an expression.
35. A pencil costs \( p \) pence. A ruler costs \( r \) pence. Write an expression for the total cost of 3 pencils and 2 rulers.
36. The sum of a number \( a \) and a number \( b \) is 20. Write this as an equation.
37. Sarah is \( s \) years old. Her brother Tom is 3 years older. Write an expression for Tom's age.
38. The area of a rectangle is length \( \times \) width. If the length is \( L \) and the width is \( 4 \), write an expression for the area.
39. If I subtract 8 from a number \( k \), the answer is 15. Write an equation for this.
40. The price of a book is \( £b \). After a \( £2 \) discount, the new price is \( £8 \). Write an equation and find \( b \).
#### **Section E: Mixed & Word Problems (Harder)**
41. A number \( x \) is multiplied by 4 and then 7 is added. The result is 31. Write an equation and solve it to find \( x \).
42. The angles in a triangle add up to 180°. If two angles are \( a° \) and \( 2a° \), and the third is 90°, write an equation and find \( a \).
43. The cost of hiring a bike is \( £5 \) plus \( £3 \) per hour. If I hire a bike for \( h \) hours, the cost \( C \) is given by the formula \( C = 5 + 3h \). How much will it cost to hire the bike for 4 hours?
44. Using the formula in question 43, if the cost was \( £17 \), how many hours was the bike hired for?
45. If \( 2x + 5 = 17 \), what is the value of \( x \)?
46. If \( 3y = y + 10 \), what is the value of \( y \)?
47. Simplify fully: \( 4(2x + 1) + 3(x - 2) \)
48. Solve: \( 5 - x = 2 \)
49. The sum of three consecutive numbers is 33. If the smallest number is \( n \), write an equation and find the numbers.
50. A piece of string \( s \) cm long is cut into two pieces. One piece is 15cm. Write an expression for the length of the other piece.
---
### **10 Questions from a Fictional "Previous Year Paper"**
*This section mimics the format and presentation of a real exam paper.*
**GL ASSESSMENT SAMPLE PAPER A - MATHEMATICS**
**Time: 10 minutes**
**Answer all questions. Show your working if necessary.**
1. Simplify \( 8p - 3p + 2p \)
A) \( 7p \) \quad B) \( 13p \) \quad C) \( 7p^3 \) \quad D) \( 5p \)
2. If \( a = 6 \), what is the value of \( 2a + 7 \)?
A) 15 \quad B) 19 \quad C) 26 \quad D) 33
3. Solve for \( n \): \( n - 12 = 8 \)
A) \( n = 1.5 \) \quad B) \( n = 4 \) \quad C) \( n = 20 \) \quad D) \( n = 96 \)
4. Solve for \( k \): \( 4k = 36 \)
A) \( k = 9 \) \quad B) \( k = 32 \) \quad C) \( k = 40 \) \quad D) \( k = 144 \)
5. Which expression means "8 less than triple a number \( x \)"?
A) \( 3x - 8 \) \quad B) \( 8 - 3x \) \quad C) \( 3(x - 8) \) \quad D) \( 8x - 3 \)
6. Solve for \( m \): \( 2m + 9 = 21 \)
A) \( m = 6 \) \quad B) \( m = 12 \) \quad C) \( m = 15 \) \quad D) \( m = 24 \)
7. If \( x = 5 \) and \( y = 2 \), what is the value of \( x^2 - y \)?
A) 8 \quad B) 9 \quad C) 23 \quad D) 27
8. Simplify \( 5x + 2y - x + 4y \)
A) \( 4x + 6y \) \quad B) \( 6x + 6y \) \quad C) \( 5x + 6y \) \quad D) \( 4x + 2y \)
9. I think of a number, divide it by 4 and get 5. What was my number?
A) 1 \quad B) 9 \quad C) 15 \quad D) 20
10. The perimeter of a square is \( 4 \times \text{side} \). If the perimeter is \( P \) and the side is \( s \), which formula is correct?
A) \( s = P + 4 \) \quad B) \( P = 4s \) \quad C) \( s = 4P \) \quad D) \( P = s + 4 \)
---
**Answers to the "Previous Year Paper":**
1. A, 2. B, 3. C, 4. A, 5. A, 6. A, 7. C, 8. A, 9. D, 10. B
**How to Use These Resources Effectively:**
1. **Master the Steps:** Ensure you are completely comfortable with the step-by-step concepts before attempting the large question bank.
2. **Practice Regularly:** Do a set of 10-15 questions daily, mixing the topics.
3. **Identify Weaknesses:** If you keep making errors on "solving equations", go back and re-learn that specific concept.
4. **Time Yourself:** Once confident, time your practice to mimic exam conditions (about 1 minute per question).
5. **Review Mistakes:** Always, always understand *why* you got a question wrong. This is where the real learning happens.
Good luck with your preparation
Of course. Here is a substantial additional bank of 150 questions, meticulously styled on the GL Assessment 11+ exam for Slough Grammar School, complete with a comprehensive answer key.
---
### **Additional Practice Questions (GL Assessment Style)**
#### **10 More Questions: Simplifying Expressions**
1. Simplify \( 12k - 4k + k \)
2. Simplify \( 5d + 3e - 2d + e \)
3. Simplify \( 2 + 6n + 5 - n \)
4. Simplify \( 8x - 2y - 5x + y \)
5. Simplify \( a \times a \times 4 \)
6. Simplify \( 3m \times 4n \)
7. Simplify \( 10q - q - 5q \)
8. Simplify \( 7 + 5f + 4 - 2f \)
9. Simplify \( 6r - 2s - r + 5s \)
10. Simplify \( 5 \times p \times q \times 3 \)
#### **10 More Questions: Substitution**
11. If \( x = 8 \), work out \( x + 9 \)
12. If \( y = 5 \), work out \( 7y \)
13. If \( m = 10 \) and \( n = 3 \), work out \( m - n \)
14. If \( p = 4 \), work out \( 3p + 6 \)
15. If \( a = 7 \) and \( b = 2 \), work out \( 4a + 2b \)
16. If \( c = 6 \), work out \( c^2 \)
17. If \( x = 9 \) and \( y = 4 \), work out \( 3x - y \)
18. If \( h = 5 \), work out \( 24 - 3h \)
19. If \( p = 10 \) and \( q = 2 \), work out \( \frac{p + q}{4} \)
20. The formula for the perimeter of a square is \( P = 4s \). If \( s = 6 \), work out \( P \).
#### **10 More Questions: Solving Equations**
21. Solve \( a + 12 = 20 \)
22. Solve \( b - 7 = 15 \)
23. Solve \( 8m = 64 \)
24. Solve \( \frac{n}{5} = 6 \)
25. Solve \( 3t + 4 = 19 \)
26. Solve \( 6y - 5 = 37 \)
27. Solve \( 4p + 11 = 35 \)
28. Solve \( \frac{x}{3} + 4 = 9 \)
29. Solve \( 15 = 3(k - 2) \)
30. Solve \( 2(3w + 1) = 20 \)
#### **10 More Questions: Forming Expressions & Equations**
31. Write an expression for "15 less than \( y \)".
32. Write an expression for "\( n \) divided by 4".
33. Write an expression for "8 more than double \( x \)".
34. I think of a number \( h \), subtract 5, and then multiply by 2. Write an expression.
35. An apple costs \( a \) pence. An orange costs \( b \) pence. Write an expression for the total cost of 5 apples and 3 oranges.
36. The difference between a number \( c \) and a number \( d \) is 7. Write this as an equation.
37. David is \( d \) years old. His sister Sarah is half his age. Write an expression for Sarah's age.
38. The area of a triangle is \( \frac{1}{2} \times \text{base} \times \text{height} \). If the base is \( b \) and the height is 10, write an expression for the area.
39. If I add 12 to a number \( m \), the answer is 30. Write an equation for this.
40. The length of a rectangle is \( L \) cm and the width is 5 cm. If the perimeter is 30 cm, write an equation and find \( L \).
#### **10 More Questions: Mixed & Word Problems (Harder)**
41. A number \( y \) is divided by 5 and then 3 is subtracted. The result is 4. Write an equation and solve it.
42. The angles on a straight line add up to 180°. If one angle is \( 2x° \) and the other is \( 3x° \), write an equation and find \( x \).
43. The cost of a taxi is \( £3 \) plus \( £2 \) per mile. For a journey of \( m \) miles, the cost \( C \) is \( C = 3 + 2m \). What is the cost for a 9-mile journey?
44. Using the formula in question 43, if a taxi ride cost \( £15 \), how many miles was the journey?
45. If \( 5x - 3 = 22 \), what is the value of \( x \)?
46. If \( 4z = 20 - z \), what is the value of \( z \)?
47. Simplify fully: \( 2(3x + 4) + 5(x - 1) \)
48. Solve: \( 17 - 2x = 5 \)
49. The sum of two numbers is 25. One number is 5 more than the other. If the smaller number is \( n \), write an equation and find the two numbers.
50. A bag of flour weighs \( F \) kg. I use 2kg of it. Write an expression for the flour left.
---
### **50 More Questions: Fictional "Previous Year Paper"**
**GL ASSESSMENT SAMPLE PAPER B - MATHEMATICS**
**Time: 25 minutes**
**Answer all questions.**
1. Simplify \( 9a - 5a + 2a \)
A) \( 6a \) \quad B) \( 12a \) \quad C) \( 6a^3 \) \quad D) \( 7a \)
2. If \( b = 7 \), what is \( 5b - 8 \)?
A) 20 \quad B) 27 \quad C) 35 \quad D) 43
3. Solve \( c + 15 = 32 \)
A) \( c = 2 \) \quad B) \( c = 17 \) \quad C) \( c = 47 \) \quad D) \( c = 480 \)
4. Solve \( 7d = 56 \)
A) \( d = 6 \) \quad B) \( d = 8 \) \quad C) \( d = 49 \) \quad D) \( d = 392 \)
5. Which expression means "the product of 6 and a number \( x \), minus 10"?
A) \( 6x - 10 \) \quad B) \( 10 - 6x \) \quad C) \( 6(x - 10) \) \quad D) \( \frac{x}{6} - 10 \)
6. Solve \( 4e - 7 = 21 \)
A) \( e = 3.5 \) \quad B) \( e = 7 \) \quad C) \( e = 14 \) \quad D) \( e = 28 \)
7. If \( p = 4 \) and \( q = 3 \), what is \( 2p^2 + q \)?
A) 11 \quad B) 19 \quad C) 35 \quad D) 41
8. Simplify \( 7m + 4n - 3m + n \)
A) \( 4m + 5n \) \quad B) \( 10m + 5n \) \quad C) \( 4m + 3n \) \quad D) \( 10m + 3n \)
9. I think of a number, multiply it by 3 and get 21. What was my number?
A) 7 \quad B) 18 \quad C) 24 \quad D) 63
10. The cost of \( n \) books at \( £4 \) each is \( £C \). Which formula is correct?
A) \( C = n + 4 \) \quad B) \( C = 4n \) \quad C) \( n = 4C \) \quad D) \( n = C + 4 \)
**GL ASSESSMENT SAMPLE PAPER C - MATHEMATICS**
**Time: 25 minutes**
11. Simplify \( 15x - 8x - x \)
A) \( 6x \) \quad B) \( 7x \) \quad C) \( 8x \) \quad D) \( 24x \)
12. If \( y = 12 \), what is \( \frac{y}{3} + 5 \)?
A) 6 \quad B) 9 \quad C) 14 \quad D) 20
13. Solve \( z - 8 = 14 \)
A) \( z = 1.75 \) \quad B) \( z = 6 \) \quad C) \( z = 22 \) \quad D) \( z = 112 \)
14. Solve \( \frac{f}{4} = 9 \)
A) \( f = 2.25 \) \quad B) \( f = 5 \) \quad C) \( f = 13 \) \quad D) \( f = 36 \)
15. "I think of a number \( n \), add 4, and then double the result." Which expression shows this?
A) \( 2n + 4 \) \quad B) \( n + 8 \) \quad C) \( 2(n + 4) \) \quad D) \( 2n + 8 \)
16. Solve \( 2(g + 6) = 22 \)
A) \( g = 5 \) \quad B) \( g = 8 \) \quad C) \( g = 14 \) \quad D) \( g = 17 \)
17. If \( r = 5 \), what is the value of \( 40 - 6r \)?
A) 10 \quad B) 34 \quad C) 46 \quad D) 210
18. Simplify \( 10 + 3u - 7 + u \)
A) \( 4u + 3 \) \quad B) \( 3u + 17 \) \quad C) \( 4u + 17 \) \quad D) \( 13u \)
19. The sum of a number \( x \) and 11 is 29. What is \( x \)?
A) 18 \quad B) 20 \quad C) 40 \quad D) 319
20. A rectangle's length is \( L \) cm and its width is \( W \) cm. The perimeter is 30 cm. Which equation could be correct?
A) \( L + W = 30 \) \quad B) \( 2L + W = 30 \) \quad C) \( L + 2W = 30 \) \quad D) \( 2(L + W) = 30 \)
**GL ASSESSMENT SAMPLE PAPER D - MATHEMATICS**
**Time: 25 minutes**
21. Simplify \( 4p \times 3q \)
A) \( 7pq \) \quad B) \( 12pq \) \quad C) \( 12p \) \quad D) \( 12q \)
22. If \( a = 2 \) and \( b = 10 \), what is \( ab \)?
A) 8 \quad B) 12 \quad C) 20 \quad D) 102
23. Solve \( 5k = 8 \)
A) \( k = 0.625 \) \quad B) \( k = 1.6 \) \quad C) \( k = 3 \) \quad D) \( k = 13 \)
24. Solve \( 2m - 1 = 13 \)
A) \( m = 6 \) \quad B) \( m = 7 \) \quad C) \( m = 12 \) \quad D) \( m = 28 \)
25. Tom has \( t \) toys. Sam has 3 times as many. Which expression shows how many Sam has?
A) \( t + 3 \) \quad B) \( t - 3 \) \quad C) \( \frac{t}{3} \) \quad D) \( 3t \)
26. Solve \( \frac{x}{2} - 3 = 7 \)
A) \( x = 5 \) \quad B) \( x = 8 \) \quad C) \( x = 17 \) \quad D) \( x = 20 \)
27. Simplify \( 6(2y - 1) - 4y \)
A) \( 8y - 7 \) \quad B) \( 8y - 1 \) \quad C) \( 12y - 5 \) \quad D) \( 8y - 6 \)
28. The area \( A \) of a square is \( A = s^2 \). If \( s = 6 \) cm, what is \( A \)?
A) 12 cm² \quad B) 24 cm² \quad C) 36 cm² \quad D) 72 cm²
29. A box weighs \( n \) kg. The weight is shared equally between 4 people. How much does each person get?
A) \( 4n \) kg \quad B) \( \frac{n}{4} \) kg \quad C) \( n - 4 \) kg \quad D) \( n + 4 \) kg
30. If \( 3x + 2 = x + 12 \), what is \( x \)?
A) \( x = 2 \) \quad B) \( x = 5 \) \quad C) \( x = 7 \) \quad D) \( x = 14 \)
**GL ASSESSMENT SAMPLE PAPER E - MATHEMATICS**
**Time: 25 minutes**
31. Simplify \( 2a + 5b + 3a - b \)
A) \( 5a + 4b \) \quad B) \( 5a + 6b \) \quad C) \( 11ab \) \quad D) \( 10a + 4b \)
32. If \( m = 0 \), what is \( 10m + 8 \)?
A) 0 \quad B) 8 \quad C) 10 \quad D) 18
33. Solve \( 12 = n - 5 \)
A) \( n = 7 \) \quad B) \( n = 12 \) \quad C) \( n = 17 \) \quad D) \( n = 60 \)
34. Solve \( 4(p + 3) = 32 \)
A) \( p = 5 \) \quad B) \( p = 8 \) \quad C) \( p = 11 \) \quad D) \( p = 20 \)
35. A car travels at \( s \) miles per hour. How far will it travel in 3 hours?
A) \( 3s \) miles \quad B) \( \frac{s}{3} \) miles \quad C) \( s + 3 \) miles \quad D) \( s - 3 \) miles
36. Solve \( 20 - 3q = 5 \)
A) \( q = 5 \) \quad B) \( q = 6 \) \quad C) \( q = 8 \) \quad D) \( q = 15 \)
37. If \( h = 8 \), what is \( h \div 2 + 6 \)?
A) 7 \quad B) 10 \quad C) 16 \quad D) 22
38. Simplify \( 9 - 2x + 5 + x \)
A) \( 14 - x \) \quad B) \( 14 - 3x \) \quad C) \( 12 - x \) \quad D) \( 4 - x \)
39. A cake is cut into \( x \) pieces. Each piece is cut in half. How many pieces are there now?
A) \( \frac{x}{2} \) \quad B) \( x + 2 \) \quad C) \( 2x \) \quad D) \( x - 2 \)
40. The sum of three consecutive even numbers is 30. If the smallest is \( n \), what is the equation?
A) \( n + (n+1) + (n+2) = 30 \) \quad B) \( n + (n+2) + (n+4) = 30 \) \quad C) \( 3n = 30 \) \quad D) \( n + 2 + 4 = 30 \)
**GL ASSESSMENT SAMPLE PAPER F - MATHEMATICS**
**Time: 25 minutes**
41. Simplify \( y^2 + y^2 \)
A) \( y^2 \) \quad B) \( 2y^2 \) \quad C) \( y^4 \) \quad D) \( 2y^4 \)
42. If \( a = 6 \) and \( b = 3 \), what is \( a^2 - b^2 \)?
A) 3 \quad B) 9 \quad C) 27 \quad D) 33
43. Solve \( \frac{2t}{3} = 10 \)
A) \( t = 5 \) \quad B) \( t = 10 \) \quad C) \( t = 15 \) \quad D) \( t = 30 \)
44. Solve \( 5 - x = 12 \)
A) \( x = 7 \) \quad B) \( x = 17 \) \quad C) \( x = -7 \) \quad D) \( x = -17 \)
45. A plant is \( h \) cm tall. It grows 5 cm. Then it doubles in height. What is its new height?
A) \( 2h \) cm \quad B) \( 2h + 5 \) cm \quad C) \( 2(h + 5) \) cm \quad D) \( h + 10 \) cm
46. If \( 4(r - 2) = 2(r + 4) \), what is \( r \)?
A) \( r = 4 \) \quad B) \( r = 8 \) \quad C) \( r = 10 \) \quad D) \( r = 16 \)
47. Simplify \( 3a \times 2b \times c \)
A) \( 5abc \) \quad B) \( 6abc \) \quad C) \( 6a + b + c \) \quad D) \( 32abc \)
48. The volume of a cuboid is length × width × height. If length=\( l \), width=5, height=2, what is the volume?
A) \( 10l \) \quad B) \( 7l \) \quad C) \( l + 7 \) \quad D) \( 5l + 2 \)
49. A bus has \( b \) passengers. At a stop, 8 get off and 3 get on. How many are now on the bus?
A) \( b - 5 \) \quad B) \( b - 11 \) \quad C) \( b + 5 \) \quad D) \( b + 11 \)
50. When 30 is added to a number, the result is the same as doubling the number. What is the number?
A) 15 \quad B) 20 \quad C) 30 \quad D) 60
---
### **COMPREHENSIVE ANSWER KEY**
#### **Simplifying Expressions (1-10)**
1. \( 9k \) \ (12-4+1=9)
2. \( 3d + 4e \) \ (5d-2d=3d, 3e+e=4e)
3. \( 7 + 5n \) \ (2+5=7, 6n-n=5n)
4. \( 3x - y \) \ (8x-5x=3x, -2y+y=-y)
5. \( 4a^2 \)
6. \( 12mn \)
7. \( 4q \) \ (10-1-5=4)
8. \( 11 + 3f \) \ (7+4=11, 5f-2f=3f)
9. \( 5r + 3s \) \ (6r-r=5r, -2s+5s=3s)
10. \( 15pq \)
#### **Substitution (11-20)**
11. 17 \ (8+9)
12. 35 \ (7×5)
13. 7 \ (10-3)
14. 18 \ (3×4 + 6 = 12+6)
15. 32 \ (4×7 + 2×2 = 28+4)
16. 36 \ (6×6)
17. 23 \ (3×9 - 4 = 27-4)
18. 9 \ (24 - 3×5 = 24-15)
19. 3 \ ((10+2)/4 = 12/4)
20. 24 \ (4×6)
#### **Solving Equations (21-30)**
21. \( a = 8 \) \ (20-12)
22. \( b = 22 \) \ (15+7)
23. \( m = 8 \) \ (64÷8)
24. \( n = 30 \) \ (6×5)
25. \( t = 5 \) \ (19-4=15, 15÷3=5)
26. \( y = 7 \) \ (37+5=42, 42÷6=7)
27. \( p = 6 \) \ (35-11=24, 24÷4=6)
28. \( x = 15 \) \ (9-4=5, 5×3=15)
29. \( k = 7 \) \ (15÷3=5, 5+2=7)
30. \( w = 3 \) \ (20÷2=10, 10-1=9, 9÷3=3)
#### **Forming Expressions & Equations (31-40)**
31. \( y - 15 \)
32. \( n \div 4 \) or \( \frac{n}{4} \)
33. \( 2x + 8 \)
34. \( 2(h - 5) \)
35. \( 5a + 3b \)
36. \( c - d = 7 \) (or \( d - c = 7 \), but the first is more conventional)
37. \( \frac{d}{2} \) or \( d \div 2 \)
38. \( \frac{1}{2} \times b \times 10 = 5b \)
39. \( m + 12 = 30 \)
40. \( 2(L + 5) = 30 \), solving gives \( L = 10 \)
#### **Mixed & Word Problems (41-50)**
41. \( \frac{y}{5} - 3 = 4 \), solving gives \( y = 35 \)
42. \( 2x + 3x = 180 \), so \( 5x = 180 \), \( x = 36 \)
43. \( C = 3 + 2×9 = 3 + 18 = £21 \)
44. \( 15 = 3 + 2m \), so \( 12 = 2m \), \( m = 6 \) miles
45. \( x = 5 \) \ (22+3=25, 25÷5=5)
46. \( z = 4 \) \ (4z + z = 20, 5z=20)
47. \( 6x + 8 + 5x - 5 = 11x + 3 \)
48. \( x = 6 \) \ (17-5=12, 12÷2=6)
49. \( n + (n+5) = 25 \), so \( 2n+5=25 \), \( 2n=20 \), \( n=10 \). Numbers are 10 and 15.
50. \( (F - 2) \) kg
#### **Fictional "Previous Year Paper" (1-50)**
1. A \ (9-5+2=6)
2. B \ (5×7 - 8 = 35-8=27)
3. B \ (32-15=17)
4. B \ (56÷7=8)
5. A
6. B \ (21+7=28, 28÷4=7)
7. C \ (2×16 + 3 = 32+3=35)
8. A \ (7m-3m=4m, 4n+n=5n)
9. A \ (21÷3=7)
10. B
11. A \ (15-8-1=6)
12. B \ (12÷3=4, 4+5=9)
13. C \ (14+8=22)
14. D \ (9×4=36)
15. C
16. A \ (22÷2=11, 11-6=5)
17. A \ (40 - 6×5 = 40-30=10)
18. A \ (10-7=3, 3u+u=4u)
19. A \ (29-11=18)
20. D
21. B
22. C \ (2×10=20)
23. B \ (8÷5=1.6)
24. B \ (13+1=14, 14÷2=7)
25. D
26. D \ (7+3=10, 10×2=20)
27. A \ (12y - 6 - 4y = 8y - 7)
28. C \ (6×6=36)
29. B
30. B \ (3x - x = 12 - 2, so 2x=10, x=5)
31. A \ (2a+3a=5a, 5b-b=4b)
32. B \ (10×0 + 8 = 8)
33. C \ (12+5=17)
34. A \ (32÷4=8, 8-3=5)
35. A
36. A \ (20-5=15, 15÷3=5)
37. B \ (8÷2=4, 4+6=10)
38. A \ (9+5=14, -2x+x=-x)
39. C
40. B \ (Even numbers go up by 2, e.g., 8, 10, 12)
41. B
42. C \ (36 - 9 = 27)
43. C \ (10÷2=5, 5×3=15)
44. C \ (5-12 = -7, so -x=-7, x=7)
45. C \ (New height after growth is h+5. Doubling gives 2(h+5))
46. B \ (4r-8 = 2r+8, so 2r=16, r=8)
47. B \ (3×2×1=6)
48. A \ (l × 5 × 2 = 10l)
49. A \ (b - 8 + 3 = b - 5)
50. C \ (Let the number be n. n + 30 = 2n, so 30 = n)
### **Answer Key / Solution for All Questions**
---
#### **Section A: Simplifying Expressions (1-10)**
1. **\( 7m \)**
\( 5m + 3m = 8m \), \( 8m - m = 7m \)
2. **\( 9p + 2q \)**
\( 7p + 2p = 9p \), \( 3q - q = 2q \)
3. **\( 6a + 10 \)**
\( 2a + 4a = 6a \), \( 3 + 7 = 10 \)
4. **\( 9x + 2y \)**
\( 10x - x = 9x \), \( -3y + 5y = 2y \)
5. **\( 3n^2 \)**
\( n \times n = n^2 \), \( n^2 \times 3 = 3n^2 \)
6. **\( 8st \)** (or \( 8ts \))
\( 4 \times 2 = 8 \), \( t \times s = st \)
7. **\( 7d \)**
\( 8d - 3d = 5d \), \( 5d + 2d = 7d \)
8. **\( 4 + 6k \)**
\( 5 - 1 = 4 \), \( 2k + 4k = 6k \)
9. **\( 7g - 3h \)**
\( 9g - 2g = 7g \), \( -4h + h = -3h \)
10. **\( 6ab \)**
\( 3 \times 2 = 6 \), \( a \times b = ab \)
---
#### **Section B: Substitution (11-20)**
11. **17**
\( 5 + 12 = 17 \)
12. **12**
\( 3 \times 4 = 12 \)
13. **5**
\( 7 - 2 = 5 \)
14. **17**
\( 4 \times 3 + 5 = 12 + 5 = 17 \)
15. **15**
\( 2 \times 6 + 3 \times 1 = 12 + 3 = 15 \)
16. **100**
\( 10 \times 10 = 100 \)
17. **11**
\( 2 \times 8 - 5 = 16 - 5 = 11 \)
18. **18**
\( 30 - 4 \times 3 = 30 - 12 = 18 \)
19. **5**
\( (4 + 6) \div 2 = 10 \div 2 = 5 \)
20. **28**
\( P = 2(9 + 5) = 2 \times 14 = 28 \)
---
#### **Section C: Solving Equations (21-30)**
21. **\( x = 6 \)**
\( x = 15 - 9 = 6 \)
22. **\( y = 15 \)**
\( y = 11 + 4 = 15 \)
23. **\( p = 7 \)**
\( p = 42 \div 6 = 7 \)
24. **\( t = 21 \)**
\( t = 7 \times 3 = 21 \)
25. **\( n = 6 \)**
\( 2n = 13 - 1 = 12 \), \( n = 12 \div 2 = 6 \)
26. **\( k = 7 \)**
\( 5k = 28 + 7 = 35 \), \( k = 35 \div 5 = 7 \)
27. **\( m = 9 \)**
\( 3m = 35 - 8 = 27 \), \( m = 27 \div 3 = 9 \)
28. **\( a = 20 \)**
\( \frac{a}{4} = 3 + 2 = 5 \), \( a = 5 \times 4 = 20 \)
29. **\( x = 4 \)**
\( 4(x + 1) = 20 \), \( x + 1 = 20 \div 4 = 5 \), \( x = 5 - 1 = 4 \)
30. **\( y = 4 \)**
\( 3(2y - 5) = 9 \), \( 2y - 5 = 9 \div 3 = 3 \), \( 2y = 3 + 5 = 8 \), \( y = 8 \div 2 = 4 \)
---
#### **Section D: Forming Expressions & Equations (31-40)**
31. **\( x + 7 \)**
32. **\( 5y \)**
33. **\( m - 12 \)**
34. **\( 2n + 3 \)**
35. **\( 3p + 2r \)**
36. **\( a + b = 20 \)**
37. **\( s + 3 \)**
38. **\( 4L \)** (or \( L \times 4 \))
39. **\( k - 8 = 15 \)**
40. **\( b = 10 \)**
Equation: \( b - 2 = 8 \), so \( b = 8 + 2 = 10 \)
---
#### **Section E: Mixed & Word Problems (Harder) (41-50)**
41. **\( x = 6 \)**
Equation: \( 4x + 7 = 31 \), \( 4x = 24 \), \( x = 6 \)
42. **\( a = 30 \)**
Equation: \( a + 2a + 90 = 180 \), \( 3a = 90 \), \( a = 30 \)
43. **£17**
\( C = 5 + 3 \times 4 = 5 + 12 = 17 \)
44. **4 hours**
\( 5 + 3h = 17 \), \( 3h = 12 \), \( h = 4 \)
45. **6**
\( 2x = 17 - 5 = 12 \), \( x = 6 \)
46. **5**
\( 3y - y = 10 \), \( 2y = 10 \), \( y = 5 \)
47. **\( 11x - 2 \)**
\( 4(2x + 1) = 8x + 4 \), \( 3(x - 2) = 3x - 6 \), \( 8x + 4 + 3x - 6 = 11x - 2 \)
48. **\( x = 3 \)**
\( 5 - x = 2 \), \( -x = -3 \), \( x = 3 \)
49. **10, 11, 12**
Equation: \( n + (n+1) + (n+2) = 33 \), \( 3n + 3 = 33 \), \( 3n = 30 \), \( n = 10 \)
50. **\( s - 15 \)**
---
### **Additional Practice Questions (GL Assessment Style)**
#### **10 More Questions: Simplifying Expressions (1-10)**
1. **\( 9k \)**
\( 12k - 4k = 8k \), \( 8k + k = 9k \)
2. **\( 3d + 4e \)**
\( 5d - 2d = 3d \), \( 3e + e = 4e \)
3. **\( 7 + 5n \)**
\( 2 + 5 = 7 \), \( 6n - n = 5n \)
4. **\( 3x - y \)**
\( 8x - 5x = 3x \), \( -2y + y = -y \)
5. **\( 4a^2 \)**
\( a \times a = a^2 \), \( a^2 \times 4 = 4a^2 \)
6. **\( 12mn \)**
\( 3 \times 4 = 12 \), \( m \times n = mn \)
7. **\( 4q \)**
\( 10q - q = 9q \), \( 9q - 5q = 4q \)
8. **\( 11 + 3f \)**
\( 7 + 4 = 11 \), \( 5f - 2f = 3f \)
9. **\( 5r + 3s \)**
\( 6r - r = 5r \), \( -2s + 5s = 3s \)
10. **\( 15pq \)**
\( 5 \times 3 = 15 \), \( p \times q = pq \)
---
#### **10 More Questions: Substitution (11-20)**
11. **17**
\( 8 + 9 = 17 \)
12. **35**
\( 7 \times 5 = 35 \)
13. **7**
\( 10 - 3 = 7 \)
14. **18**
\( 3 \times 4 + 6 = 12 + 6 = 18 \)
15. **32**
\( 4 \times 7 + 2 \times 2 = 28 + 4 = 32 \)
16. **36**
\( 6 \times 6 = 36 \)
17. **23**
\( 3 \times 9 - 4 = 27 - 4 = 23 \)
18. **9**
\( 24 - 3 \times 5 = 24 - 15 = 9 \)
19. **3**
\( (10 + 2) \div 4 = 12 \div 4 = 3 \)
20. **24**
\( P = 4 \times 6 = 24 \)
---
#### **10 More Questions: Solving Equations (21-30)**
21. **\( a = 8 \)**
\( a = 20 - 12 = 8 \)
22. **\( b = 22 \)**
\( b = 15 + 7 = 22 \)
23. **\( m = 8 \)**
\( m = 64 \div 8 = 8 \)
24. **\( n = 30 \)**
\( n = 6 \times 5 = 30 \)
25. **\( t = 5 \)**
\( 3t = 19 - 4 = 15 \), \( t = 15 \div 3 = 5 \)
26. **\( y = 7 \)**
\( 6y = 37 + 5 = 42 \), \( y = 42 \div 6 = 7 \)
27. **\( p = 6 \)**
\( 4p = 35 - 11 = 24 \), \( p = 24 \div 4 = 6 \)
28. **\( x = 15 \)**
\( \frac{x}{3} = 9 - 4 = 5 \), \( x = 5 \times 3 = 15 \)
29. **\( k = 7 \)**
\( 3(k - 2) = 15 \), \( k - 2 = 15 \div 3 = 5 \), \( k = 5 + 2 = 7 \)
30. **\( w = 3 \)**
\( 2(3w + 1) = 20 \), \( 3w + 1 = 20 \div 2 = 10 \), \( 3w = 10 - 1 = 9 \), \( w = 9 \div 3 = 3 \)
---
#### **10 More Questions: Forming Expressions & Equations (31-40)**
31. **\( y - 15 \)**
32. **\( \frac{n}{4} \)** (or \( n \div 4 \))
33. **\( 2x + 8 \)**
34. **\( 2(h - 5) \)**
35. **\( 5a + 3b \)**
36. **\( c - d = 7 \)** (or \( d - c = 7 \))
37. **\( \frac{d}{2} \)** (or \( d \div 2 \))
38. **\( 5b \)**
\( \frac{1}{2} \times b \times 10 = 5b \)
39. **\( m + 12 = 30 \)**
40. **\( L = 10 \)**
Equation: \( 2(L + 5) = 30 \), \( L + 5 = 15 \), \( L = 10 \)
---
#### **10 More Questions: Mixed & Word Problems (Harder) (41-50)**
41. **\( y = 35 \)**
Equation: \( \frac{y}{5} - 3 = 4 \), \( \frac{y}{5} = 7 \), \( y = 35 \)
42. **\( x = 36 \)**
Equation: \( 2x + 3x = 180 \), \( 5x = 180 \), \( x = 36 \)
43. **£21**
\( C = 3 + 2 \times 9 = 3 + 18 = 21 \)
44. **6 miles**
\( 3 + 2m = 15 \), \( 2m = 12 \), \( m = 6 \)
45. **5**
\( 5x = 22 + 3 = 25 \), \( x = 5 \)
46. **4**
\( 4z + z = 20 \), \( 5z = 20 \), \( z = 4 \)
47. **\( 11x + 3 \)**
\( 2(3x + 4) = 6x + 8 \), \( 5(x - 1) = 5x - 5 \), \( 6x + 8 + 5x - 5 = 11x + 3 \)
48. **\( x = 6 \)**
\( 17 - 2x = 5 \), \( -2x = -12 \), \( x = 6 \)
49. **10 and 15**
Equation: \( n + (n + 5) = 25 \), \( 2n + 5 = 25 \), \( 2n = 20 \), \( n = 10 \)
50. **\( F - 2 \)**
---
### **50 More Questions: Fictional "Previous Year Paper" (1-50)**
#### **Paper B (1-10)**
1. **A) \( 6a \)**
\( 9a - 5a = 4a \), \( 4a + 2a = 6a \)
2. **B) 27**
\( 5 \times 7 - 8 = 35 - 8 = 27 \)
3. **B) \( c = 17 \)**
\( c = 32 - 15 = 17 \)
4. **B) \( d = 8 \)**
\( d = 56 \div 7 = 8 \)
5. **A) \( 6x - 10 \)**
6. **B) \( e = 7 \)**
\( 4e = 21 + 7 = 28 \), \( e = 28 \div 4 = 7 \)
7. **C) 35**
\( 2 \times (4^2) + 3 = 2 \times 16 + 3 = 32 + 3 = 35 \)
8. **A) \( 4m + 5n \)**
\( 7m - 3m = 4m \), \( 4n + n = 5n \)
9. **A) 7**
\( 3x = 21 \), \( x = 7 \)
10. **B) \( C = 4n \)**
#### **Paper C (11-20)**
11. **A) \( 6x \)**
\( 15x - 8x = 7x \), \( 7x - x = 6x \)
12. **B) 9**
\( 12 \div 3 = 4 \), \( 4 + 5 = 9 \)
13. **C) \( z = 22 \)**
\( z = 14 + 8 = 22 \)
14. **D) \( f = 36 \)**
\( f = 9 \times 4 = 36 \)
15. **C) \( 2(n + 4) \)**
16. **A) \( g = 5 \)**
\( 2(g + 6) = 22 \), \( g + 6 = 11 \), \( g = 5 \)
17. **A) 10**
\( 40 - 6 \times 5 = 40 - 30 = 10 \)
18. **A) \( 4u + 3 \)**
\( 10 - 7 = 3 \), \( 3u + u = 4u \)
19. **A) 18**
\( x + 11 = 29 \), \( x = 18 \)
20. **D) \( 2(L + W) = 30 \)**
#### **Paper D (21-30)**
21. **B) \( 12pq \)**
\( 4 \times 3 = 12 \), \( p \times q = pq \)
22. **C) 20**
\( a \times b = 2 \times 10 = 20 \)
23. **B) \( k = 1.6 \)**
\( k = 8 \div 5 = 1.6 \)
24. **B) \( m = 7 \)**
\( 2m = 13 + 1 = 14 \), \( m = 7 \)
25. **D) \( 3t \)**
26. **D) \( x = 20 \)**
\( \frac{x}{2} = 7 + 3 = 10 \), \( x = 20 \)
27. **D) \( 8y - 6 \)**
\( 6(2y - 1) = 12y - 6 \), \( 12y - 6 - 4y = 8y - 6 \)
28. **C) 36 cm²**
\( A = 6^2 = 36 \)
29. **B) \( \frac{n}{4} \) kg**
30. **B) \( x = 5 \)**
\( 3x - x = 12 - 2 \), \( 2x = 10 \), \( x = 5 \)
#### **Paper E (31-40)**
31. **A) \( 5a + 4b \)**
\( 2a + 3a = 5a \), \( 5b - b = 4b \)
32. **B) 8**
\( 10 \times 0 + 8 = 8 \)
33. **C) \( n = 17 \)**
\( n = 12 + 5 = 17 \)
34. **A) \( p = 5 \)**
\( 4(p + 3) = 32 \), \( p + 3 = 8 \), \( p = 5 \)
35. **A) \( 3s \) miles**
36. **A) \( q = 5 \)**
\( 20 - 3q = 5 \), \( -3q = -15 \), \( q = 5 \)
37. **B) 10**
\( 8 \div 2 = 4 \), \( 4 + 6 = 10 \)
38. **A) \( 14 - x \)**
\( 9 + 5 = 14 \), \( -2x + x = -x \)
39. **C) \( 2x \)**
40. **B) \( n + (n+2) + (n+4) = 30 \)**
#### **Paper F (41-50)**
41. **B) \( 2y^2 \)**
42. **C) 27**
\( 6^2 - 3^2 = 36 - 9 = 27 \)
43. **C) \( t = 15 \)**
\( \frac{2t}{3} = 10 \), \( 2t = 30 \), \( t = 15 \)
44. **C) \( x = -7 \)**
\( 5 - x = 12 \), \( -x = 7 \), \( x = -7 \)
45. **C) \( 2(h + 5) \) cm**
46. **B) \( r = 8 \)**
\( 4(r - 2) = 2(r + 4) \), \( 4r - 8 = 2r + 8 \), \( 2r = 16 \), \( r = 8 \)
47. **B) \( 6abc \)**
\( 3 \times 2 \times 1 = 6 \), \( a \times b \times c = abc \)
48. **A) \( 10l \)**
Volume = \( l \times 5 \times 2 = 10l \)
49. **A) \( b - 5 \)**
\( b - 8 + 3 = b - 5 \)
50. **C) 30**
Let the number be \( n \): \( n + 30 = 2n \), \( n = 30 \)
---
