Teach Me Maths

Forming and Solving Equations

Question 11 mark

Priya thinks of a number. She adds 9 to her number. Her answer is 23

What number did Priya think of?

Hint

Call the number n and write an equation, or work backwards from 23.

Worked solution

Priya thinks of a number. She adds 9 to her number. Her answer is 23

What number did Priya think of?

  1. Let the number be n. Then n+9=23.
  2. Subtract 9 from both sides: n=23-9=14.
  3. Priya thought of 14.

Question 21 mark

Leo thinks of a number. He multiplies his number by 4. His answer is 52

What number did Leo think of?

Hint

The opposite of multiplying by 4 is dividing by 4.

Worked solution

Leo thinks of a number. He multiplies his number by 4. His answer is 52

What number did Leo think of?

  1. Let the number be n. Then 4n=52.
  2. Divide both sides by 4: n=52\div4=13.
  3. Leo thought of 13.

Question 32 marks

Mina thinks of a number. She adds 6 to her number and then multiplies the result by 3. Her answer is 33

What number did Mina think of?

Hint

Undo the last thing Mina did first.

Worked solution

Mina thinks of a number. She adds 6 to her number and then multiplies the result by 3. Her answer is 33

What number did Mina think of?

  1. Let the number be n. Then 3(n+6)=33.
  2. Divide both sides by 3: n+6=11.
  3. Subtract 6 from both sides: n=5.
  4. Check: 5+6=11 and 11\times3=33.

Question 42 marks

A taxi company charges £3 plus £2 for each mile.

A taxi journey costs £17

How many miles long is the journey?

Hint

Take off the fixed charge first, then share what is left by the cost of each mile.

Worked solution

A taxi company charges £3 plus £2 for each mile.

A taxi journey costs £17

How many miles long is the journey?

  1. Let the number of miles be m. Then 3+2m=17.
  2. Subtract 3 from both sides: 2m=14.
  3. Divide both sides by 2: m=7.
  4. The journey is 7 miles long.

Question 53 marks

Tom has x sweets.

Ella has 6 more sweets than Tom.

(a)

Write down an expression, in terms of x, for the number of sweets Ella has.

1 mark

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

(b)

Tom and Ella have 30 sweets altogether.

Work out how many sweets Tom has.

2 marks

Hint

Write Ella's number of sweets using x, then use the total to make an equation.

Worked solution

Tom has x sweets.

Ella has 6 more sweets than Tom.

Part (a)

Write down an expression, in terms of x, for the number of sweets Ella has.

  1. Ella has 6 more than x, so Ella has x+6 sweets.

Part (b)

Tom and Ella have 30 sweets altogether.

Work out how many sweets Tom has.

  1. Altogether: x+(x+6)=30, so 2x+6=30.
  2. Subtract 6 from both sides: 2x=24.
  3. Divide both sides by 2: x=12.
  4. Check: Ella has 18, and 12+18=30. Tom has 12 sweets.

Question 62 marks

A, C and B are points on a straight line.

AC=(2x+3) cm and CB=(x+6) cm.

AB=30 cm.

Work out the value of x.

Hint

AC and CB together make the whole line AB.

Worked solution

A, C and B are points on a straight line.

AC=(2x+3) cm and CB=(x+6) cm.

AB=30 cm.

Work out the value of x.

  1. AC+CB=AB, so (2x+3)+(x+6)=30.
  2. Collect like terms: 3x+9=30.
  3. Subtract 9: 3x=21. Divide by 3: x=7.

Question 73 marks

The diagram shows a rectangle.

The perimeter of the rectangle is 44 cm.

Work out the value of x.

Hint

A rectangle has two lengths and two widths. Add all four sides and make the total 44.

Worked solution

The diagram shows a rectangle.

The perimeter of the rectangle is 44 cm.

Work out the value of x.

  1. Perimeter =2(2x+1)+2(x+3)=4x+2+2x+6=6x+8.
  2. So 6x+8=44.
  3. Subtract 8: 6x=36. Divide by 6: x=6.
  4. Check: the sides are 13 cm and 9 cm, and 13+9+13+9=44.

Question 82 marks

The diagram shows an isosceles triangle. The two marked sides are equal in length.

Work out the value of x.

Hint

The two marked sides are equal, so make their expressions equal.

Worked solution

The diagram shows an isosceles triangle. The two marked sides are equal in length.

Work out the value of x.

  1. The marked sides are equal: 3x-2=x+8.
  2. Subtract x from both sides: 2x-2=8.
  3. Add 2: 2x=10. Divide by 2: x=5.
  4. Check: 3\times5-2=13 and 5+8=13.

Question 92 marks

The diagram shows three angles on a straight line.

Work out the value of x.

Hint

Angles on a straight line add up to 180°.

Worked solution

The diagram shows three angles on a straight line.

Work out the value of x.

  1. Angles on a straight line add up to 180^\circ: x+2x+(x+40)=180.
  2. Collect like terms: 4x+40=180.
  3. Subtract 40: 4x=140. Divide by 4: x=35.

Question 102 marks

Three whole numbers follow on from each other in order, for example 8, 9, 10

The three numbers add up to 72

What is the largest of the three numbers?

Hint

Call the smallest number n. What are the other two in terms of n?

Worked solution

Three whole numbers follow on from each other in order, for example 8, 9, 10

The three numbers add up to 72

What is the largest of the three numbers?

  1. Let the smallest number be n. The numbers are n, n+1 and n+2.
  2. n+(n+1)+(n+2)=72, so 3n+3=72.
  3. Subtract 3: 3n=69. Divide by 3: n=23.
  4. The numbers are 23, 24 and 25. The largest is 25.

Question 113 marks

Ali, Bea and Cara collect stickers.

Bea has 3 more stickers than Ali.

Cara has twice as many stickers as Ali.

Altogether they have 51 stickers.

How many stickers does Cara have?

Hint

Let Ali have x stickers. Write Bea's and Cara's numbers in terms of x.

Worked solution

Ali, Bea and Cara collect stickers.

Bea has 3 more stickers than Ali.

Cara has twice as many stickers as Ali.

Altogether they have 51 stickers.

How many stickers does Cara have?

  1. Let Ali have x stickers. Bea has x+3 and Cara has 2x.
  2. x+(x+3)+2x=51, so 4x+3=51.
  3. Subtract 3: 4x=48. Divide by 4: x=12.
  4. Cara has 2\times12=24 stickers. Check: 12+15+24=51.

Question 123 marks

Sam makes cakes to sell at a school fair.

The ingredients cost Sam £18 altogether. He sells each cake for £2.50

After paying for the ingredients, Sam has £42 left from selling the cakes.

How many cakes did Sam sell?

Hint

Money from the cakes take away the cost of the ingredients is 42.

Worked solution

Sam makes cakes to sell at a school fair.

The ingredients cost Sam £18 altogether. He sells each cake for £2.50

After paying for the ingredients, Sam has £42 left from selling the cakes.

How many cakes did Sam sell?

  1. Let Sam sell n cakes. He takes £2.50n.
  2. 2.50n-18=42.
  3. Add 18: 2.50n=60.
  4. Divide by 2.50: n=24. Sam sold 24 cakes.

Question 133 marks

The diagram shows a rectangle.

(a)

Work out the value of x.

2 marks

(b)

Work out the perimeter of the rectangle.

1 mark

Hint

Opposite sides of a rectangle are the same length.

Worked solution

The diagram shows a rectangle.

Part (a)

Work out the value of x.

  1. The top and bottom are equal: 5x-4=3x+10.
  2. Subtract 3x: 2x-4=10. Add 4: 2x=14.
  3. Divide by 2: x=7.

Part (b)

Work out the perimeter of the rectangle.

  1. Top side: 5\times7-4=31 cm. (Bottom: 3\times7+10=31 cm.)
  2. Perimeter =31+8+31+8=78 cm.

Question 143 marks

The diagram shows a triangle.

Work out the size of the largest angle of the triangle.

Hint

The angles in a triangle add up to 180°. Find x first.

Worked solution

The diagram shows a triangle.

Work out the size of the largest angle of the triangle.

  1. Angles in a triangle add up to 180^\circ: x+(2x+10)+(3x-10)=180.
  2. Collect like terms: 6x=180, so x=30.
  3. The angles are 30^\circ, 2\times30+10=70^\circ and 3\times30-10=80^\circ.
  4. The largest angle is 80^\circ.

Question 153 marks

Pens cost x pence each. A ruler costs 35 pence.

Mia buys 3 pens and 1 ruler. She pays £1.85 altogether.

Mia writes this equation: 3x+35=1.85

Mia's equation is wrong.

(a)

What mistake has Mia made?

1 mark

(b)

Work out the cost of one pen. Give your answer in pence.

2 marks

Hint

Every amount in an equation must be in the same units.

Worked solution

Pens cost x pence each. A ruler costs 35 pence.

Mia buys 3 pens and 1 ruler. She pays £1.85 altogether.

Mia writes this equation: 3x+35=1.85

Mia's equation is wrong.

Part (a)

What mistake has Mia made?

  1. She used pence on one side and pounds on the other side
  2. She should have multiplied the 35 by 3
  3. She should have written 3 + x + 35
  4. She should have subtracted 35 instead of adding it
  1. 3x and 35 are in pence, but 1.85 is in pounds.
  2. Both sides must use the same units. £1.85 is 185 pence.

Part (b)

Work out the cost of one pen. Give your answer in pence.

  1. 3x+35=185.
  2. Subtract 35: 3x=150.
  3. Divide by 3: x=50. One pen costs 50 pence.

Question 16Challenge4 marks

The diagram shows an L-shape. All the corners are right angles.

All measurements are in centimetres.

The perimeter of the shape is 62 cm.

Work out the value of x.

Hint

Work out the two sides that have no label first, in terms of x.

Worked solution

The diagram shows an L-shape. All the corners are right angles.

All measurements are in centimetres.

The perimeter of the shape is 62 cm.

Work out the value of x.

  1. Missing across side: 3x-(x+4)=2x-4.
  2. Missing up side: (2x+1)-5=2x-4.
  3. Perimeter: 3x+5+(2x-4)+(2x-4)+(x+4)+(2x+1)=10x+2.
  4. 10x+2=62, so 10x=60 and x=6.
  5. Check: the sides are 18, 5, 8, 8, 10 and 13, which add up to 62.

Question 17Challenge3 marks

Leah had some money.

She spent \frac{3}{8} of her money on a coat. She then spent £14 on a scarf.

She had £26 left.

How much money did Leah have at the start?

Hint

Call the starting amount £m. The coat, the scarf and what is left add up to m.

Worked solution

Leah had some money.

She spent \frac{3}{8} of her money on a coat. She then spent £14 on a scarf.

She had £26 left.

How much money did Leah have at the start?

  1. Let Leah start with £m. Coat + scarf + left =m: \frac{3}{8}m+14+26=m.
  2. So 40=m-\frac{3}{8}m=\frac{5}{8}m.
  3. \frac{1}{8}m=40\div5=8, so m=8\times8=64.
  4. Check: the coat costs £24, and 24+14+26=64. Leah had £64.

Question 18Challenge3 marks

A dance school has two ways to pay.

Plan A: a joining fee of £30, then £3 for each class.

Plan B: a joining fee of £12, then £5 for each class.

For how many classes do the two plans cost the same?

Hint

Write the cost of n classes for each plan, then make the two costs equal.

Worked solution

A dance school has two ways to pay.

Plan A: a joining fee of £30, then £3 for each class.

Plan B: a joining fee of £12, then £5 for each class.

For how many classes do the two plans cost the same?

  1. For n classes, plan A costs 30+3n and plan B costs 12+5n.
  2. 30+3n=12+5n.
  3. Subtract 3n: 30=12+2n. Subtract 12: 18=2n, so n=9.
  4. Check: plan A costs 30+27=57 and plan B costs 12+45=57. The answer is 9 classes.

Question 19Challenge4 marks

Two rectangular flower beds, A and B, are shown in the diagram.

Bed A covers 34 m² more ground than bed B.

Work out the value of x.

Hint

Write the area of each bed in terms of x. The area of A take away the area of B is 34.

Worked solution

Two rectangular flower beds, A and B, are shown in the diagram.

Bed A covers 34 m² more ground than bed B.

Work out the value of x.

  1. Area of A =6(x+5)=6x+30. Area of B =4(x+2)=4x+8.
  2. (6x+30)-(4x+8)=34, so 2x+22=34.
  3. Subtract 22: 2x=12. Divide by 2: x=6.
  4. Check: A is 6\times11=66 m² and B is 4\times8=32 m², and 66-32=34.

Question 20Challenge4 marks

The diagram shows an equilateral triangle and a square.

The perimeter of the triangle is equal to the perimeter of the square.

Work out the area of the square.

Hint

All three sides of the triangle are equal, and all four sides of the square are equal.

Worked solution

The diagram shows an equilateral triangle and a square.

The perimeter of the triangle is equal to the perimeter of the square.

Work out the area of the square.

  1. Perimeter of the triangle =3(x+6)=3x+18. Perimeter of the square =4(x+1)=4x+4.
  2. 3x+18=4x+4. Subtract 3x: 18=x+4, so x=14.
  3. Each side of the square is 14+1=15 cm.
  4. Area of the square =15\times15=225 cm².

More on this topic: Forming and Solving Equations worksheet with full solutions

All Edexcel GCSE Foundation practice questions