What is A21: Forming equations?

Choose a variable and state exactly what it represents, including units where helpful.

How many marks are A21 questions worth?

In verified past-paper appearances, A21 questions on Edexcel GCSE Maths 1MA1 have been worth 1–7 marks.

Which past exam papers has A21 appeared in?

A21 has appeared in 51 verified past-paper questions indexed on this page, with direct links to the official question paper and mark scheme for each.

A21

Translate simple situations or procedures into algebraic expressions or formulae; derive an equation (or two simultaneous equations), solve the equation(s) and interpret the solution

Forming equations

Worked answers, methods and verified real exam appearances for A21 on Edexcel GCSE Maths 1MA1.

Explanation

Choose a variable and state exactly what it represents, including units where helpful.
Translate each related quantity into an expression in that variable, then use the stated relationship to form an equation or a pair of simultaneous equations.
Solve using appropriate algebra and interpret the mathematical values in the original situation.
A valid algebraic root may still be impossible as a length, age or count, so reject it with a contextual reason.
The examiner awards marks for forming the model as well as solving it; an unlabelled value of $x$ is not a complete answer when dimensions, prices or numbers of items were requested.

Worked example

A rectangle has width $x$ cm and length $(x+3)$ cm. Its perimeter is $34$ cm. Find both dimensions.

1.Form the perimeter equation: $2x+2(x+3)=34$ .
2.Expand and solve: $4x+6=34$ , so $x=7$ .
3.Interpret the expressions: width $=7$ cm and length $=7+3=10$ cm.

Answer: Width $7$ cm and length $10$ cm.

Common mistakes

Don't form $x+(x+3)=34$ and count only half of the rectangle's perimeter.
Don't stop at $x=7$ without finding and labelling both requested dimensions.
Don't keep a negative algebraic root even though the quantity is a length or age.

Exam tip

Define the variable before forming the equation and finish with a sentence interpreting every required quantity.

Worked practice

Tier 1 · Easy

A rectangle has width

x

cm and length

(x+3)

cm. Its perimeter is

34

cm. Form and solve an equation to find both dimensions.

(3)

(Total for Question 1 is 3 marks)

Mark scheme

Mark scheme for question 1
Question	Answer	Mark	Mark scheme
1	Width $=7$ cm Length $=10$ cm	3	The perimeter equation is $2x+2(x+3)=34$ . Hence $4x+6=34$ , so $4x=28$ and $x=7$ . The width is $7$ cm and the length is $7+3=10$ cm.

Tier 2 · Standard

A club sells

38

tickets. Adult tickets cost £7 and junior tickets cost £4. The total received is £203. Form two equations and find how many tickets of each type were sold.

(4)

(Total for Question 2 is 4 marks)

Mark scheme

Mark scheme for question 2
Question	Answer	Mark	Mark scheme
2	$17$ adult tickets and $21$ junior tickets	4	Let $a$ and $j$ be the numbers of adult and junior tickets. Then $a+j=38$ and $7a+4j=203$ . Subtract $4(a+j)=152$ from the money equation to get $3a=51$ , so $a=17$ and $j=21$ .

Tier 3 · Hard

Mira is

4

years older than Theo. In

3

years, the product of their ages will be

192

. Form an equation and find their current ages.

(5)

(Total for Question 3 is 5 marks)

Mark scheme

Mark scheme for question 3
Question	Answer	Mark	Mark scheme
3	Theo is $9$ years old and Mira is $13$ years old.	5	Let Theo's current age be $x$ , so Mira's is $x+4$ . In three years their ages are $x+3$ and $x+7$ , giving $(x+3)(x+7)=192$ . Thus $x^2+10x-171=0=(x-9)(x+19)$ . The solutions are $x=9$ and $x=-19$ ; reject the negative age. Theo is $9$ and Mira is $13$ .

Tier 1 · Easy

A coach has

r

rows with

s

seats in each row and

6

extra seats at the back. Write down an expression, in terms of

r

and

s

, for the total number of seats on the coach.

(1)

(Total for Question 4 is 1 mark)

Mark scheme

Mark scheme for question 4
Question	Answer	Mark	Mark scheme
4	$rs+6$	1	The $r$ rows contain $rs$ seats altogether. Including the $6$ extra seats at the back gives $rs+6$ .

Tier 2 · Standard

52

cm ribbon is cut into pieces of lengths

x

cm,

(x+4)

cm and

2x

cm. Form and solve an equation to find the length of each piece.

(3)

(Total for Question 5 is 3 marks)

Mark scheme

Mark scheme for question 5
Question	Answer	Mark	Mark scheme
5	$12$ cm, $16$ cm and $24$ cm	3	Let the first piece have length $x$ cm. Then the pieces have lengths $x$ , $x+4$ and $2x$ , so $x+(x+4)+2x=52$ . Hence $4x+4=52$ , giving $x=12$ . The three lengths are $12$ cm, $16$ cm and $24$ cm.

Tier 3 · Hard

A right-angled triangle has perpendicular sides of lengths

x

cm and

(x+1)

cm. Its hypotenuse has length

29

cm. Form and solve an equation to find the lengths of the two perpendicular sides.

(4)

(Total for Question 6 is 4 marks)

Mark scheme

Mark scheme for question 6
Question	Answer	Mark	Mark scheme
6	$20$ cm and $21$ cm	4	Pythagoras' theorem gives $x^2+(x+1)^2=29^2$ . Expanding and simplifying gives $x^2+x-420=0=(x+21)(x-20)$ . Hence $x=-21$ or $x=20$ . A length cannot be negative, so the perpendicular sides are $20$ cm and $21$ cm.

Tier 2 · Standard

A repair company charges a fixed call-out fee of £18 and then £7 for each half-hour of work. A bill is £67. Form and solve an equation to work out the time spent on the repair.

(3)

(Total for Question 7 is 3 marks)

Mark scheme

Mark scheme for question 7
Question	Answer	Mark	Mark scheme
7	$3.5$ hours	3	Let $h$ be the number of half-hours. The equation is $18+7h=67$ , so $7h=49$ and $h=7$ . Seven half-hours is $3.5$ hours.

Tier 3 · Hard

A rectangular play area has length

(4x-3)

metres and width

(x+2)

metres. Its perimeter is

68

metres. Form and solve an equation to work out the length and width of the play area.

(4)

(Total for Question 8 is 4 marks)

Mark scheme

Mark scheme for question 8
Question	Answer	Mark	Mark scheme
8	The length is $25$ m and the width is $9$ m.	4	The perimeter equation is $2(4x-3)+2(x+2)=68$ . Expanding and collecting gives $8x-6+2x+4=68$ , so $10x-2=68$ and $x=7$ . Therefore the length is $4(7)-3=25$ m and the width is $7+2=9$ m. The check $2(25)+2(9)=68$ confirms the result.

Tier 3 · Hard

A two-digit number has tens digit

x

and units digit

(x+3)

. The number is

6

more than four times the sum of its digits. Form and solve an equation to find the number.

(4)

(Total for Question 9 is 4 marks)

Mark scheme

Mark scheme for question 9
Question	Answer	Mark	Mark scheme
9	$58$	4	The number is $10x+(x+3)=11x+3$ , and the sum of its digits is $2x+3$ . Therefore $11x+3=4(2x+3)+6$ . This gives $11x+3=8x+18$ , so $3x=15$ and $x=5$ . The units digit is $8$ , so the number is $58$ .

Q10

Tier 3 · Hard

A positive integer is multiplied by the integer that is

5

greater than it. The product is

176

. Form and solve an equation to find the two positive integers.

(4)

(Total for Question 10 is 4 marks)

Mark scheme

Mark scheme for question 10
Question	Answer	Mark	Mark scheme
10	$11$ and $16$	4	Let the smaller positive integer be $n$ . Then $n(n+5)=176$ , so $n^2+5n-176=0$ . Factorising gives $(n+16)(n-11)=0$ , so $n=-16$ or $n=11$ . Reject the negative value because $n$ is positive. The integers are $11$ and $16$ .

Verified exam appearances

Series	Paper	Question	Marks	Calculator	Tier	Links
2024-11	1F	Q16	5	Non-calculator	Foundation	QP MS
2019-06	1F	Q28	4	Non-calculator	Foundation	QP MS
2023-06	1H	Q18	4	Non-calculator	Higher	QP MS
2023-06	2F	Q24	4	Allowed	Foundation	QP MS
2023-11	1H	Q7	5	Non-calculator	Higher	QP MS
2021-11	1H	Q14	3	Non-calculator	Higher	QP MS
2024-06	3H	Q10	3	Allowed	Higher	QP MS
2019-06	1H	Q22	5	Non-calculator	Higher	QP MS
2023-06	1F	Q18	3	Non-calculator	Foundation	QP MS
2023-06	1H	Q10	4	Non-calculator	Higher	QP MS
2022-11	1F	Q16	4	Non-calculator	Foundation	QP MS
2024-11	1F	Q23	4	Non-calculator	Foundation	QP MS
2024-06	3F	Q16	3	Allowed	Foundation	QP MS
2022-06	3F	Q28	3	Allowed	Foundation	QP MS
2023-06	2H	Q5	4	Allowed	Higher	QP MS
2024-11	1H	Q6	4	Non-calculator	Higher	QP MS
2023-11	3F	Q18	5	Allowed	Foundation	QP MS
2021-11	1H	Q11	4	Non-calculator	Higher	QP MS
2022-06	3H	Q3	5	Allowed	Higher	QP MS
2023-11	1F	Q24	5	Non-calculator	Foundation	QP MS
2024-11	3H	Q19	4	Allowed	Higher	QP MS
2024-11	2H	Q7	5	Allowed	Higher	QP MS
2021-11	3F	Q25	3	Allowed	Foundation	QP MS
2019-06	1F	Q13	1	Non-calculator	Foundation	QP MS
2021-11	1F	Q26	4	Non-calculator	Foundation	QP MS
2022-06	2H	Q15	4	Allowed	Higher	QP MS
2019-06	3H	Q5	4	Allowed	Higher	QP MS
2021-11	1F	Q16	4	Non-calculator	Foundation	QP MS
2022-11	2F	Q14	7	Allowed	Foundation	QP MS
2019-11	1H	Q14	4	Non-calculator	Higher	QP MS
2024-11	3F	Q12	2	Allowed	Foundation	QP MS
2023-06	3H	Q24	5	Allowed	Higher	QP MS
2019-11	1F	Q28	3	Non-calculator	Foundation	QP MS
2024-11	2H	Q22	5	Allowed	Higher	QP MS
2024-06	2H	Q7	4	Allowed	Higher	QP MS
2024-06	2F	Q26	4	Allowed	Foundation	QP MS
2023-06	3F	Q11	3	Allowed	Foundation	QP MS
2024-06	1H	Q21	5	Non-calculator	Higher	QP MS
2021-11	3F	Q11	3	Allowed	Foundation	QP MS
2019-06	1H	Q7	4	Non-calculator	Higher	QP MS
2021-11	3H	Q5	3	Allowed	Higher	QP MS
2024-11	2F	Q26	5	Allowed	Foundation	QP MS
2019-11	3F	Q14	2	Allowed	Foundation	QP MS
2024-11	1H	Q13	5	Non-calculator	Higher	QP MS
2019-06	2F	Q7	1	Allowed	Foundation	QP MS
2022-06	3F	Q24	5	Allowed	Foundation	QP MS
2019-06	1H	Q17	4	Non-calculator	Higher	QP MS
2021-11	1H	Q7	4	Non-calculator	Higher	QP MS
2021-11	3H	Q13	5	Allowed	Higher	QP MS
2019-11	2F	Q18	4	Allowed	Foundation	QP MS
2022-06	1H	Q16	4	Non-calculator	Higher	QP MS

Other points in A Algebra · sequences and functions

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