Solutions — Full Answer Key

Mathematics

Year 7 · 7.3 Introduction to Algebra

Solutions · Full Answer Key

Pack A answers · Pack B answers · Problem-solving worked solutions

Pack A — Answers

Bronze

b = 3

; because

3 \times 4 = 12

and

12 + 3 = 15

✓

3n

is correct; in algebra the coefficient is written before the variable

3.E.g.

5x + 3x

and

4x + 4x

; both simplify to

8x

✓

4.E.g.

x + 4 = 12

; solving:

x = 12 − 4 = 8

✓

4n = 28

... wait —

n = 7

chairs per row; equation from context: "total = rows × per row"

3x + 12

; when

x = 2

: bracket gives

3(6) = 18

; expanded gives

6 + 12 = 18

✓

x = 6

; check:

2(6) + 5 = 17

✓

4n - 3

and

4 \times n - 3

are the same; in algebra,

4n

means

4 \times n

9.Yes, both have

y

;

9y - 4y = 5y

because

9 - 4 = 5

10.Coefficient of

x

is 7 (multiplies

x

); constant is 3 (never changes)

11.

a = 0

. Multiplying anything by

0

gives

0

, and

5

on its own is not zero, so

a

has to be the

0

Silver

12.18

13.

7x + 2y

14.

4x + 17

15.

x = 6

16.Expression:

5p

; cost = £40

17.48

18.

8a + 2b

19.

(2x + 14)

20.

8x - 20

21.

x = 15

22.

a = 4

. The

3

isn't zero, so whatever's in the bracket has to be the zero — meaning

4 - a = 0

, so

a = 4

Gold

23.

6x

24.8

25.

5x + 16

26.

x = -2

27.

3p + 4q

pence (cannot simplify further as

p

and

q

are different)

28.

-18

29.

x = 4

30.

x = 4

31.

x - 12

32.53

Platinum

33.

n = 2

34.15, 16, 17

35.

x = 4

36.The difference is always 5.

37.

C = 12 + 3g

;

g = 7

38.

p^2 - p = p(p-1)

— always even.

39.

x = 9,\ y = 5

40.

x = 5

; sides: 13 cm, 14 cm, 10 cm

41.

c = 15

;

4(x-2) + 6x = 10x - 8

42.Input for output 7:

x = 5

. Fixed point:

x = 4

Pack B — Answers

Bronze

b = 3

; because

3 \times 6 = 18

and

18 + 3 = 21

✓

n^2

is correct; repeated multiplication of the same variable is written using index notation

3.E.g.

7x + 4x

and

6x + 5x

; both simplify to

11x

✓

4.E.g.

3x = 21

; solving:

x = 21 ÷ 3 = 7

✓

5n = 45

...

n = 9

chairs per row; equation from context: "total = rows × per row"

5x + 10

; when

x = 3

: bracket gives

5(5) = 25

; expanded gives

15 + 10 = 25

✓

x = 5

; check:

3(5) + 4 = 19

✓

6n - 5

and

6 \times n - 5

are the same;

6n

is the conventional shorthand

9.Yes, both have

y

;

8y - 3y = 5y

because

8 - 3 = 5

10.Coefficient of

x

is 4 (multiplies

x

); constant is 9 (never changes)

11.

a = 0

. Multiplying anything by

0

gives

0

, and

7

on its own is not zero, so

a

has to be the

0

Silver

12.16

13.

10x + 2y

14.

3x + 22

15.

x = 5

16.Expression:

7p

; cost = £42

17.50

18.

9a + 3b

19.

(2x + 22)

20.

12x - 21

21.

x = 20

22.

a = 2

. The

5

isn't zero, so the bracket has to be zero — meaning

2 - a = 0

, so

a = 2

Gold

23.

7x

24.21

25.

7x + 18

26.

x = -2

27.

5p + 2q

pence

28.23

29.

x = 3

30.

x = 4

31.

6x - 8

32.32

Platinum

33.

n = -2

34.20, 21, 22

35.

x = 7

36.The difference is always 5.

37.

C = 15 + 4g

;

g = 9

38.

p^2 - p = p(p-1)

— always even.

39.

x = 13,\ y = 7

40.

x = 8

; sides: 26 cm, 13 cm, 7 cm

41.

c = 24

;

3(x-5) + 7x = 10x - 15

42.Input for output 6:

x = 4

. Fixed point:

x = \frac{10}{3}

Problem-solving — Worked Solutions

1Problem 1

Answer

Full working

Let the number be

n

. Triple it:

3n

. Add 5:

3n + 5

. Subtract twice the original:

3n + 5 - 2n = n + 5

. Set equal to 11:

n + 5 = 11

. Subtract 5:

n = 6

. Check: triple 6 = 18; add 5 = 23; subtract twice 6 = 23 − 12 = 11 ✓.

2Problem 2

Answer

x = 5

; area = 128 cm²

Full working

Perimeter

= 2(\text{length} + \text{width}) = 2((3x+1)+(x+3)) = 2(4x+4) = 8x+8

. Set equal to 48:

8x + 8 = 48 \Rightarrow 8x = 40 \Rightarrow x = 5

. Length

= 3(5)+1 = 16

cm; width

= 5+3 = 8

cm. Check perimeter:

2(16+8) = 48

✓. Area

= 16 \times 8 = 128

cm².

3Problem 3

Answer

Priya: 11, brother: 16, mother: 33

Full working

Priya:

x

. Brother:

x + 5

. Mother:

3x

. Sum:

x + (x+5) + 3x = 5x + 5

. Set equal to 60:

5x + 5 = 60 \Rightarrow 5x = 55 \Rightarrow x = 11

. Priya is 11, her brother is

11+5=16

, her mother is

3 \times 11=33

. Check:

11+16+33=60

✓.

4Problem 4

Answer

(a) −1 (b) −1 (c) Proof:

(n-1)(n+1) - n^2 = n^2 - 1 - n^2 = -1

, always.

Full working

(a) Outer two: 5 and 7.

5 \times 7 = 35

. Middle squared:

6^2 = 36

35 - 36 = \mathbf{-1}

✓.

(b) Outer two:

(-3)

and

(-1)

(-3) \times (-1) = 3

. Middle squared:

(-2)^2 = 4

3 - 4 = \mathbf{-1}

✓. (This also interleaves Y7.2 directed-number multiplication.)

(c) Let the three consecutive integers be

n-1

n

n+1

. Multiply the outer two:

(n-1)(n+1) = n^2 - 1

(using the difference-of-two-squares pattern). Subtract the middle squared:

(n^2 - 1) - n^2 = \mathbf{-1}

. This is independent of

n

, so the result is always −1.

5Problem 5

Answer

£36

Full working

Let the meal cost

\pounds m

. Total bill including service charge:

m + 6

. Split three ways:

\dfrac{m+6}{3} = 14

. Multiply both sides by 3:

m + 6 = 42

. Subtract 6:

m = 36

. The meal cost £36. Check:

36 + 6 = 42

;

42 \div 3 = 14

✓.

6Problem 6

Answer

Full working

Let the number be

n

. "Subtract 8":

n - 8

. "Multiply by 3":

3(n-8)

. Set equal to

-6

3(n-8) = -6

. Divide both sides by 3:

n - 8 = -2

. Add 8:

n = 6

. Check:

3(6-8) = 3(-2) = -6

✓.

7Problem 7

Answer

(a) 11 (b) 3 (c)

x = 5

(d) Inputs above 5 grow without bound; inputs below 5 decrease without bound.

Full working

(a)

3 \times 7 - 10 = 21 - 10 = \mathbf{11}

.

(b) We need

3x - 10 = -1

. Add 10:

3x = 9

. Divide by 3:

x = \mathbf{3}

. Check:

3(3)-10 = -1

✓.

(c) Fixed point: output = input, so

3x - 10 = x

. Subtract

x

2x - 10 = 0

. Add 10:

2x = 10

. Divide by 2:

x = \mathbf{5}

. Check:

3(5)-10 = 5

✓.

(d) Try

x = 6

(above 5):

3(6)-10=8

;

3(8)-10=14

; the outputs grow larger. Try

x = 4

(below 5):

3(4)-10=2

;

3(2)-10=-4

;

3(-4)-10=-22

; the outputs decrease (and become very negative). The fixed point

x=5

is an **unstable** equilibrium — inputs close to it move away from it under repeated application.

8Problem 8

Answer

t = -3

°C

Full working

At 6 am:

t + 7

. At noon (doubled from 6 am):

2(t + 7)

. Set equal to 8:

2(t+7) = 8

. Divide by 2:

t + 7 = 4

. Subtract 7:

t = -3

. The midnight temperature was

-3

°C. Check:

-3 + 7 = 4

;

4 \times 2 = 8

✓.

9Problem 9

Answer

(a)

4n + 1

; (b) pattern 10; (c) No

Full working

(a) The sequence 5, 9, 13, … increases by 4 each time. This is linear:

\text{tiles} = 4n + 1

(check:

n=1 \Rightarrow 5

✓,

n=2 \Rightarrow 9

✓). (b) Set

4n + 1 = 41

4n = 40

n = 10

✓. (c) Set

4n + 1 = 100

4n = 99

n = 24.75

. Since

n

must be a whole number, no pattern has exactly 100 tiles.

10Problem 10

Answer

(a) Three ways for 15: see working (b) 16 is not a staircase number (c) See algebraic derivation (d) No power of 2 is a staircase number.

Full working

(a)

15 = 7 + 8

(two consecutive).

15 = 4 + 5 + 6

(three consecutive).

15 = 1 + 2 + 3 + 4 + 5

(five consecutive). ✓

(b) Two consecutive:

n + (n+1) = 2n+1

— always odd. 16 is even, so no. Three consecutive:

n + (n+1) + (n+2) = 3n + 3 = 16 \Rightarrow 3n = 13

— not an integer. Four consecutive:

4n + 6 = 16 \Rightarrow 4n = 10

— not an integer. Five consecutive:

5n + 10 = 16 \Rightarrow 5n = 6

— not an integer. **16 is not a staircase number.**

(c) Consecutive integers from

n

n+(k-1)

: sum =

n + (n+1) + \cdots + (n+k-1)

. There are

k

terms each contributing

n

, giving

kn

, plus the extras

0 + 1 + 2 + \cdots + (k-1) = \dfrac{(k-1)k}{2}

. Total:

kn + \dfrac{k(k-1)}{2}

.

(d) Powers of 2: 1, 2, 4, 8, 16, 32, … Testing each (as in part b), none can be written as a staircase. The sum

kn + \frac{k(k-1)}{2} = \frac{k(2n+k-1)}{2}

. For a power of 2, both factors

k

and

(2n+k-1)

must be powers of 2 — but they have opposite parity (one odd, one even), so the product can only produce a power of 2 if one of them equals 1, giving trivial cases. **Conjecture: no power of 2 is a staircase number.** (This is a famous result in number theory.)

11Problem 11

Answer

Each box: 1.25 kg; total each side: 8.75 kg

Full working

Let the mass of one box be

m

kg. Balance equation:

3m + 5 = 7m

. Subtract

3m

5 = 4m

. Divide by 4:

m = 1.25

. Total on each side:

7 \times 1.25 = 8.75

kg. Check left:

3 \times 1.25 + 5 = 3.75 + 5 = 8.75

✓.

12Problem 12

Answer

(a) 12 (even); (b) proof below; (c)

n = 6

and

n = -5

Full working

(a)

(-3)^2 - (-3) = 9 + 3 = 12

✓ (even — interleaves Y7.2 directed numbers). (b) Factorise:

n^2 - n = n(n-1)

n

and

n-1

are consecutive integers, so one is always even. The product of an even and any integer is even. Therefore

n^2 - n

is always even ✓. (c)

n(n-1) = 30

. Look for factor pairs of 30 where one factor is one more than the other:

6 \times 5 = 30

✓ (so

n = 6

);

(-5) \times (-6) = 30

✓ (so

n = -5

). Check:

6^2 - 6 = 30

✓;

(-5)^2 - (-5) = 25 + 5 = 30

✓.