🧠 Grade 7 Practice Paper 3 — Challenge

💻 Calculator Permitted ⏱ 60 Minutes 📝 60 Marks 30 Questions
Instructions: Answer all questions. A calculator is permitted for this paper. Show your working where asked — method marks may be awarded even if your final answer is incorrect. Marks for each question are shown in brackets. Give answers to the degree of accuracy stated.
Name:
Class:
Date:
Section A: Number (Questions 1–8)
1
The mass of a hydrogen atom is 1.67 × 10−27 kg. The mass of an oxygen atom is 2.66 × 10−26 kg.
How many times heavier is an oxygen atom than a hydrogen atom? Give your answer in standard form to 2 significant figures.
[2]
2
A = 23 × 32 × 5    B = 22 × 3 × 7
Find (a) the HCF of A and B, and (b) the LCM of A and B.
(a) HCF =    (b) LCM =
[3]
3
A bacteria population doubles every hour. It starts at 500.
(a) Write an expression for the population after n hours.
(b) Find the population after 8 hours.
(a) Population =
(b) After 8 hours =
[2]
4
Show that 0.1̅8̅ = 2/11. (Let x = 0.181818… and multiply appropriately to eliminate the recurring part.)
Show full algebraic working in the box below. Answer: 2/11 ✓
[2]
5
Between which two consecutive integers does √200 lie? Explain your reasoning without using a calculator.
and
[2]
6
A number is decreased by 30%, then the result is increased by 40%. What is the overall percentage change from the original? State whether it is an increase or decrease.
Overall change:
[2]
7
The value of a car decreases by 15% per year. After 3 years it is worth £9,752.50. Find the original value to the nearest pound.
£
[3]
8
Write all values of n such that n is prime and 20 < n2 < 200.
[2]
Section B: Algebra (Questions 9–16)
9
Expand and simplify: 3(2x − 1) − 2(x − 4) + 5
[2]
10
The perimeter of a rectangle is 46 cm. The length is (3x + 2) cm and the width is (x − 1) cm.
Form an equation, solve for x, then find the area of the rectangle.
x =    Area = cm²
[3]
11
Solve:   (2x − 1)⁄3 = (x + 1)⁄2
x =
[2]
12
The nth term of a sequence is 2n2 + n − 3. Find the first term that exceeds 100.
First term exceeding 100 is term number:   with value:
[2]
13
Plan A costs £5 per month plus 8p per minute. Plan B costs £12 per month plus 3p per minute.
Form an equation and find the number of minutes at which both plans cost the same.
minutes
[3]
14
Consider the lines y = 2x + 1 and y = 8 − x.
(a) Find the coordinates of their intersection point algebraically.
(b) Verify your answer satisfies both equations.
(a) Intersection: x =   y =
[2]
15
Two inequalities: x > −1   AND   x ≤ 4.
(a) List all integers satisfying both.
(b) Describe how to represent this on a number line (open/closed circles).
(a) Integers:
(b)
[2]
16
Given f(x) = 3x − 2, find:
(a) f(5)    (b) The value of x when f(x) = 16
(a) f(5) =    (b) x =
[2]
Section C: Fractions, Decimals & Percentages (Questions 17–22)
17
A recipe uses 2½ cups of flour. You need to make 1¾ times the recipe. How much flour do you need? Give your answer as a mixed number.
[2]
18
After two successive discounts of 10% and 20%, the final price of an item is £43.20. Find the original price.
£
[3]
19
Three fractions are in the ratio 1 : 2 : 3 and their sum is 1. Find the three fractions.
Fractions:
[2]
20
Express 5/7 as a recurring decimal. Clearly indicate the repeating block using dot notation or an overline description.
[2]
21
A salary is £24,000. It increases by 5%, then by a further 3%. What is the new salary?
£
[2]
22
A tank is ⅔ full. After removing 45 litres it is ¼ full. Find the total capacity of the tank.
litres
[3]
Section D: Geometry (Questions 23–27)
23
Two circles share the same centre. The smaller circle has radius 4 cm and the larger has radius 7 cm. Find the area of the ring (annulus) between them. Give your answer to 1 decimal place.
cm²
[2]
24
A cylinder has the same volume as a cube with side length 6 cm. The cylinder's height equals its diameter (h = 2r). Find the radius of the cylinder to 2 decimal places.
r = cm
[3]
25
In a diagram, a pair of parallel lines are cut by a transversal. Alternate angles give:
(4x − 15)° = (2x + 25)°
Solve for x and find the size of the angle.
x =    Angle = °
[2]
26
A sector has radius 12 cm and arc length 15.7 cm. Find the angle at the centre to the nearest degree.
(Arc length = rθ where θ is in radians, or use arc length = (θ/360) × 2πr)
Angle = °
[2]
27
The total surface area of a cube is 150 cm². Find its volume.
cm³
[2]
Section E: Ratio, Proportion & Statistics (Questions 28–30)
28
The mean of 8 numbers is 12.5. One number is removed, making the new mean 11.8 (for the remaining 7 numbers). What was the removed number?
[2]
29
A map has a scale of 1 : 50,000. A rectangular field on the map measures 2.4 cm by 3.6 cm. Find the real area of the field in km².
km²
[3]
30
A back-to-back stem-and-leaf diagram shows test scores for two classes (scores are in the 40s, 50s, 60s):

Class A (leaves) Stem Class B (leaves)
7 5 248 9
9 8 3 152 5 7
6 4 061 3 5 8
(a) Find the median score for each class.
(b) Make one comparison about the distributions of the two classes.
(a) Median Class A =    Median Class B =

(b) Comparison:
[3]
60:00

✅ Mark Scheme — Paper 3 (Challenge)

11.6 × 101 (approx 15.9 times) [2] 2.66×10⁻²⁶ ÷ 1.67×10⁻²⁷ = 15.93… ≈ 1.6×10
2aHCF = 60 [1] 2²×3×5 = 60
2bLCM = 2520 [2] 2³×3²×5×7 = 2520
3a500 × 2n [1]
3b128,000 [1] 500 × 2⁸ = 500 × 256
42/11 [2] 100x=18.18…, 99x=18, x=18/99=2/11
514 and 15 [2] 14²=196<200<225=15²
6−2% decrease [2] 0.7×1.4=0.98, so 2% decrease
7£16,000 [3] 9752.50 ÷ (0.85)³ = 9752.50 ÷ 0.614125
85, 7, 11, 13 [2] 5²=25, 7²=49, 11²=121, 13²=169; 17²=289>200
94x + 16 [2] 6x−3−2x+8+5 = 4x+16
10x = 5.5; Area = 83.25 cm² [3] 2(4x+1)=46 → 4x=22 → x=5.5; l=18.5cm, w=4.5cm; Area=18.5×4.5=83.25
11x = 5 [2] Multiply by 6: 2(2x−1)=3(x+1) → 4x−2=3x+3 → x=5
12n = 7, value = 102 [2] n=6: 72+6−3=75; n=7: 98+7−3=102
13140 minutes [3] 5+0.08m=12+0.03m → 0.05m=7 → m=140
14x = 7/3 ≈ 2.33, y = 17/3 ≈ 5.67 [2] 2x+1=8−x → 3x=7
15a0, 1, 2, 3, 4 [1]
15bOpen circle at −1, closed/filled circle at 4 [1]
16af(5) = 13 [1] 3(5)−2
16bx = 6 [1] 3x−2=16 → 3x=18
174⅜ cups (4 and 3/8) [2] 5/2 × 7/4 = 35/8 = 4⅜
18£60 [3] P × 0.9 × 0.8 = 43.20 → P = 43.20 ÷ 0.72 = 60
191/6, 1/3, 1/2 [2] 6k=1 → k=1/6; fractions: 1/6, 2/6, 3/6
200.̇714285̇ (repeating block: 714285) [2]
21£25,956 [2] 24000 × 1.05 × 1.03 = 25200 × 1.03
22108 litres [3] (2/3 − 1/4)C = 45 → 5/12 C = 45 → C = 108
23103.7 cm² [2] π(49−16) = 33π ≈ 103.67
24r ≈ 3.25 cm [3] Volume cube=6³=216 cm³; h=2r so πr²(2r)=216; 2πr³=216; r³=108/π≈34.38; r=∛34.38≈3.25 cm
25x = 20, angle = 65° [2] 4(20)−15 = 65°
2675° [2] θ=15.7/12≈1.308 rad → ×180/π ≈ 74.9° ≈ 75°
27125 cm³ [2] 6s²=150 → s=5 → V=125
2817.4 [2] Sum 8 numbers=100; Sum 7=82.6; removed=100−82.6=17.4
292.16 km² [3] 2.4×50000=120,000cm=1,200m=1.2km; 3.6×50000=1,800m=1.8km; Area=1.2×1.8=2.16 km²
30aMedian A = 55.5, Median B = 57 [2] A(10 values): 5th&6th = 53,58 → median=(53+58)/2=55.5; B(9 values): 5th value = 57
30bAccept: similar medians; Class B slightly higher median; Class B has more high scores (60s); Class A more spread in 50s [1]