June 2022 Paper 3 Foundation

Per cent means out of 100, so $35\% = \frac{35}{100}$.

This simplifies to $\frac{7}{20}$ by dividing top and bottom by 5.

To find $\frac{1}{4}$ of an amount, divide by 4.

$28 \div 4 = 7$

Factors of 12 are the numbers that divide into it with no remainder: 1, 2, 3, 4, 6 and 12.

Any two of these, for example 3 and 4, are correct.

Multiply the numbers: $2 \times 3 = 6$, and keep the letter.

$2m \times 3 = 6m$

Look for a number that multiplies by itself to give 1.69.

$1.3 \times 1.3 = 1.69$, so $\sqrt{1.69} = 1.3$

A rectangle and a rhombus have order 2 but also have lines of symmetry, so avoid them.

A general parallelogram (slanted, with unequal adjacent sides) has order 2 and no lines of symmetry. Draw one on the grid.

Apples: $86 + 75 + 92 = 253$

Oranges: $68 + 80 + 76 = 224$

Difference: $253 - 224 = 29$

(a) The terms go up by 5: $23 + 5 = 28$, $28 + 5 = 33$

(b) The terms go 3, 8, 13, 18, ... so every term ends in 3 or 8.

50 ends in 0, so it is not a term (48 and 53 are consecutive terms).

(a) Two triangular ends plus three rectangular sides: $2 + 3 = 5$ faces.

(b) Each triangle has 3 edges and three more join the triangles: $3 + 3 + 3 = 9$ edges.

(a) There is no 7 in the list, so it is impossible: probability 0.

(b) 6, 6, 8, 9 are greater than 5: $\frac{4}{8} = \frac{1}{2}$

(c) Even numbers: 2, 2, 6, 6, 8, which is 5 out of 8, so $\frac{5}{8}$

Nails needed: $35 \times 4 = 140$

Nails available: $3 \times 48 = 144$

$144 > 140$, so yes, she has enough (4 to spare).

Write it as a fraction: $\frac{60}{1000}$

Divide top and bottom by 20: $\frac{3}{50}$

(a) Measure: AB ≈ 5 cm, BC ≈ 4 cm, AC ≈ 7 cm.

Parveen: $5 + 4 = 9$ cm, Susan: 7 cm, so a difference of 2 cm.

$2 \times 150 = 300$ metres

(b) At C, measure clockwise from North round to the line CA: about $288°$.

(a) 12 values, so the median is halfway between the 6th and 7th: both are 7, so median = 7

(b) $9 - 4 = 5$

(c) Boys' median 7 > girls' 6, so boys' shoe sizes are larger on average.

Boys' range 5 > girls' 3, so boys' sizes are more spread out.

Top: $2.75 \times 14.6 = 40.15$

Bottom: $10 - 1.97 = 8.03$

$40.15 \div 8.03 = 5$

Area of a triangle is $\frac{1}{2} \times b \times h$, so $b \times h = 24$.

Choose base 6 cm and height 4 cm.

Put the top vertex directly above the middle of the base (3 cm along), so the two slanted sides are equal.

(a) $3 \times 4 = 12$ and $3 \times -2x = -6x$, giving $12 - 6x$

(b) $\frac{3y}{4} = 12$, so $3y = 48$ and $y = 16$

(c) The highest common factor of 4 and 6 is 2: $4p + 6 = 2(2p + 3)$

(a) The first two significant figures are 2 and 5; the next digit 3 rounds down: 2500

(b) The first significant figure is 8 (hundredths); the next digit 7 rounds up: 0.09

Red: $\frac{3}{8} \times 400 = 150$

Green: $400 - 150 - 82 = 168$

Percentage: $\frac{168}{400} \times 100 = 42\%$

Angle $QPR = 56°$ (vertically opposite angles are equal).

Angle $PQR = (180 - 56) \div 2 = 62°$ (base angles of an isosceles triangle are equal; angles in a triangle add to 180°).

Angle $PRQ = 62°$, and $CQD \parallel APR$, so angle $CQR = 180 - 62 = 118°$ (co-interior angles add to 180°).

$24 = 2^3 \times 3$ and $56 = 2^3 \times 7$

LCM $= 2^3 \times 3 \times 7 = 168$

(Check: multiples of 56 are 56, 112, 168, and $168 = 7 \times 24$.)

The hypotenuse is 8.5 cm.

$x^2 = 8.5^2 - 4^2 = 72.25 - 16 = 56.25$

$x = \sqrt{56.25} = 7.5$

(a) $(-3)^2 = 9$, so $T = 4 \times 9 - 11 = 36 - 11 = 25$

(b) $d = 3p + 4$

Subtract 4: $d - 4 = 3p$

Divide by 3: $p = \frac{d - 4}{3}$

Let Rick have $n$. Selma has $2n$, Tony has $2n - 6$.

$n + 2n + 2n - 6 = 54 \Rightarrow 5n = 60 \Rightarrow n = 12$

Tony has $24 - 6 = 18$

$12 : 18 = 1 : 1.5$, so $p = 1.5$

Chic Decor: $15 \div 3 = 5$ lots of £36, so $5 \times 36 = £180$

Style Papers: $3 \times 70 = £210$, then $210 \times 0.88 = £184.80$

£180 < £184.80, so buy from Chic Decor.

Look at the frequency axis: it goes 30 then 50, so 40 is missing and the scale is not even.

Points should be at class midpoints (5, 15, 25, 35, 45). The last point is at (50, 5) but should be at (45, 5).

Stage 1: $6 \times \frac{15}{60} = 1.5$ miles

Stage 2: $9 \times \frac{40}{60} = 6$ miles

Total = 7.5 miles

Amy: $7.5 \div 0.75 = 10$ mph

$QU = (3x + 5) - 2x = x + 5$ and $RV = 5$; the height is $4x$.

$A = \frac{1}{2}(QU + RV) \times 4x = \frac{1}{2}(x + 10) \times 4x$

$A = 2x(x + 10) = 2x^2 + 20x$

In one hour: $30 \times 3600 = 108\,000$ metres

$108\,000 \div 1000 = 108$ km/h

After a 15% decrease the car is worth 85% of the original.

$0.85 \times \text{original} = 13\,600$

Original $= 13\,600 \div 0.85 = £16\,000$