June 2022 Paper 2 Foundation

Look at the units digit: it is 6, which is 5 or more, so round up.

$1476 \to 1480$

One whole is $\frac{10}{10}$.

$\frac{10}{10} - \frac{3}{10} = \frac{7}{10}$

The mode is the value that appears most often.

3 appears four times, more than any other number, so the mode is 3.

Multiples of 5 always end in 0 or 5.

So any number from 100 to 995 ending in 0 or 5 works, for example $725$.

To change a decimal to a percentage, multiply by 100.

$0.4 \times 100 = 40\%$

The most negative number is the smallest.

Negatives: $-11 < -7 < -2$, then positives: $3 < 8 < 10$.

Order: $-11, -7, -2, 3, 8, 10$

(a) The shape has 6 sides, so it is a hexagon.

(b) $CD$ is horizontal. The other horizontal side is $AF$.

(c) $AF$ is horizontal, so a perpendicular side is vertical: $AB$ or $EF$.

(a) Go across 3, then up 2: $A = (3, 2)$.

(b) Go 4 left from $O$, then 3 up, and mark a cross labelled B.

(c) Set compasses to 4 cm (4 squares). Put the point on $(1, -1)$ and draw the circle; it passes through $(5, -1)$, $(-3, -1)$, $(1, 3)$ and $(1, -5)$.

(a) The February cross is at 23.

(b) January: 10 houses. June: 56 houses.

Ratio January : June $= 10 : 56 = 5 : 28$

Amount left after the deposit: $1428 - 150 = 1278$

Each payment: $1278 \div 6 = 213$

Each monthly payment is £213.

Angle $ACB = 180 - 116 - 25 = 39^\circ$ (angles in a triangle add up to $180^\circ$).

$x$ and angle $ACB$ are vertically opposite, so $x = 39^\circ$ (vertically opposite angles are equal).

(a) $28 \div 7 = 4$, then $4 + 5 = 9$.

(b) Work backwards from 154: $154 \div 11 = 14$.

The missing step turns 8 into 14, so it is $+6$.

Put in the given values: total 198, without mattress 59, double total 45, single without 17, king total 83, king with 67.

King without: $83 - 67 = 16$. Single total: $198 - 45 - 83 = 70$, so single with $= 70 - 17 = 53$.

Double without: $59 - 17 - 16 = 26$, so double with $= 45 - 26 = 19$.

With mattress total: $198 - 59 = 139$ (check: $53 + 19 + 67 = 139$).

(i) Same denominator, and $5 > 2$, so $\frac{5}{8} > \frac{2}{8}$.

(ii) $-2 \times -3 = 6$ and $-3 + 9 = 6$, so they are equal.

(a) $0\times3 + 1\times57 + 2\times84 + 3\times75 + 4\times81 = 0 + 57 + 168 + 225 + 324 = 774$

(b) There are 300 students, so the median is between the 150th and 151st values.

Running totals: 3, 60, 144, 219. The 150th and 151st are both in the '3' row, so the median is 3.

Scale: $62 \div 12.4 = 5$, so 1 cm on the drawing is 5 m in real life.

Real width: $9.4 \times 5 = 47$ m.

Make a table of values using $y = 4 - x$:

$x$: $-2, -1, 0, 1, 2, 3, 4$

$y$: $6, 5, 4, 3, 2, 1, 0$

Plot the points and join them with a ruler from $(-2, 6)$ to $(4, 0)$.

Change hours to minutes: $25.3 \times 60 = 1518$ minutes.

Mean: $1518 \div 115 = 13.2$ minutes.

Number of bags: $3000 \div 150 = 20$

Total needed with 35% profit: $17.60 \times 1.35 = 23.76$

Price per bag: $23.76 \div 20 = 1.188$

Round up so the profit is at least 35%: £1.19

(a) Each pair of branches adds to 1.

Work not late: $1 - 0.13 = 0.87$. Home not late: $1 - 0.06 = 0.94$ on both pairs.

(b) Late and late: multiply along the branches.

$0.13 \times 0.06 = 0.0078$

(a) Multiply the powers: $(x^3)^5 = x^{15}$

(b) $4x + 12 + 28 - 14x = 40 - 10x$

(c) The HCF of $15x^3$ and $3x^2y$ is $3x^2$.

$15x^3 + 3x^2y = 3x^2(5x + y)$

The shape is the same size and way round, so it is a translation.

Pick a corner: $(1, -5)$ on S moves to $(-4, 1)$ on T.

That is 5 left and 6 up: translation by $\begin{pmatrix}-5\\6\end{pmatrix}$.

Rounded to the nearest metre, so go half a metre either side.

Lower bound: $90 - 0.5 = 89.5$

Upper bound: $90 + 0.5 = 90.5$

$89.5 \le \text{length} < 90.5$

(a) Field B: $700 \times 2000 = 1\,400\,000\text{ m}^2$

B per person: $1\,400\,000 \div 6750 = 207.4...$

A per person: $80\,000 \div 425 = 188.2...$

Difference: $207.4 - 188.2 = 19.17... \approx 19\text{ m}^2$

(b) A square metre is $100\text{ cm} \times 100\text{ cm} = 10\,000\text{ cm}^2$, so $300\text{ cm}^2 = 0.03\text{ m}^2$.

From L to M: $x$ goes up $4 - (-3) = 7$, $y$ goes up $9 - 1 = 8$. This is 2 parts.

One part: $x$ up 3.5, $y$ up 4.

MN is 3 parts: $x$ up 10.5, $y$ up 12.

$N = (4 + 10.5,\ 9 + 12) = (14.5, 21)$

The multiplier for a 4% decrease is $0.96$.

$679 \times 0.96^3 = 600.7357...$

Value after 3 years: £600.74

Spain in £: $27 \times 0.85 = 22.95$, so per litre $22.95 \div 18 = 1.275$.

Wales: $8 \times 4.5 = 36$ litres, so per litre $40.80 \div 36 = 1.133...$

£1.13 < £1.275, so petrol is cheaper in Wales. Sam is not correct.

Double the first equation: $10x + 4y = 54$

Subtract the second: $(10x + 4y) - (6x + 4y) = 54 - 28$, so $4x = 26$ and $x = 6.5$

Substitute: $5(6.5) + 2y = 27$, so $2y = -5.5$ and $y = -2.75$