June 2024 Paper 3 Foundation

Per cent means 'out of 100', so $23\% = \frac{23}{100}$.

There are 100 cm in 1 m.

$800 \div 100 = 8$ m

In 62 837 the digits are: 6 ten-thousands, 2 thousands, 8 hundreds, 3 tens, 7 units.

So the 3 is worth 3 tens, i.e. $30$.

$7a + a = 8a$

$8a - 5a = 3a$

Convert to decimals: $\frac12 = 0.5$, $\frac23 \approx 0.67$, $\frac14 = 0.25$.

Smallest first: $\frac14,\ \frac12,\ \frac23$

(a) Each 1 cm is 4 km, so $8 \times 4 = 32$ km.

(b) $10 \div 4 = 2.5$, so the road is $2.5$ cm long on the map.

(a) The tallest bar is Bus (9 students), so the mode is Bus.

(b) Walk = 8, Cycle = 3, so $8 - 3 = 5$ more.

Aisha's 18th birthday: $1993 + 18 = 2011$.

Elections every 5 years: 2011, 2016, 2021, 2026, 2031.

2030 is not in the list, so No, there will not be an election in 2030.

Ski hire: $5 \times 26 = £130$

Lift pass: $4 \times 45 = £180$

Lessons: $8 \times 23.50 = £188$

Total: $130 + 180 + 188 = £498$, and $498 < 500$, so Lucia has enough money.

(a) A radius joins the centre of the circle to any point on the edge.

(b) A straight line joining two points on the circumference (not through the centre) is a chord.

$8 \times 45 = 360$ minutes

$360 \div 60 = 6$ hours

Check each odd number from 21 to 39 for factors:

21, 25, 27, 33, 35, 39 all have factors other than 1 and themselves.

The primes are $23, 29, 31, 37$, so any three of these.

Spanish Year 11: $67 - 34 = 33$

French Year 11: $113 - 45 - 33 = 35$

Year 10 total: $207 - 113 = 94$

German Year 10: $94 - 33 - 34 = 27$

French total: $33 + 35 = 68$; German total: $27 + 45 = 72$ (check: $68 + 72 + 67 = 207$)

Total people: $30 + 10 + 50 = 90$

Each person: $360 \div 90 = 4°$

Coffee $30 \times 4 = 120°$, Hot chocolate $10 \times 4 = 40°$, Tea $50 \times 4 = 200°$

Draw each angle with a protractor and label each sector with the drink.

$15\%$ of $88 = 0.15 \times 88 = 13.2$

$20\%$ of $62 = 0.2 \times 62 = 12.4$

$13.2 > 12.4$, so 15% of 88 is greater.

(a) $m$ multiplied by itself 4 times is $m^4$.

(b) Wins score $5 \times x = 5x$, draws score $2 \times y = 2y$, losses score 0.

Total: $5x + 2y$

Scale factor: $30 \div 20 = 1.5$

Butter: $120 \times 1.5 = 180$ g

Flour: $200 \times 1.5 = 300$ g

Sugar: $50 \times 1.5 = 75$ g

(a) A 180° rotation about $(0,0)$ sends $(x, y)$ to $(-x, -y)$.

$(1,4)\to(-1,-4)$, $(5,4)\to(-5,-4)$, $(4,2)\to(-4,-2)$, $(2,2)\to(-2,-2)$. Join these points.

(b) $x = 3$ is a vertical line through 3 on the $x$-axis. Mike's image is below A, so he reflected in the horizontal line $y = 3$.

Make a table using $y = 3x - 2$:

| $x$ | −2 | −1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|
| $y$ | −8 | −5 | −2 | 1 | 4 | 7 |

Plot the points and join them with a ruler from $x = -2$ to $x = 3$.

In triangle ABC, $AB = BC$ so angle BCA = angle CAB = $81°$.

Angle ABC $= 180 - 2 \times 81 = 18°$

Angle BCD $= 4 \times 18 = 72°$

In triangle BCD, $BC = CD$ so angle CBD = angle CDB $= (180 - 72) \div 2 = 54°$

$18 : 54 = 1 : 3$

(a) The HCF of 6 and 15 is 3: $6x - 15 = 3(2x - 5)$

(b) Both terms contain $m$: $m^2 + 5m = m(m + 5)$

$63 = 3 \times 3 \times 7$

$105 = 3 \times 5 \times 7$

Common prime factors: $3 \times 7 = 21$

(a)(i) Move the decimal point 4 places right: $53\,000$

(a)(ii) Move the decimal point 5 places left: $0.000\,074$

(b) $9\,700\,000 + 24\,500\,000 = 34\,200\,000 = 3.42 \times 10^7$

(a) Rana drew a 7 cm by 4 cm rectangle. The 4 cm sides of the triangle slope, so the prism is not 4 cm tall. Its height is $\sqrt{4^2 - 3^2} = \sqrt7 \approx 2.6$ cm.

(b) From above you see the 6 cm by 7 cm base. The top edge (ridge) is above the middle of the 6 cm side, so draw a 7 cm by 6 cm rectangle with a line along its length 3 cm from each long side.

Multiplier for a 6% increase: $1.06$

$25\,000 \times 1.06^3 = 29\,775.4$

So about $29\,775$ workers.

Volume of tin $= \frac{\text{mass}}{\text{density}} = 600 \div 0.6 = 1000$ cm³

Volume of salt $= 1000 - 700 = 300$ cm³

Density of salt $= 600 \div 300 = 2$ g/cm³

(a) $1 - 0.6 = 0.4$ and $1 - 0.55 = 0.45$. Each coin B pair of branches is $0.55$ (heads) and $0.45$ (not heads).

(b) $P(\text{both heads}) = 0.6 \times 0.55 = 0.33$

Volume $= \pi r^2 h = \pi \times 100^2 \times 30 = 942\,478$ cm³

Time $= 942\,478 \div 250 = 3769.9$ seconds

$3769.9 \div 60 = 62.8$, so about $63$ minutes.

(a) $w = 40 - (-5)^2 = 40 - 25 = 15$

(b) $p = \frac{h - 5}{3}$

Multiply by 3: $3p = h - 5$

Add 5: $h = 3p + 5$