1. 1
   A school orders exercise books in packs of $24$ and pencils in packs of $18$. The office manager wants to order the smallest matching numbers of each, and also wants to split both packs into identical bundles as large as possible with none left over.

   (a)

   (a) What is $24$ expressed as a product of its prime factors?

   [1 mark]

   • A$2^3 \times 3$
   • B$2^2 \times 3^2$
   • C$2 \times 3^3$
   • D$2^4 \times 3$

   (b)

   (b) What is the lowest common multiple (LCM) of $24$ and $18$?

   [1 mark]

   • A36
   • B72
   • C144
   • D432

   (c)

   (c) What is the highest common factor (HCF) of $24$ and $18$?

   [1 mark]

   • A2
   • B3
   • C6
   • D12

   Total for question 1: 3 marks

2. 2
   An image sensor contains $1.2 \times 10^{7}$ pixels arranged evenly over an area of $4.8 \times 10^{-5}$ m$^2$.

   (a)

   (a) Calculate the number of pixels per m$^2$ on this sensor, giving your answer in standard form.

   [2 marks]

   (b)

   (b) A separate calculation for the sensor's lens curvature requires simplifying $\sqrt{75} - \sqrt{27}$. Simplify this fully, giving your answer in the form $k\sqrt{3}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A recurring decimal is defined as $x = 0.4\dot{5}$, meaning $x = 0.455555...$, where the digit $5$ recurs forever.

   (a)

   (a) Show that $x = \frac{41}{90}$.

   [3 marks]

   (b)

   (b) A student claims that, written as a percentage and rounded to 1 decimal place, $\frac{41}{90}$ is $45.6\%$. Show that this claim is correct.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A market stall holder buys a crate of $150$ mangoes for a total cost price of $£54.00$. She sells each mango for $85$p. By the end of the day she has sold $132$ mangoes and discards the rest.

   (a)

   (a) Calculate the stall holder's profit for the day, in pounds.

   [4 marks]

   (b)

   (b) Calculate her profit as a percentage of the cost price, giving your answer correct to 1 decimal place.

   [4 marks]

   (c)

   (c) The stall holder wants to estimate, without a calculator, her likely profit for a full week (7 similar trading days). By rounding the day's profit to the nearest £10, estimate her weekly profit. Then state whether this estimate is an overestimate or an underestimate, giving a reason.

   [5 marks]

   Total for question 4: 13 marks

5. 5
   A submarine's onboard computer logs four depth changes in metres, where a positive number means rising and a negative number means descending: $-18$, $+7$, $-24$, $+12$.

   (a)

   (a) What is the submarine's net change in depth after these four readings?

   [1 mark]

   • A$-61$ m
   • B$-24$ m
   • C$5$ m
   • D$-23$ m

   (b)

   (b) Using the correct order of operations, evaluate $(-18 + 7) \times 2 - (-24 \div 12)$.

   [1 mark]

   • A$-20$
   • B$-24$
   • C$-22$
   • D$20$

   (c)

   (c) Which of the four readings has the greatest magnitude (furthest from zero)?

   [1 mark]

   • A$-18$ m
   • B$-24$ m
   • C$+7$ m
   • D$+12$ m

   Total for question 5: 3 marks

6. 6
   Light travels at a speed of $3 \times 10^{8}$ m/s. A particular star is a distance of $4.08 \times 10^{16}$ m from Earth.

   (a)

   (a) Calculate the time, in seconds, for light from this star to reach Earth, giving your answer in standard form.

   [2 marks]

   (b)

   (b) A separate calculation for the telescope's mirror area involves simplifying $\sqrt{20} \times \sqrt{45}$. Simplify this fully.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A charity's annual report states that its donations increased from $£24{,}000$ to $£31{,}200$ over one year.

   (a)

   (a) Show that this represents a percentage increase of $30\%$.

   [3 marks]

   (b)

   (b) The charity's treasurer claims that, if donations increase by a further $30\%$ the following year, the overall two-year increase (from the original $£24{,}000$) is not $60\%$ but is instead $69\%$. Show that the treasurer is correct.

   [3 marks]

   Total for question 7: 6 marks

8. 8
   A small business buys a spool of filament weighing $1000$ g for a cost price of $£18.50$. Each printed keyring uses $12.5$ g of filament. In one batch the business prints $64$ keyrings, but $6\frac{1}{4}\%$ of them are rejected due to printing faults and cannot be sold. Each successful keyring sells for $£1.20$.

   (a)

   (a) Calculate the number of keyrings that are rejected, and hence the number that are sold.

   [4 marks]

   (b)

   (b) Calculate the profit made on this batch as a percentage of the cost price of the filament used, correct to 1 decimal place.

   [4 marks]

   (c)

   (c) To check this, the business rounds the cost of filament used to the nearest pound and the revenue to the nearest £10, and uses these to estimate the profit. Find this estimate, compare it with the exact profit from part (b), and state whether the estimate is reasonable.

   [5 marks]

   Total for question 8: 13 marks

End of questions