Westminster International School | Term 1, 2026 | Total: 38 marks
1. The Earth has a surface area of approximately 510 100 000 km².
Write this surface area in standard form.
(1 mark)
$5.101 \times 10^8$ km²
Standard form means writing a number as $a \times 10^n$, where $a$ must be between 1 and 10 (but not equal to 10).
STEP 1 – Find where to put the decimal point
We need just one non-zero digit before the decimal point.
$510\,100\,000 \rightarrow 5.101\,00\,000$
STEP 2 – Count how many places the decimal moved
The decimal moved 8 places to the left, so the power of 10 is $+8$.
ANSWER
$5.101 \times 10^8$ km²
We need just one non-zero digit before the decimal point.
$510\,100\,000 \rightarrow 5.101\,00\,000$
STEP 2 – Count how many places the decimal moved
The decimal moved 8 places to the left, so the power of 10 is $+8$.
ANSWER
$5.101 \times 10^8$ km²
2. Work out $\dfrac{5}{12}$ of $132.
(2 marks)
$\dfrac{5}{12} \times \dfrac{132}{1}$
$= \dfrac{5 \times 132}{12 \times 1}$
$= \dfrac{660}{12}$
$= 55$
$= \dfrac{5 \times 132}{12 \times 1}$
$= \dfrac{660}{12}$
$= 55$
$\$55$
3. Lenny makes 400 pizzas for a large event.
$\dfrac{3}{8}$ of the pizzas are Sicilian style.
23% of the pizzas are four cheese style.
The rest of the pizzas are all ‘Lenny’s special deluxe’ style.
Calculate how many pizzas are Lenny’s special deluxe style. (3 marks)
$\dfrac{3}{8}$ of the pizzas are Sicilian style.
23% of the pizzas are four cheese style.
The rest of the pizzas are all ‘Lenny’s special deluxe’ style.
Calculate how many pizzas are Lenny’s special deluxe style. (3 marks)
Find how many pizzas belong to each style first, then subtract both from the total to find the deluxe amount. “The rest” always means total minus everything else.
STEP 1 – Find the number of Sicilian pizzas
$\dfrac{3}{8} \times \dfrac{400}{1} = \dfrac{3 \times 400}{8 \times 1} = \dfrac{1200}{8} = 150$ pizzas
STEP 2 – Find the number of four cheese pizzas
$\dfrac{23}{100} \times \dfrac{400}{1} = \dfrac{23 \times 400}{100 \times 1} = \dfrac{9200}{100} = 92$ pizzas
STEP 3 – Subtract both from the total to find the deluxe amount
$400 – 150 – 92 = 158$ pizzas
$\dfrac{3}{8} \times \dfrac{400}{1} = \dfrac{3 \times 400}{8 \times 1} = \dfrac{1200}{8} = 150$ pizzas
STEP 2 – Find the number of four cheese pizzas
$\dfrac{23}{100} \times \dfrac{400}{1} = \dfrac{23 \times 400}{100 \times 1} = \dfrac{9200}{100} = 92$ pizzas
STEP 3 – Subtract both from the total to find the deluxe amount
$400 – 150 – 92 = 158$ pizzas
158 pizzas
4. Write $4.59 as a percentage of $3.40.
(1 mark)
“Write A as a percentage of B” always means: divide A by B, then multiply by 100. The first number goes on top of the fraction.
STEP 1 – Write as a fraction
$\dfrac{4.59}{3.40}$
STEP 2 – Multiply by 100 to convert to a percentage
$\dfrac{4.59}{3.40} \times 100 = 1.35 \times 100 = 135\%$
$\dfrac{4.59}{3.40}$
STEP 2 – Multiply by 100 to convert to a percentage
$\dfrac{4.59}{3.40} \times 100 = 1.35 \times 100 = 135\%$
135%
5. Write $24.60 as a fraction of $2870. Give your answer in its lowest terms.
(2 marks)
“Fraction of” means divide the first number by the second. Remove the decimal by multiplying both top and bottom by 100, then simplify by dividing both by the same number until you cannot go further.
STEP 1 – Write as a fraction and remove the decimal
Multiply top and bottom by 10 to remove the decimal point:
$\dfrac{24.60}{2870} = \dfrac{24.60 \times 10}{2870 \times 10} = \dfrac{246}{28700}$
STEP 2 – Divide both by 2 (both are even)
$\dfrac{246 \div 2}{28700 \div 2} = \dfrac{123}{14350}$
STEP 3 – Find a common factor of 123 and 14350
Factorise the numerator: $123 = 3 \times 41$
Check if 3 divides 14350: $1+4+3+5+0 = 13$, not divisible by 3. ✗
Check if 41 divides 14350: $41 \times 350 = 14350$. ✓
$\dfrac{123 \div 41}{14350 \div 41} = \dfrac{3}{350}$
STEP 4 – Check it is fully simplified
The numerator is 3. Check if 3 divides 350: $3+5+0 = 8$, not divisible by 3. ✗
So $\dfrac{3}{350}$ is already in its lowest terms.
Multiply top and bottom by 10 to remove the decimal point:
$\dfrac{24.60}{2870} = \dfrac{24.60 \times 10}{2870 \times 10} = \dfrac{246}{28700}$
STEP 2 – Divide both by 2 (both are even)
$\dfrac{246 \div 2}{28700 \div 2} = \dfrac{123}{14350}$
STEP 3 – Find a common factor of 123 and 14350
Factorise the numerator: $123 = 3 \times 41$
Check if 3 divides 14350: $1+4+3+5+0 = 13$, not divisible by 3. ✗
Check if 41 divides 14350: $41 \times 350 = 14350$. ✓
$\dfrac{123 \div 41}{14350 \div 41} = \dfrac{3}{350}$
STEP 4 – Check it is fully simplified
The numerator is 3. Check if 3 divides 350: $3+5+0 = 8$, not divisible by 3. ✗
So $\dfrac{3}{350}$ is already in its lowest terms.
$\dfrac{3}{350}$
6. In 2018, Gretal earned $32 000. She paid tax of 24% on these earnings.
Work out the amount she paid in tax in 2018.
(2 marks)
To find a percentage of an amount, convert the percentage to a decimal (divide by 100) and then multiply. For example, 24% becomes 0.24.
STEP 1 – Convert 24% to a decimal
$24\% = \dfrac{24}{100} = 0.24$
STEP 2 – Multiply by the earnings
Tax $= 0.24 \times 32000 = 7680$
$24\% = \dfrac{24}{100} = 0.24$
STEP 2 – Multiply by the earnings
Tax $= 0.24 \times 32000 = 7680$
$\$7\,680$
7. The Earth has a surface area of approximately 510 100 000 km². Water covers 70.8% of the Earth’s surface.
Work out the area of the Earth’s surface covered by water.
(2 marks)
This is the same method as Question 6 — convert the percentage to a decimal and multiply. With large numbers, a calculator is the best tool. Make sure you include the correct units (km²) in your answer.
STEP 1 – Convert 70.8% to a decimal
$70.8\% = \dfrac{70.8}{100} = 0.708$
STEP 2 – Multiply by the total surface area
Area covered by water $= 0.708 \times 510\,100\,000 = 361\,150\,800$ km²
$70.8\% = \dfrac{70.8}{100} = 0.708$
STEP 2 – Multiply by the total surface area
Area covered by water $= 0.708 \times 510\,100\,000 = 361\,150\,800$ km²
$361\,150\,800$ km²
8. Write $4^{-3}$ as a decimal.
(1 mark)
A negative power means “1 divided by the positive version.” The rule is: $a^{-n} = \dfrac{1}{a^n}$
STEP 1 – Apply the negative index rule
$4^{-3} = \dfrac{1}{4^3}$
STEP 2 – Work out $4^3$
$4^3 = 4 \times 4 \times 4 = 64$
STEP 3 – Convert to decimal
$\dfrac{1}{64} = 0.015625$
$4^{-3} = \dfrac{1}{4^3}$
STEP 2 – Work out $4^3$
$4^3 = 4 \times 4 \times 4 = 64$
STEP 3 – Convert to decimal
$\dfrac{1}{64} = 0.015625$
$0.015625$
9. Calculate $0.125^{-\frac{2}{3}}$.
(1 mark)
For a fractional power $a^{\frac{m}{n}}$: the denominator (bottom number) tells you which root to take, and the numerator (top number) tells you the power. A negative in front means flip the base first.
STEP 1 – Convert 0.125 to a fraction
$0.125 = \dfrac{1}{8}$
STEP 2 – Handle the negative power by flipping
$\left(\dfrac{1}{8}\right)^{-\frac{2}{3}} = \left(\dfrac{8}{1}\right)^{\frac{2}{3}} = 8^{\frac{2}{3}}$
STEP 3 – Apply the fractional power
The denominator is 3, so take the cube root first: $\sqrt[3]{8} = 2$
The numerator is 2, so square the result: $2^2 = 4$
$0.125 = \dfrac{1}{8}$
STEP 2 – Handle the negative power by flipping
$\left(\dfrac{1}{8}\right)^{-\frac{2}{3}} = \left(\dfrac{8}{1}\right)^{\frac{2}{3}} = 8^{\frac{2}{3}}$
STEP 3 – Apply the fractional power
The denominator is 3, so take the cube root first: $\sqrt[3]{8} = 2$
The numerator is 2, so square the result: $2^2 = 4$
$4$
10. Simplify $\sqrt{32} + \sqrt{98}$.
(2 marks)
To simplify a surd like $\sqrt{32}$, find the largest square number that divides into it. Square numbers are 4, 9, 16, 25, 36, 49 … Once both surds have the same $\sqrt{\phantom{x}}$ part, you can add them like letters in algebra.
STEP 1 – Simplify $\sqrt{32}$
Largest square factor of 32 is 16: $32 = 16 \times 2$
$\sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2}$
STEP 2 – Simplify $\sqrt{98}$
Largest square factor of 98 is 49: $98 = 49 \times 2$
$\sqrt{98} = \sqrt{49 \times 2} = \sqrt{49} \times \sqrt{2} = 7\sqrt{2}$
STEP 3 – Add the like surds
$4\sqrt{2} + 7\sqrt{2} = (4 + 7)\sqrt{2} = 11\sqrt{2}$
(This works just like $4x + 7x = 11x$.)
Largest square factor of 32 is 16: $32 = 16 \times 2$
$\sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2}$
STEP 2 – Simplify $\sqrt{98}$
Largest square factor of 98 is 49: $98 = 49 \times 2$
$\sqrt{98} = \sqrt{49 \times 2} = \sqrt{49} \times \sqrt{2} = 7\sqrt{2}$
STEP 3 – Add the like surds
$4\sqrt{2} + 7\sqrt{2} = (4 + 7)\sqrt{2} = 11\sqrt{2}$
(This works just like $4x + 7x = 11x$.)
$11\sqrt{2}$
11. Write $1.92 as a percentage of $1.60.
(1 mark)
$\dfrac{1.92}{1.60} \times 100 = 120\%$
120%
12. At the start of a particular year, an animal shelter had a total of 528 kg of rabbit food.
$\dfrac{5}{16}$ of the food was used in January.
$\dfrac{4}{11}$ of the remaining food was used in February.
The rest of the food was all used in March.
Calculate the percentage of the 528 kg of rabbit food that was used in March. (4 marks)
$\dfrac{5}{16}$ of the food was used in January.
$\dfrac{4}{11}$ of the remaining food was used in February.
The rest of the food was all used in March.
Calculate the percentage of the 528 kg of rabbit food that was used in March. (4 marks)
Work through one month at a time. Be careful — February’s fraction is of the remaining food after January, not the original total.
STEP 1 – January
$\dfrac{5}{16} \times \dfrac{528}{1} = \dfrac{5 \times 528}{16 \times 1} = \dfrac{2640}{16} = 165$ kg
STEP 2 – Food remaining after January
$528 – 165 = 363$ kg
STEP 3 – February (fraction of the 363 kg remaining)
$\dfrac{4}{11} \times \dfrac{363}{1} = \dfrac{4 \times 363}{11 \times 1} = \dfrac{1452}{11} = 132$ kg
STEP 4 – Food used in March
$363 – 132 = 231$ kg
STEP 5 – Express March as a percentage of the original 528 kg
$\dfrac{231}{528} \times 100 = 43.75\%$
$\dfrac{5}{16} \times \dfrac{528}{1} = \dfrac{5 \times 528}{16 \times 1} = \dfrac{2640}{16} = 165$ kg
STEP 2 – Food remaining after January
$528 – 165 = 363$ kg
STEP 3 – February (fraction of the 363 kg remaining)
$\dfrac{4}{11} \times \dfrac{363}{1} = \dfrac{4 \times 363}{11 \times 1} = \dfrac{1452}{11} = 132$ kg
STEP 4 – Food used in March
$363 – 132 = 231$ kg
STEP 5 – Express March as a percentage of the original 528 kg
$\dfrac{231}{528} \times 100 = 43.75\%$
43.75%
13. Work out $\dfrac{7}{11}$ of 198 kg.
(2 marks)
$\dfrac{7}{11} \times \dfrac{198}{1}$
$= \dfrac{7 \times 198}{11 \times 1}$
$= \dfrac{1386}{11}$
$= 126$ kg
$= \dfrac{7 \times 198}{11 \times 1}$
$= \dfrac{1386}{11}$
$= 126$ kg
126 kg
14. Write 15 060 in standard form.
(1 mark)
Standard form: write the number as $a \times 10^n$ where $1 \leq a < 10$. Count how many places the decimal point moves to the left — that number is your power of 10.
STEP 1 – Place the decimal point after the first digit
$15060 \rightarrow 1.5060$
We can drop the trailing zero: $1.506$
STEP 2 – Count how many places the decimal moved
$1\underbrace{5060}_{4 \text{ places}}$ — the decimal moved 4 places to the left.
STEP 3 – Write in standard form
$15060 = 1.506 \times 10^4$
$15060 \rightarrow 1.5060$
We can drop the trailing zero: $1.506$
STEP 2 – Count how many places the decimal moved
$1\underbrace{5060}_{4 \text{ places}}$ — the decimal moved 4 places to the left.
STEP 3 – Write in standard form
$15060 = 1.506 \times 10^4$
$1.506 \times 10^4$
15. In a cycling club, the number of members are in the ratio males : females = 8 : 3. The club has 342 females.
i) Find the total number of members. [2]
ii) Find the percentage of the total number of members that are female. [1] (3 marks)
i) Find the total number of members. [2]
ii) Find the percentage of the total number of members that are female. [1] (3 marks)
In a ratio problem, first find the value of “one part” by dividing the known quantity by its ratio number. Then use that to find anything else.
i)
STEP 1 – Find the total number of parts
Males : Females = 8 : 3, so total parts $= 8 + 3 = 11$
STEP 2 – Find the value of one part
Females = 3 parts = 342 members
1 part $= \dfrac{342}{3} = 114$ members
STEP 3 – Find the total
Total $= 11 \times 114 = 1254$ members
Males : Females = 8 : 3, so total parts $= 8 + 3 = 11$
STEP 2 – Find the value of one part
Females = 3 parts = 342 members
1 part $= \dfrac{342}{3} = 114$ members
STEP 3 – Find the total
Total $= 11 \times 114 = 1254$ members
Total members = 1254
ii)
STEP 1 – Percentage of females
$\dfrac{\text{females}}{\text{total}} \times 100 = \dfrac{342}{1254} \times 100$
STEP 2 – Simplify (females are 3 out of 11 parts)
$= \dfrac{3}{11} \times 100 = 27.27\ldots\% \approx 27.3\%$
$\dfrac{\text{females}}{\text{total}} \times 100 = \dfrac{342}{1254} \times 100$
STEP 2 – Simplify (females are 3 out of 11 parts)
$= \dfrac{3}{11} \times 100 = 27.27\ldots\% \approx 27.3\%$
27.3%
16. The earth’s oceans cover an area of approximately 361 900 000 km².
Write this area in standard form.
(1 mark)
$361\,900\,000 = 3.619 \times 10^8$ km²
(The decimal moves 8 places to the left, so the power is 8.)
(The decimal moves 8 places to the left, so the power is 8.)
$3.619 \times 10^8$ km²
17. The price of a gaming computer is $2440. In a sale, the price is reduced by 15%.
Find the sale price of the gaming computer.
(3 marks)
A percentage reduction means the customer pays less than 100% of the original price. Subtract the percentage from 100% first, then multiply. This is quicker than finding 15% separately and subtracting.
STEP 1 – Find the percentage the customer actually pays
$100\% – 15\% = 85\%$
STEP 2 – Calculate 85% of the original price
Sale price $= 85\% \times \$2440$
$= 0.85 \times 2440$
$= \$2074$
$100\% – 15\% = 85\%$
STEP 2 – Calculate 85% of the original price
Sale price $= 85\% \times \$2440$
$= 0.85 \times 2440$
$= \$2074$
$\$2074$
18. Work out $\left(\dfrac{125}{27}\right)^{-\frac{2}{3}}$.
(1 mark)
When the base is a fraction and the power is negative and fractional, follow three steps in order: (1) flip the fraction to remove the negative, (2) take the cube root (denominator = 3), (3) square the result (numerator = 2).
STEP 1 – Remove the negative power by flipping the fraction
$\left(\dfrac{125}{27}\right)^{-\frac{2}{3}} = \left(\dfrac{27}{125}\right)^{\frac{2}{3}}$
STEP 2 – Take the cube root (the denominator is 3)
$\sqrt[3]{27} = 3$ because $3 \times 3 \times 3 = 27$
$\sqrt[3]{125} = 5$ because $5 \times 5 \times 5 = 125$
So: $\left(\dfrac{27}{125}\right)^{\frac{1}{3}} = \dfrac{3}{5}$
STEP 3 – Square the result (the numerator is 2)
$\left(\dfrac{3}{5}\right)^{2} = \dfrac{3^2}{5^2} = \dfrac{9}{25}$
$\left(\dfrac{125}{27}\right)^{-\frac{2}{3}} = \left(\dfrac{27}{125}\right)^{\frac{2}{3}}$
STEP 2 – Take the cube root (the denominator is 3)
$\sqrt[3]{27} = 3$ because $3 \times 3 \times 3 = 27$
$\sqrt[3]{125} = 5$ because $5 \times 5 \times 5 = 125$
So: $\left(\dfrac{27}{125}\right)^{\frac{1}{3}} = \dfrac{3}{5}$
STEP 3 – Square the result (the numerator is 2)
$\left(\dfrac{3}{5}\right)^{2} = \dfrac{3^2}{5^2} = \dfrac{9}{25}$
$\dfrac{9}{25}$
19. Kristian receives $72. Kristian spends 45% of his $72 on a computer game.
Calculate the price of the computer game.
(2 marks)
Price $= 45\% \times 72$
$= 0.45 \times 72$
$= 32.40$
$= 0.45 \times 72$
$= 32.40$
$\$32.40$
20. Work out $\dfrac{7}{8} \div 4\dfrac{2}{3}$.
You must show all your working and give your answer as a fraction in its simplest form.
(3 marks)
To divide by a fraction, flip the second fraction and multiply instead. A mixed number must first be changed into an improper fraction before you can do this.
STEP 1 – Convert the mixed number to an improper fraction
$4\dfrac{2}{3}$: multiply the whole number by the denominator, then add the numerator.
$= \dfrac{(4 \times 3) + 2}{3} = \dfrac{12 + 2}{3} = \dfrac{14}{3}$
STEP 2 – Rewrite the division
$\dfrac{7}{8} \div \dfrac{14}{3}$
STEP 3 – Flip the second fraction and multiply
$= \dfrac{7}{8} \times \dfrac{3}{14}$
STEP 4 – Multiply the numerators and denominators
$= \dfrac{7 \times 3}{8 \times 14} = \dfrac{21}{112}$
STEP 5 – Simplify by dividing by 7 (the HCF of 21 and 112)
$21 \div 7 = 3$ and $112 \div 7 = 16$
$= \dfrac{3}{16}$
$4\dfrac{2}{3}$: multiply the whole number by the denominator, then add the numerator.
$= \dfrac{(4 \times 3) + 2}{3} = \dfrac{12 + 2}{3} = \dfrac{14}{3}$
STEP 2 – Rewrite the division
$\dfrac{7}{8} \div \dfrac{14}{3}$
STEP 3 – Flip the second fraction and multiply
$= \dfrac{7}{8} \times \dfrac{3}{14}$
STEP 4 – Multiply the numerators and denominators
$= \dfrac{7 \times 3}{8 \times 14} = \dfrac{21}{112}$
STEP 5 – Simplify by dividing by 7 (the HCF of 21 and 112)
$21 \div 7 = 3$ and $112 \div 7 = 16$
$= \dfrac{3}{16}$
$\dfrac{3}{16}$
