Standard form – GCSE Maths exam questions with mark scheme

1Grade 4–5Non-calculator2 marks

(a) Write $43\,000$ in standard form. [1]

(b) Write $6.08 \times 10^{-3}$ as an ordinary number. [1]

(a) $4.3 \times 10^4$B1
(b) $0.00608$B1
Answer: (a) $4.3 \times 10^4$ (b) 0.00608
Examiner tip: The first number must be at least 1 and less than 10: $43 \times 10^3$ is not standard form.

2Grade 4–5Non-calculator4 marks

Work out the following. Give each answer in standard form.

(a) $(3 \times 10^5) \times (4 \times 10^7)$ [2]

(b) $\dfrac{8.4 \times 10^6}{2.1 \times 10^{-2}}$ [2]

(a) for $12 \times 10^{12}$ or $3 \times 4 = 12$ and $10^{5+7}$M1
(a) $1.2 \times 10^{13}$A1
(b) for $4 \times 10^{n}$ or $10^{6-(-2)} = 10^8$ seenM1
(b) $4 \times 10^8$A1
Answer: (a) $1.2 \times 10^{13}$ (b) $4 \times 10^8$
Examiner tip: $12 \times 10^{12}$ is not in standard form – adjust to $1.2 \times 10^{13}$ for the final mark.

3Grade 4–5Non-calculator2 marks

Work out $6.2 \times 10^5 + 3.9 \times 10^4$

Give your answer in standard form. [2]

for both written as ordinary numbers ($620\,000 + 39\,000$) or with the same power of 10, e.g. $6.2 \times 10^5 + 0.39 \times 10^5$ or $62 \times 10^4 + 3.9 \times 10^4$; also award for $659\,000$ or $65.9 \times 10^4$ seen (correct value not in standard form)M1
$6.59 \times 10^5$A1
Answer: $6.59 \times 10^5$
Examiner tip: You can only add the front numbers when the powers of 10 are the same – $10.1 \times 10^5$ and $10.1 \times 10^9$ are common wrong answers.

4Grade 4–5Calculator2 marks

The mass of one atom of carbon is $1.99 \times 10^{-23}$ grams.

Work out the number of carbon atoms in 12 grams of carbon. Give your answer in standard form correct to 3 significant figures. [2]

for $12 \div (1.99 \times 10^{-23})$M1
$6.03 \times 10^{23}$A1
Answer: $6.03 \times 10^{23}$
Examiner tip: Use the ×10ˣ (EXP) key so the calculator treats 1.99 × 10⁻²³ as one number.

5Grade 4–5Calculator3 marks

The distance from the Earth to the Sun is $1.5 \times 10^8$ km. Light travels at $3.0 \times 10^5$ km per second.

How long does light take to travel from the Sun to the Earth? Give your answer in minutes and seconds. [3]

for $\frac{1.5 \times 10^8}{3.0 \times 10^5}$M1
for $500$ (seconds) or $5 \times 10^2$A1
8 minutes 20 seconds (ft their time in seconds, correctly converted)B1
Answer: 8 minutes 20 seconds
Examiner tip: 8.33 minutes is NOT 8 min 33 s – convert the remainder: 500 − 480 = 20 seconds.

6Grade 8–9HigherNon-calculator4 marks

$x = 4 \times 10^{n}$ and $y = 8 \times 10^{2n}$, where $n$ is an integer.

(a) Find $xy$. Give your answer in standard form in terms of $n$. [2]

(b) Find $\dfrac{y}{x^2}$. Give your answer in standard form. [2]

(a) for $32 \times 10^{3n}$M1
(a) $3.2 \times 10^{3n+1}$A1
(b) for $x^2 = 16 \times 10^{2n}$ or $\frac{8}{16}$ or $0.5$ seenM1
(b) $5 \times 10^{-1}$A1
Answer: (a) $3.2 \times 10^{3n+1}$ (b) $5 \times 10^{-1}$
Examiner tip: When squaring $4 \times 10^n$, square BOTH parts: $16 \times 10^{2n}$, not $4 \times 10^{2n}$.

7Grade 1–3FoundationNon-calculator3 marks

(a) Here are four numbers.

$$0.8 \times 10^{4} \qquad 15 \times 10^{-2} \qquad 2.7 \times 10^{5} \qquad 2.7 \times 5^{10}$$

Which one of these numbers is written in standard form? [1]

(b) Write $0.000\,039$ in standard form. [1]

(a) $2.7 \times 10^{5}$ caoB1
(b) $3.9 \times 10^{-5}$ caoB1
(c) $20\,500\,000$ cao (accept with or without spaces/commas)B1
Answer: (a) $2.7 \times 10^{5}$ (b) $3.9 \times 10^{-5}$ (c) $20\,500\,000$
Examiner tip: In (b), move the decimal point 5 places to the right so that it sits just after the 3 (giving 3.9), so the power is $-5$, not $-6$ or $-4$.

8Grade 4–5Non-calculator2 marks

Write these numbers in order of size. Start with the smallest number.

$$4.1 \times 10^{-3} \qquad 0.000\,52 \qquad 3.9 \times 10^{-2} \qquad 38 \times 10^{-4}$$ [2]

converts at least three of the numbers to a common form (e.g. all ordinary numbers 0.0041, 0.00052, 0.039, 0.0038, or all in standard form), or a correct order with only one pair interchangedM1
$0.000\,52,\ 38 \times 10^{-4},\ 4.1 \times 10^{-3},\ 3.9 \times 10^{-2}$ (accept in any equivalent form; allow reverse order for M1 only)A1
Answer: $0.000\,52,\ 38 \times 10^{-4},\ 4.1 \times 10^{-3},\ 3.9 \times 10^{-2}$
Examiner tip: $38 \times 10^{-4}$ is not in standard form: rewrite it as $3.8 \times 10^{-3}$ before comparing, or it looks bigger than it is.

9Grade 4–5Non-calculator3 marks

Work out the following. Give each answer in standard form.

(a) $(5.3 \times 10^{-4}) - (8 \times 10^{-5})$ [2]

(b) $(2 \times 10^{-3})^{3}$ [1]

(a) writes both numbers with the same power of 10 or as ordinary numbers, e.g. $0.000\,53 - 0.000\,08$ or $53 \times 10^{-5} - 8 \times 10^{-5}$, or obtains $0.000\,45$M1
(a) $4.5 \times 10^{-4}$ caoA1
(b) $8 \times 10^{-9}$ caoB1
Answer: (a) $4.5 \times 10^{-4}$ (b) $8 \times 10^{-9}$
Examiner tip: You cannot subtract 8 from 5.3 directly: the powers of 10 are different, so line them up first.

10Grade 4–5Calculator3 marks

A country has a population of $6.7 \times 10^{7}$ people.

The area of the country is $2.43 \times 10^{5}$ km².

(a) Work out the average number of people per km². Give your answer correct to the nearest whole number. [2]

(b) Write your answer to part (a) in standard form. [1]

(a) $(6.7 \times 10^{7}) \div (2.43 \times 10^{5})$ or 275.7… seenM1
(a) 276 caoA1
(b) $2.76 \times 10^{2}$ ft from their whole-number answer to (a)B1
Answer: (a) 276 (b) $2.76 \times 10^{2}$
Examiner tip: Divide people by area, not area by people: 0.0036… people per km² should tell you the division is the wrong way round.

11Grade 4–5Non-calculator3 marks

(a) Ravi works out $(4 \times 10^{3}) \times (5 \times 10^{6})$.

He writes:

$$4 \times 5 = 20 \quad \text{so the answer is } 20 \times 10^{9} = 2 \times 10^{8}$$

Explain what Ravi has done wrong and write down the correct answer in standard form. [2]

(b) Sasha says

“$3 \times 10^{-4}$ is bigger than $2 \times 10^{-3}$ because 3 is bigger than 2.”

Is Sasha correct? Give a reason for your answer. [1]

(a) explanation, e.g. $20 = 2 \times 10$ so the power should go up by 1 (not down), or he divided by 10 instead of multiplyingC1
(a) $2 \times 10^{10}$ caoB1
(b) No, with a correct reason, e.g. $3 \times 10^{-4} = 0.0003$ and $2 \times 10^{-3} = 0.002$, or compare the powers first: $10^{-3}$ is bigger than $10^{-4}$ (do not accept “No” on its own)C1
Answer: (a) He made the power smaller instead of larger; $2 \times 10^{10}$ (b) No: $0.0003 < 0.002$
Examiner tip: When you change 20 to 2 you have divided by 10, so you must multiply by 10 again, which means the power goes up.

12Grade 4–5Calculator4 marks

The table gives the masses of four planets.

Planet	Mass (kg)
Mercury	$3.30 \times 10^{23}$
Earth	$5.97 \times 10^{24}$
Mars	$6.42 \times 10^{23}$
Jupiter	$1.90 \times 10^{27}$

(a) Which planet in the table has the smallest mass? [1]

(b) Work out the total mass of Mercury and Mars. Give your answer in standard form. [1]

(a) MercuryB1
(b) $9.72 \times 10^{23}$ (kg) caoB1
(c) $(1.90 \times 10^{27}) \div (5.97 \times 10^{24})$ or 318.2… seenM1
(c) 318 caoA1
Answer: (a) Mercury (b) $9.72 \times 10^{23}$ kg (c) 318
Examiner tip: In (a) compare powers of 10 first: Mercury and Mars share $10^{23}$, so then compare 3.30 with 6.42.

13Grade 4–5Non-calculator4 marks

The diagram shows a rectangular forest.

The forest is $1.6 \times 10^{4}$ m long and $4.5 \times 10^{3}$ m wide.

(a) Work out the area of the forest. Give your answer in m² in standard form. [2]

(b) Work out the perimeter of the forest. Give your answer in m in standard form. [2]

(a) $1.6 \times 4.5 = 7.2$ and $10^{4} \times 10^{3} = 10^{7}$, or $16\,000 \times 4500$, or $72\,000\,000$ seenM1
(a) $7.2 \times 10^{7}$ (m²) caoA1
(b) $2 \times (16\,000 + 4500)$ or $16\,000 + 4500 + 16\,000 + 4500$ or 41 000 seenM1
(b) $4.1 \times 10^{4}$ (m) caoA1
Answer: (a) $7.2 \times 10^{7}$ m² (b) $4.1 \times 10^{4}$ m
Examiner tip: For the perimeter you are adding, so you cannot just add the powers: change both lengths to ordinary numbers (or the same power of 10) first.

14Grade 8–9HigherNon-calculator5 marks

(a) Work out $\sqrt{\dfrac{3.6 \times 10^{7}}{4 \times 10^{-4}}}$

Give your answer in standard form. [3]

(b) $k$ is a positive integer.

Write $\left(5 \times 10^{k}\right)^{-1}$ in standard form. Give your answer in terms of $k$. [2]

(a) $0.9 \times 10^{11}$ or $9 \times 10^{10}$ or $90\,000\,000\,000$M1
(a) writes with an even power of 10 and square roots both parts, e.g. $\sqrt{9} \times \sqrt{10^{10}}$M1
(a) $3 \times 10^{5}$ caoA1
(b) $\dfrac{1}{5 \times 10^{k}}$ or $0.2 \times 10^{-k}$M1
(b) $2 \times 10^{-k-1}$ oe, e.g. $2 \times 10^{-(k+1)}$A1
Answer: (a) $3 \times 10^{5}$ (b) $2 \times 10^{-k-1}$
Examiner tip: Dividing by $10^{-4}$ means multiplying by $10^{4}$, so the power becomes $7 - (-4) = 11$, not 3.

15Grade 4–5Calculator4 marks

A memory card can store $2 \times 10^{12}$ bytes of data.

Each photo uses $4.8 \times 10^{6}$ bytes.

(a) Work out the greatest number of photos that can be stored on the empty memory card. [2]

Mia stores $3.5 \times 10^{5}$ of these photos on the empty memory card.

(b) What percentage of the memory card's storage is still free? [2]

(a) $(2 \times 10^{12}) \div (4.8 \times 10^{6})$ or 416 666.6… seenM1
(a) 416 666 cao (accept $4.16666 \times 10^{5}$; do not accept 416 667 or any value rounded up)A1
(b) $3.5 \times 10^{5} \times 4.8 \times 10^{6}$ $(= 1.68 \times 10^{12})$, or $\dfrac{3.5 \times 10^{5}}{416\,666.6…} \times 100$ (= 84)M1
(b) 16(%) caoA1
Answer: (a) 416 666 (b) 16%
Examiner tip: The answer to (a) must be rounded down: 416 667 photos would need more space than the card has.

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