Standard form and calculator entry

(a) $4.5 \times 10^4$

1. Place the decimal point after the first digit: $4.5$. Moving it back to $45\,000$ multiplies by $10$ four times.

• B1 $4.5 \times 10^4$.

(b) $0.0032$

1. $10^{-3}$ means divide by $1000$: $3.2 \div 1000 = 0.0032$.

• B1 The correct answer, $0.0032$.

(c) $5.9 \times 10^6$

1. Compare the powers of 10 first: $10^6$ is bigger than $10^5$, so $5.9 \times 10^6$ is larger ($5\,900\,000 > 600\,000$).

• B1 $5.9 \times 10^6$.

(a) $279$

1. Divide the population by the area.
   $$\frac{6.7 \times 10^7}{2.4 \times 10^5} = 279.166\ldots$$
2. To the nearest whole number: $279$.

• M1 Dividing population by area.
• A1 The correct answer, $279$.

(b) $67\,000\,000$

1. Multiply $6.7$ by $10$ seven times: $67\,000\,000$.

• B1 The correct answer, $67\,000\,000$.

(a) $8.00\times10^{2}$ bytes/s

1. Divide distance by time, dividing coefficients and subtracting indices.
   $$\frac{4\times 10^{5}}{5\times 10^{2}}$$
2. Therefore $8.00\times10^{2}$ bytes/s.

• P1 Divide distance by time, dividing coefficients and subtracting indices.
• A1 Correct answer: $8.00\times10^{2}$ bytes/s

(b) A coefficient from 1 up to but not including 10 gives one consistent power of ten for the size of a positive number.

1. A coefficient from 1 up to but not including 10 gives one consistent power of ten for the size of a positive number.

• C1 Correct conclusion with supporting reasoning: A coefficient from 1 up to but not including 10 gives one consistent power of ten for the size of a positive number.

(a) $5.6\times10^{6}$

1. Write both quantities with the same power of ten.
   $$5\times 10^{6}+3/5\times 10^{6}$$
2. Therefore $5.6\times10^{6}$.

• M1 Write both quantities with the same power of ten.
• A1 Correct answer: $5.6\times10^{6}$

(b) $28$

1. Divide total size by capacity and round up if needed.
   $$5600000/(2\times 10^{5})$$
2. Therefore $28$.

• P1 Divide total size by capacity and round up if needed.
• A1 Correct answer: $28$

(a) $2.8\times10^{3}$ g

1. Multiply coefficients and add the indices.
   $$4\times 7\times 10^{-6+8}$$
2. Therefore $2.8\times10^{3}$ g.

• M1 Multiply coefficients and add the indices.
• A1 Correct answer: $2.8\times10^{3}$ g

(b) $2.8$ kg

1. Divide grams by 1000.
   $$2800/1000$$
2. Therefore $2.8$ kg.

• M1 Divide grams by 1000.
• A1 Correct answer: $2.8$ kg

(a) $500$ seconds

1. Divide the distance by the distance travelled each second.
   $$\frac{1.5\times 10^{8}}{3\times 10^{5}}$$
2. Therefore $500$ seconds.

• M1 Divide the distance by the distance travelled each second.
• A1 Correct answer: $500$ seconds

(b) $5\times10^{2}$

1. Therefore $5\times10^{2}$.

• B1 Correct answer: $5\times10^{2}$

(a) $2.8 \times 10^8$

1. Convert tonnes to kilograms.
   $$18 \text{ tonnes} = 18\,000 \text{ kg} = 1.8 \times 10^4 \text{ kg}$$
2. Divide by the mass of one grain.
   $$\frac{1.8 \times 10^4}{6.5 \times 10^{-5}} = 276\,923\,076.9\ldots$$
3. To 2 significant figures: $2.8 \times 10^8$.

• P1 Converting 18 tonnes to $18\,000$ kg.
• P1 Dividing the total mass by $6.5 \times 10^{-5}$.
• A1 $2.8 \times 10^8$.

(b) 500 seconds

1. Time = distance ÷ speed.
   $$\frac{1.5 \times 10^8}{3.0 \times 10^5} = 0.5 \times 10^3 = 500 \text{ seconds}$$
2. That is 8 minutes 20 seconds.

• M1 Dividing distance by speed to get 500 seconds.
• A1 The correct answer, 500 seconds.

(a) $2.4\times10^{2}$

1. $$3.6\times 8/1.2$$
2. $$10^{-4+7-2}$$
3. Combine the coefficients: 3.6 $\times$ 8 $\div$ 1.2 = 24.
4. Combine powers: 10^($-4$ + 7 $-$ 2) = $10^{1}.$
5. 24 $\times$ $10^{1}$ = 2.4 $\times$ $10^{2}$ in standard form.

• M1 Establishing $3.6\times 8/1.2$ or an equivalent valid method.
• M1 Establishing $10^{-4+7-2}$ or an equivalent valid method.
• A1 Correct answer: $2.4\times10^{2}$