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The mass of the Earth is \(5.97 \times 10^{24}\) kg. The mass of the Moon is \(7.35 \times 10^{22}\) kg.
(a) Work out how many times heavier the Earth is than the Moon. Give your answer correct to 3 significant figures.[2]
(b) Hana estimates the answer to part (a). She writes \(6 \div 7 \approx 0.86\) and \(10^{24} \div 10^{22} = 10^{\frac{24}{22}} \approx 10\), so the Earth is about 8.6 times heavier. Explain the mistake in Hana's working.[1]
Show the answer and mark scheme
(a)Answer: 81.2
M1 for \((5.97 \times 10^{24}) \div (7.35 \times 10^{22})\)
A1 for 81.2
Worked solution: \(\frac{5.97 \times 10^{24}}{7.35 \times 10^{22}} = 0.81224\ldots \times 10^2 = 81.224\ldots\), so 81.2 times.
(b)Answer: To divide powers of 10 you subtract the indices: \(10^{24} \div 10^{22} = 10^{2} = 100\), not \(10^{\frac{24}{22}}\).
C1 for explaining that the indices should be subtracted, \(10^{24} \div 10^{22} = 10^2\) (= 100)
Worked solution: \(10^{24} \div 10^{22} = 10^{24 - 22} = 10^2 = 100\), so the estimate should be about \(0.86 \times 100 = 86\).
Question 3Hard3 marks
Work out \(\frac{8 \times 10^{8} + 4 \times 10^{7}}{2 \times 10^{-2}}\) Give your answer in standard form.[3]
Show the answer and mark scheme
Answer: \(4.2 \times 10^{10}\)
M1 for writing the numerator as a single number, e.g. \(8.4 \times 10^{8}\) or \(840{,}000{,}000\)
M1 for a correct method to divide by \(2 \times 10^{-2}\), e.g. \(8.4 \div 2 = 4.2\) and \(10^{8} \div 10^{-2} = 10^{10}\)