The mass of the Earth is approximately 5.97 times 10 to the power of 24
kilograms. The mass of Mercury is approximately 3.30 times 10 to the power of 26 grams. Work out an estimate for the number of times heavier the Earth is than Mercury.
The keyword here is the word “estimate.” When we estimate the answer to a question, we round each part to one significant
figure. The first significant figure in the number 5.97 times 10 to the power of 24 is
five. And the nine is the deciding number. Since nine is greater than five, we round the five up to a six, giving us six times
10 to the power of 24 kilograms.
Similarly, in the number 3.30 times 10 to the power of 26, the first three is the
first significant figure. The second three is the deciding number. And since this second three is less than five, we’re going to round down, giving us
three times 10 to the power 26 grams.
However, since this number is in grams, we’re going to need to convert it to
kilograms before we perform any further calculations. One kilogram is equal to 1000 grams. Another way of saying that though is 10 to the power of three grams.
To change from grams to kilograms then, we’re going to divide by 10 to the power of
three. Since multiplication and division are commutative, which means we can perform them in
any order, we’ll do the division first. Remember, when we divide a number with a power, as long as the base is the same — in
our example, that’s the number 10 — we subtract the powers.
26 minus three is 23. So 10 to the power of 26 divided by 10 to the power of three is 10 to the power of
23. That means that three times 10 to the power of 26 grams is the same as three times 10
to the power of 23 kilograms.
To find the number of times heavier the Earth is than Mercury, we’ll divide the mass
of the Earth by the mass of Mercury. Six divided by three is two. And using our rule of indices from earlier, 10 to the power of 24 divided by 10 to
the power of 23 is 10 to the power of one, or just 10. That means then that six times 10 to the power of 24 divided by three times 10 to the
power of 23 is two times 10. Two times 10 is simply 20. So an estimate for the number of times heavier the Earth is than Mercury is 20
Is your answer an overestimate or an underestimate? Give a reason for your answer. To answer this question, let’s look at the rounding we performed at the start. We rounded the mass of the Earth up, and we rounded the mass of Mercury down. That means when we estimated and performed the division calculation, we divided a
slightly bigger number than the original by a slightly smaller number than the
original. This means our estimated answer will be larger than the correct answer. It’s an overestimate, since 5.97 was rounded up and 3.30 was rounded down.