Arithmetic and numerical computation

This is the definition trap. Asked to round to significant figures, candidates round to that many decimal places instead. Significant figures count from the first non-zero digit; decimal places count from the decimal point. On $0.0384$ they part company: 2 s.f. gives $0.038$, 2 d.p. gives $0.04$. Fix: find the first significant figure before rounding, then count from there.

Standard form errors are stereotyped. The value is out of range: $a$ must be $1 \le a < 10$, so $42 \times 10^{3}$ is wrong even though it multiplies out — write $4.2 \times 10^{4}$. The power has the wrong sign: a number below 1 takes a negative power, so $0.00056 = 5.6 \times 10^{-4}$. Or it is off by one from miscounting the moves. Check by multiplying back out.

The command estimate has a meaning candidates ignore: they compute the exact answer and then round it, when the mark is for rounding the inputs first. In one case the exact answer was 239 but the intended estimate, rounding first, was 240. Computing exactly is not estimating and loses the mark. Fix: write each number to 1 significant figure and use only those.

Written as a decimal, a proportion must lie between 0 and 1, because the part is smaller than the whole. If 3 of 8 people help, the decimal is $3 \div 8 = 0.375$. Dividing the wrong way, $8 \div 3$, gives a value above 1, which cannot be a proportion. Keep the leading zero too: write $0.375$, not $375$, so it is not mistaken for a whole number.

When a large number is rounded to significant figures, the places after the rounded digits become zeros that hold its size — they are not dropped. $24\,680$ to 3 s.f. is $24\,700$: the significant figures are 2, 4 and 6, the next digit (8) rounds the 6 up to 7, and the rest become 0. Writing 247 shrinks the number by 100.

In an estimate, round every input to 1 significant figure, not just one. To estimate $3870 \div 21$, round both: $4000 \div 20 = 200$. Leaving the divisor as 21 keeps the division as awkward as the original and defeats the point. Round to 1 significant figure, not 2 — the aim is numbers easy enough to work in your head as a check.

Two linked misconceptions. First, an estimate need not equal the exact answer; it is a rough figure that checks a result is in the right ballpark, so being a little out is expected. Second, more significant figures is not always better. A mean of $0.30$ seconds is often better than $0.298\,342$ seconds, because the extra digits imply a precision the measurement never had.