General arithmetic and estimation strategies

What the exam asks

Most Section III arithmetic is a single multiplication or division whose inputs are handed to you in the stem, and the marks are lost in execution rather than in setup. Expect calculation items where the options are spread across decades, which are pure estimation, and a smaller number where two options sit close together, which are a deliberate test of whether you noticed. The trap that catches most candidates is estimating into a gap too narrow for the estimate: they round 4.8 to 5, land between options at 4.5 and 5.2, and pick by feel. Before rounding, ask what error your rounding introduces and whether the options can survive it. If two options are within about ten percent of each other, one-significant-figure arithmetic cannot separate them and you have to do the sum.

There is no calculator in Section III, and nobody expects long division under time. What is expected is that you can get an answer accurate enough to pick the right option, and that you know when 'accurate enough' has stopped being good enough. Those are two separate skills and most lost marks come from the second one.

Estimation is a decision, not a habit. Look at the options first. If they are an order of magnitude apart, round everything to one significant figure and you will be done in ten seconds with room to spare. If they sit within a few percent of each other, the question has deliberately closed the gap and you must actually compute, because one-figure rounding carries an error larger than the spacing you are trying to resolve. Reading the options before starting the arithmetic is the whole technique.

The rest is a small kit of moves that turn awkward numbers into easy ones: round the numerator and the denominator in the same direction so the errors partly cancel, double one factor and halve the other, rearrange the formula before any number goes into it, and take ratios whenever the question asks for a factor rather than a value. That last one is the biggest single time saver in the paper, because in a ratio almost every constant cancels and never has to be computed at all.

What to hold

• Read the options before doing any arithmetic, because their spacing tells you how much precision the question actually requires.
• Options an order of magnitude apart can be settled by rounding every input to one significant figure; options a few percent apart cannot, and must be computed.
• Rounding the numerator and denominator in the same direction makes the two errors partly cancel; rounding them in opposite directions makes the errors add.
• Get the power of ten right first and the leading digits second, because a magnitude error is fatal while a ten percent error usually is not.
• Doubling one factor and halving the other leaves a product unchanged, which turns 25 times 36 into 50 times 18 into 100 times 9.
• Dividing by 5 is multiplying by 2 and dividing by 10; multiplying by 25 is multiplying by 100 and dividing by 4.
• A percentage can be split into parts you can do in your head: 15 percent is 10 percent plus half of it.
• A rise of 10 percent followed by a fall of 10 percent is a net fall of 1 percent, because percentage changes multiply rather than add.
• Rearrange the formula to isolate the unknown before substituting anything, so the algebra stays readable and a unit check is still possible.
• When a question asks by what factor something changes, form the ratio of the two cases: every constant common to both cancels and never needs a value.
• Worth carrying: the square root of 2 is about 1.41, the square root of 3 about 1.73, the square root of 10 about 3.16, and pi is close enough to 3 for a one-figure estimate.
• Worth carrying: one third is 0.33, one sixth is 0.17, one seventh is 0.14, one eighth is 0.125, one ninth is 0.11.