GAMSAT Section 3 maths: an estimation toolkit, not a formula list

Section 3 calls for less maths than a long formula list implies, and more judgement about how much precision each step deserves. The toolkit below has five moves: size the answer first, handle exponents separately from the digits, check the units, cancel before multiplying, and test the direction of change. Each one is a way of deciding whether an exact calculation is needed at all.

Confirm the official scope separately

This article does not say what Section 3 officially covers, how much maths it contains, or what you may use while working through it. Those are facts for the Australian Council for Educational Research (ACER) and its published information for candidates, and they are worth checking before you decide what to revise. The moves below are written to hold whatever that information says about calculators: estimation is a useful check if arithmetic is assisted, and a necessary skill if it is not.

Move 1: decide how precise the answer needs to be

Before calculating anything, read the answer options and ask how far apart they are. If they differ by a factor of ten, a one-digit estimate already separates them. If they differ by a few per cent, an estimate may leave options standing that it cannot separate, and you will need more care. Precision comes first as a decision because the arithmetic that follows is only as long as you choose to make it.

Here is an invented example, not taken from any examination. Suppose a setup gives (4.1 × 10³ × 2.9 × 10⁻²) ÷ (6.2 × 10¹). Round each figure to one digit and you have (4 × 10³ × 3 × 10⁻²) ÷ (6 × 10¹). Multiply the digits, 4 × 3 = 12, and combine the powers of ten, 10³ × 10⁻² = 10¹, so the top line is 12 × 10¹. Dividing by 6 × 10¹ leaves 2. Working it in full gives about 1.9. If the options were 0.19, 1.9, 19 and 190, the estimate alone was enough to choose. If they were 1.9 and 2.1, it was not.

Move 2: treat exponents and logs as counting powers of ten

Split every number into its digits and its power of ten, and handle the two separately. Exponents add when you multiply and subtract when you divide, so the power of ten in the example above took one line and never needed the digits. A base-ten log is the same bookkeeping run backwards, because it reports the power of ten. The log of 2000 is 3 plus the log of 2, and the log of 2 is about 0.3, so the answer sits near 3.3. A change of one unit in a base-ten log is a tenfold change in the quantity, which gives you something to test any option against that implies a different size.

Move 3: let the units check the setup

Write the units beside each quantity and cancel them as you would cancel numbers. Suppose you want an amount of substance and you hold a concentration in mol per litre and a volume in litres. Multiplying them gives mol, and dividing that by a time in seconds gives mol per second. If your units come out as mol per litre squared, a volume was divided in where it should have been multiplied, and careful arithmetic will not rescue that. A unit check tests the setup rather than the arithmetic, so it exposes a different kind of error from an estimate of size.

Move 4: cancel before you multiply

If the same quantity sits in the numerator and the denominator, remove it before doing any arithmetic. In a ratio problem the value of a shared quantity may not be needed at all, and a long calculation that ends in a short ratio is a prompt to look for a term that cancels. Cancelling also shortens what you have to hold in your head, which leaves less to lose if the arithmetic is the part that wobbles.

Move 5: check the direction of change before the size

Ask what happens to the result if one input rises while the others hold. If the quantity sits in the denominator, the result falls. If it appears squared, the result changes by a larger factor than the input did. Options that move the wrong way can be set aside before any number is computed, and fewer remaining options means less to separate. The check assumes you can hold the other quantities fixed. When two inputs change together it tells you less.

Sorting your misses: arithmetic, setup or decision

A wrong answer on a calculation question can come from several places, and the fix differs for each. These are patterns to check for when you review, not verdicts on what happened in your head.

• A slipped sign or exponent inside a setup that was otherwise sound is an arithmetic slip. A rough size estimate made before calculating is the check that would have exposed it, so ask whether you made one.
• Three significant figures worked out when the options sat a factor of ten apart is a decision about precision. The answer may be right and the time still wasted, which a correct mark hides.
• Units that do not resolve to what the question asks for are a setup error. The arithmetic can be flawless and the answer still wrong.
• A result that moves the wrong way without being noticed points to a missing direction check.
• A relationship you could not retrieve at all is a recall gap, and it needs a different fix: write the relationship from memory on a later day and see whether it holds.

This is the distinction drawn in our piece on correct answers and reasoning quality. An answer can be correct with weak reasoning behind it, and wrong with sound reasoning and one slip in it.

A practice routine, and where it stops working

Our suggestion for your next batch of calculation questions is to write three things beside each one before any arithmetic: the units the answer must have, the direction the result should move, and a rough size. Then decide whether to calculate further. Afterwards, compare your rough size with what you found, and note the time each decision took, so you can see where the exact route cost you something the estimate would not have.

The routine has costs. Three written lines take time, and on a question where the exact route is short they are slower than simply calculating. Estimation cannot separate options that sit close together, and a direction check weakens when several inputs move at once. None of it replaces knowing the science relationships a question assumes. Comparing estimates with another person can also help: a study group that stays on how each person reasoned can show you a rougher, faster route than the one you took.

A marked practice question cannot show which of these moves was missing. The score looks identical whether the loss was a sign slip, a precision decision or a setup that never resolved. Verantrix can record how you reasoned through a practice question, not only which option you chose, and can identify patterns across your attempts and name the reasoning failure they share. It does not predict a score, and it does not replace the checks above. For the next ten calculation questions, write units, direction and size first, label each miss as arithmetic, setup or precision, and see whether one label keeps coming back.