Math: the syllabus
You cannot prepare for a question type you do not know exists. This is every concept the math section draws from, taught with worked examples, so nothing on test day is a surprise. Learn a topic, then drill it.
Split-and-round multiplication
Most multiplications on these tests involve one awkward number and one friendly one. The core method is to split the awkward number into a round part and a leftover, multiply each part, and add the results. To find 12 times 15, think of 12 as 10 plus 2. Ten fifteens is 150, two fifteens is 30, and 150 plus 30 is 180. You never held a hard multiplication in your head, only two easy ones and an addition.
The split works on either factor, so pick the split that gives the friendliest pieces. For 6 times 4.5, split the 4.5 instead: 6 times 4 is 24, and 6 times 0.5 is 3, so the answer is 27. With practice you should recognize which side to break apart within a second of reading the question.
There is a second family of shortcuts called compensation: round a number up to something easy, then subtract the overshoot. To find 500 times 3.8, compute 500 times 4, which is 2000, then take back the extra 500 times 0.2, which is 100, leaving 1900. Compensation shines whenever a factor sits just below a round number, such as 3.8, 19, 49, or 98.
Division by recognizing multiplication
Division under time pressure is easiest when you flip it into a multiplication question: instead of asking what 3600 divided by 450 is, ask how many 450s fit into 3600. Double 450 to get 900, then notice 900 goes into 3600 four times, so 450 goes in eight times. Doubling and halving turn ugly divisors into ones you already know.
Another reliable move is scaling both numbers by the same factor, because a division does not change when you multiply or divide top and bottom alike. So 9000 divided by 15 is the same as 900 divided by 1.5, which is 600. Strip zeros, double both sides, or halve both sides until the numbers look like a times-table fact.
When a division does not come out evenly, the test is usually asking for an estimate, and the answer options will be far apart. Do the division roughly, note whether your rounding pushed the result high or low, and pick the option on the correct side of your estimate.
Times-table fluency targets
Every mental shortcut leans on instant recall of basic facts. Your target before test day: all products up to 12 times 12 in under two seconds each, with no counting on fingers and no skip-counting. If any fact makes you pause, drill that fact specifically until it is automatic.
Beyond the classic table, memorize the multiples that fireground numbers love: 15, 25, 45, and 150. Know cold that 4 times 15 is 60, 4 times 25 is 100, 8 times 45 is 360, and 6 times 150 is 900. These appear constantly in hose lengths, flow rates, and pressure problems.
Also drill powers of ten. Multiplying by 10, 100, or 1000 just shifts the decimal point, and dividing shifts it the other way. Being fluent with the shift means a number like 40,000 centimetres converts to 400 metres without any real calculation.
Worked example
A pumper carries 12 lengths of supply hose and each length is 15 metres. What is the total length of hose?
- Split the 12 into 10 plus 2, because multiplying by 10 is instant.
- Compute 10 times 15: shift the decimal, giving 150.
- Compute 2 times 15: double 15 to get 30.
- Add the pieces: 150 plus 30 is 180 metres.
180 metres
Key facts to know cold
- To multiply an awkward number, split it into a round part plus a leftover, multiply each, and add.
- Compensation: round up to an easy number, multiply, then subtract the overshoot (500 times 3.8 is 2000 minus 100).
- A division does not change if you scale both numbers the same way: 9000 over 15 equals 900 over 1.5, which is 600.
- Multiplying by 10, 100, or 1000 only shifts the decimal point.
- Fluency target: every product up to 12 times 12 recalled in under two seconds.
- Know the fireground multiples cold: 4 times 25 is 100, 8 times 45 is 360, 6 times 150 is 900.