Integrated and Cross-Topic Problem Solving
What this domain tests
Multi-step problems that combine two or more domains: rates with algebra, ratios with geometry, functions read from a table, or a diagram that must be modelled by an equation. The mathematics is familiar; the difficulty is choosing the order of steps.
Core rules
- Name the unknown in words and symbols before writing anything else, and state its unit.
- Convert every quantity to consistent units at the start, not halfway through.
- For work or filling rates, add rates rather than times: if A finishes in a hours and B in b hours, together they complete 1/a + 1/b per hour, and the time is the reciprocal of that sum.
- For mixture and ratio problems, track the amount of the substance, not just the total.
- Finish by re-reading the question: it may ask for a time in hours and minutes, a perimeter rather than a side, or a total cost rather than a unit price.
Worked example
Pipe A fills a tank in 6 hours and pipe B in 4 hours. How long together? Combined rate = 1/6 + 1/4 = 2/12 + 3/12 = 5/12 of the tank per hour. Time = 12/5 = 2.4 hours = 2 hours 24 minutes. Check: in 2.4 h, A supplies 0.4 of the tank and B supplies 0.6, totalling 1.
Traps that cost marks
Averaging the two times to get 5 hours; leaving the answer as 2.4 hours when minutes are requested; adding times instead of rates; forgetting that a shared dimension cancels in a ratio.