Unit 7
Finding Procedures
About this unit
Master multi-step problem-solving strategies: working backwards, the supposition method, rate-and-worker problems, and percentage applications.
What types of questions will you face?
- 1Supposition problems — two types of items each contribute a different amount to a total; assume all are one type, find the gap, divide by the per-swap difference to count the other type
- 2Combined rate problems — two pipes or workers each complete a task in different times; add their rates (1÷time each) to get the combined rate, then take the reciprocal for the total time
- 3Working backwards — a sequence of operations is applied to an unknown starting number to produce a known result; reverse each operation in reverse order to find the original number
- 4Percentage discount and profit — calculate a discount percentage from original and sale prices, or a selling price from cost price and a given profit percentage
- 5Ratio and proportion — share a total in a given ratio, scale a recipe or quantity proportionally, or find one part after another fraction has been taken off first
Skills you will build
- Applying the supposition shortcut: assume all items are type A, compute expected total, divide the gap by (value_B − value_A) to count type B items — without setting up two equations
- Converting time into rate (rate = 1 ÷ time), adding rates as fractions with a common denominator, and taking the reciprocal to get the combined time
- Reversing a chain of operations step by step — undo the last operation first (multiply → divide, add → subtract) — to recover the original value in working-backwards problems
- Computing discount percentage as (original − sale) ÷ original × 100, and selling price as cost × (1 + profit%/100)
- Dividing a quantity in a given ratio by finding the unit value (total ÷ sum of ratio parts) and multiplying by each part
By the end of this unit, you will be able to
- Solve any two-type counting problem (legs, coins, containers) using the supposition method in three arithmetic steps
- Find the combined time for two pipes or workers using the rate-addition formula — never by averaging times
- Work backwards through any sequence of operations to recover the starting number
- Calculate percentage discounts, profit percentages, and selling prices accurately
- Share quantities in any ratio and scale recipes or measurements proportionally
Difficulty profile
Questions in this unit range from Easy to Difficult. Simple supposition problems and direct proportion questions are Easy. Combined rate and multi-step working-backwards problems are Medium. Rate problems with a drain pipe or partial completion reach Difficult.
Exam tip: Finding Procedures
For supposition problems, always start with the lower-value type — assume all are ducks (2 legs), not cats (4 legs). For combined rate problems, never average the times — convert each time to a rate (1÷time), add the fractions, and invert the result. These two reflexes prevent the two most common wrong answers on this unit.
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