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.
Sample Questions
Fair-money puzzles are working backwards: start from what is left and undo each spend in reverse order until you reach the starting amount.
Half-then-fixed-spend items appear in the easy band of Selective MR — reliable marks when you reverse the steps instead of guessing from the options.
The examiner checks whether you can undo sequential operations (halving, subtracting a fixed amount) to recover the original value.
Someone spends half their money, then a fixed dollar amount, and has a stated amount left. You find how much they started with.
Best approach: After the fixed spend they had (left + fixed). Before halving they had double that. Check forward: half, subtract, equals the given remainder.
Question
Connor goes to the fair.
- He spends half his money on rides.
- He then spends $4 on food.
- He has $6 left.
How much money did Connor start with?
- A$14
- B$16
- C$20
- D$24
- E$28
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92 people, 2 large buses of 17, small buses of 8 for the rest. Three steps: subtract to find leftover people, divide by 8, round UP (you can't leave anyone behind), then add the 2 large buses back.
Transport and packaging problems requiring 'division with ceiling rounding' appear regularly in NSW Selective MR. The key move is rounding the quotient UP to the next whole number whenever there is any remainder — even a remainder of 1 requires a full extra bus.
The examiner tests whether students (a) correctly subtract 2 × 17 = 34 from 92 to get 58 remaining people; (b) divide 58 ÷ 8 = 7.25 and round UP to 8 small buses; and (c) add 2 + 8 = 10 total buses. The trap answer C (9) comes from forgetting to round up (using 7 small buses).
A group of people must be transported in two types of vehicles with different capacities. Some people fill the large vehicles first; the rest go in small vehicles. Since you can't have a fraction of a bus, any remainder requires an extra vehicle.
Best approach: Step 1: People in large vehicles = 2 × 17 = 34. Step 2: Remaining = 92 − 34 = 58. Step 3: Small buses = ⌈58 ÷ 8⌉ = ⌈7.25⌉ = 8. Step 4: Total = 2 + 8 = 10.
Question
92 people are going on a trip.
First they will fill two large minibuses, each with space for 17 people.
The rest of the people will travel in small minibuses, each with space for 8 people.
How many minibuses do they need in total?
- A7
- B8
- C9
- D10
- E11
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Combined-work questions use rates, not averages: each worker completes 1÷time of the job per day — add the rates, then take the reciprocal for total time.
Two-worker “working together” items appear in Selective MR — the trap is averaging the individual times instead of adding 1/time_A + 1/time_B.
The examiner tests whether you convert days-per-job into jobs-per-day, add rates for simultaneous work, and divide 1 by the combined rate.
Worker A finishes alone in one number of days, Worker B in another. Both work together on the same total task. You find how many days until completion.
Best approach: Rate A = 1/30, Rate B = 1/20 per day. Combined = 1/30 + 1/20 = 1/12. Days = 1 ÷ (1/12) = 12. Scale if a total quantity is given (here 3600 parts ÷ rate).
Question
A factory has a task to produce 3600 parts. If it is given to Worker A only, it will take 30 days. If it is given to Worker B only, it will take 20 days. Now they are working together. In how many days can they finish the task?
- A8
- B10
- C12
- D15
- E25
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The robot goes up (10s) and down (8s) in a repeating 18-second cycle. Convert 2 minutes to seconds, divide by the cycle length to get complete cycles, then check whether the leftover time is enough for another trip to the top.
Cycle-counting questions (how many times does X happen in Y minutes?) appear regularly in the medium band of Selective MR. The critical skill is handling the remainder — a partial cycle may or may not complete the key event (reaching the top).
The examiner tests whether students (1) convert 2 minutes to 120 seconds, (2) find the correct cycle length (10 + 8 = 18s, not just 10s or 8s), (3) divide 120 ÷ 18 = 6 remainder 12, and (4) check whether 12 remaining seconds ≥ 10s up-time, giving a 7th reach-the-top. Trap answer 6 (ignoring the remainder) and 6×2=12 (only counting complete up-down pairs) are common.
A machine or person repeats a two-phase cycle (e.g. up + down). A total time is given. Students must find how many times the key phase (e.g. reaching the top) is completed, including any partial cycles.
Best approach: Step 1: Convert total time to consistent units (seconds). Step 2: Full cycle = up + down. Step 3: Complete cycles = total ÷ cycle (integer division). Step 4: Remainder = leftover seconds. Step 5: If remainder ≥ up-time, add 1 more top-reach.
Question
A robot climbs all the way up and down a ladder repeatedly, without stopping.
It takes the robot 10 seconds to go up and 8 seconds to come down each time.
The robot starts at the bottom of the ladder each time.
How many times will the robot reach the top of the ladder in the next 2 minutes?
- A6
- B7
- C11
- D12
- E13
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