7.3 Rates, Time, and Quantity Word Problems

Rate stems are unit problems first

A Core Abilities numeracy word problem is usually a rate wearing a fireground story: litres per minute through a pump, kilometres per hour between station and incident, or firefighters per crew completing a stores count. Pearson OnVUE still gives you no calculator and no scrap paper. The working that survives is a short chain of compatible numbers plus unit cancellation so litres cannot masquerade as minutes.

QFD does not publish how many rate items appear on the Online Cognitive Ability Test, how they are weighted inside Core Abilities, or a percentage cut score. The April 2024 pack states that candidates are not expected to answer every question correctly. Independent OpenExamPrep examples below use original flows and distances. They are study quantities, not official QFD operational doctrine and not copies of the local practice bank.

The three identities you actually need

If time is given in minutes and speed in kilometres per hour, convert before you multiply. 36 minutes is 36/60 = 0.6 h, not 36 h and not 0.36 h.

Unit cancellation as a mental checklist

Say the units out loud (quietly — you are on camera) as you combine them:

• L/min × min → the minutes cancel → L.
• L ÷ (L/min) → litres cancel → min.
• km ÷ (km/h) → kilometres cancel → h.
• km/h × h → hours cancel → km.
• crew members × lengths/person × m/length → m of hose.

If the leftover unit is not the unit in the question, you inverted a division. That check is cheaper than restarting the arithmetic.

Flow, time, and remaining tank water

Worked example. A pump delivers 180 L/min for 15 minutes. Volume = 180 × 15. Use 180 × 10 = 1,800 and 180 × 5 = 900, sum 2,700 L. Check: 200 × 15 = 3,000, minus 20 × 15 = 300, so 2,700 L.

Worked example. 2,400 L remain in a tank. Flow is 160 L/min. Time to empty = 2,400 ÷ 160. Halve both: 1,200 ÷ 80, halve again: 600 ÷ 40 = 15 minutes. Check: 160 × 10 = 1,600, 160 × 5 = 800, sum 2,400 L in 15 minutes.

Worked example. A 2,100 L remainder at 175 L/min. 175 × 10 = 1,750, remainder 350, and 175 × 2 = 350, total 12 minutes. Check: 175 × 12 = 175 × 10 + 175 × 2 = 1,750 + 350 = 2,100.

Worked example. After 8 minutes at 250 L/min, how much has left a 3,000 L tank, and how much remains? Used = 250 × 8 = 2,000 L. Remaining = 3,000 − 2,000 = 1,000 L. If the stem then asks how long the remainder lasts at the same flow, 1,000 ÷ 250 = 4 minutes. Finish the used-volume step before you start the remaining-time step.

Combined pumps

When two appliances fill or empty together, add the rates, then divide the volume by the combined rate. When the stem later says one pump shuts down, drop back to the surviving rate.

Worked example. Pumps at 90 L/min and 135 L/min together fill 1,800 L. Combined rate = 90 + 135 = 225 L/min. Time = 1,800 ÷ 225. 225 × 8 = 1,800, so 8 minutes. Check: 225 × 10 = 2,250, which is 450 L too much; 450 ÷ 225 = 2 minutes less than 10, so 8 minutes.

Worked example. Pumps at 180 L/min and 120 L/min fill a 3,600 L tank together. Combined = 300 L/min. 3,600 ÷ 300 = 12 minutes. If the 120 L/min pump stops after they have already run together for 6 minutes, volume in 6 minutes = 300 × 6 = 1,800 L, remaining 1,800 L, now at 180 L/min only, extra time = 1,800 ÷ 180 = 10 minutes. Total time = 6 + 10 = 16 minutes. The trap is to divide 3,600 by 180 from the start and skip the six combined minutes.

Distance, speed, and time

Worked example. 42 km at 70 km/h. Time = 42 ÷ 70 = 0.6 h. 0.6 × 60 = 36 minutes. Fraction path: 42/70 = 6/10 = 3/5 h, and 3/5 of 60 minutes = 36 minutes. A candidate who treats 0.6 h as 6 minutes understates the travel by half an hour.

Worked example. Walk 4 km at 6 km/h from an appliance to a rural hydrant. Time = 4 ÷ 6 = 2/3 h = 40 minutes. Check: 6 km/h is 1 km every 10 minutes, so 4 km is 40 minutes.

Worked example. An appliance covers 18 km in 24 minutes. Speed = distance ÷ time in hours. 24 minutes = 24/60 = 0.4 h. 18 ÷ 0.4 = 45 km/h. Minute path: 18 km in 24 min is 18/24 km per minute = 0.75 km/min, × 60 = 45 km/h.

Multi-step quantity then rate

Worked example. A tanker starts with 2,800 L. It dumps 650 L at the first stop and 350 L at the second. Remaining = 2,800 − 1,000 = 1,800 L. It then delivers that remainder at 150 L/min. Time = 1,800 ÷ 150 = 12 minutes. Pair the dumps (650 + 350 = 1,000) before you touch the flow. Mixing 2,800 ÷ 150 first answers a different stem: time to empty the start quantity, not the remainder.

Worked example. Six firefighters pack stores at 4 boxes per person per minute for 5 minutes. Boxes = 6 × 4 × 5 = 120 boxes. If a seventh firefighter joins for another 3 minutes at the same personal rate, extra boxes = 7 × 4 × 3 = 84, total 204 boxes. Restarting as 7 people for the whole 8 minutes over-counts the first five minutes.

Traps on rate wording

• Leaving speed in km/h and time in minutes, then multiplying 70 × 36 and calling it kilometres.
• Adding times when you should add rates (two pumps), or adding rates when the pumps run in sequence, not together.
• Dividing volume by only one pump after the stem introduced two.
• Treating remaining time as start volume ÷ flow after a dump has already left the tank.
• Forcing a number sequence onto 180, 225, 270 L/min when the stem is just listing two pumps and their sum. Sequences are Chapter 8.

Pacing when the arithmetic branches

A three-step remaining-then-rate stem is still one item. Compute the remaining litres, say the unit (L), then divide by L/min to get min. If the first remaining-litre total already disagrees with a sense-check (a 2,800 L tank cannot have 3,100 L left), stop and redo the dumps; do not feed a broken remainder into a perfect division.

QFD does not publish item counts or a percentage cut score for this battery. Invitation emails control the live Core Abilities window; the Pearson listing dated 9 September 2026 named 23 September 2026 to 13 October 2026 as the 2027-campaign example only. Your job in the room is accurate litres and minutes, not an invented percentage target.

Use the diagram as a working-memory order, not as a published scoring rubric. Convert units, choose the identity, add rates only when the stem says the pumps work together, cancel, then check the leftover unit. Independent OpenExamPrep four-option quizzes below train that order with original numbers.