7.1 Mental Arithmetic Without a Calculator

Why Core Abilities numeracy is a no-paper skill

The Queensland Fire Department (QFD) Recruit Firefighter Online Cognitive Ability Test is delivered through Pearson OnVUE. Pearson’s published candidate rules for this sitting ban calculators, pens, pencils, note paper, and phones. You work at a laptop with a webcam and an offsite proctor. Every arithmetic step has to live in working memory.

Core Abilities is one of two named cognitive components in the current Firefighter Recruitment Candidate Information Pack (April 2024, still linked from the recruit page last updated 19 August 2026). The pack names four Core Abilities elements: literacy, numeracy, problem solving, and analytical aptitude. QFD does not publish how many numeracy items appear, how numeracy is weighted against literacy or Mechanical Reasoning, or a percentage cut score. Independent OpenExamPrep practice in this chapter helps you study the numeracy skills the pack names. It is not an official QFD paper and does not claim an official relationship with QFD or Pearson VUE.

“I will write the working on a pad” is not available on OnVUE. The method that survives is: rewrite the numbers into compatible partners, hold one running total, and sense-check the answer against the starting quantity.

What mental arithmetic means on this paper

Mental arithmetic is not a licence to guess when the stem asks for an exact litre count. It is exact arithmetic done with shortcuts instead of a written algorithm. You still need the precise remainder in the tank. You just cannot stack a long subtraction in the margin.

Use this split:

• Exact items — remaining litres, crew shares that must add to a headcount, minutes until a tank is empty. Use breaking, factors, doubling, and compensation.
• Estimate-first items — a stem that asks which quantity is closest, or a multi-step word problem where an early round number keeps you in range. Round, compute, then adjust back to the true numbers.

Number sequences (for example 2, 6, 12, 20, …) are a different Core Abilities skill. They sit with problem solving in Chapter 8. Do not hunt for the next term inside a remaining-tank stem.

A compact toolkit

Memorise the move, not a script of litres. Live stems change the numbers.

Compatible numbers and pairing

Compatible numbers are partners that make a round total. On a hose-length or water-use stem, scan the list for pairs that hit 10, 50, 100, or 1,000 before you add left to right.

Worked example. Four short dumps into a collecting tank: 185 L, 215 L, 340 L, and 160 L. Pair 185 with 215 (400). Pair 340 with 160 (500). Total 900 L. Left-to-right is 185 + 215 = 400, then 400 + 340 = 740, then 740 + 160 = 900 — same answer, more load if you refuse to pair.

Worked example. Travel log: 18 km to the incident, 7 km to a hospital rendezvous, 14 km back to station, then 11 km to a second job. 18 + 7 = 25, 25 + 14 = 39, 39 + 11 = 50 km. Landing on 49 or 51 means a dropped kilometre.

Breaking apart and compensation

Breaking apart splits a number by place value. Compensation changes a number to a friendlier one, computes, then puts the change back.

Worked example. An appliance tank holds 2,400 L. The first line uses 675 L. Remaining? Break 675 into 600 + 75. 2,400 − 600 = 1,800. 1,800 − 75 = 1,725 L. Compensation path: 2,400 − 700 = 1,700, then add back the extra 25 you subtracted: 1,725 L. Both paths must agree.

Worked example. 1,850 + 375. Compensation: 1,850 + 400 = 2,250, then subtract the extra 25: 2,225 L. Breaking: 1,850 + 300 = 2,150, then +75 = 2,225.

Hold one total. Do not keep 1,850, 375, 400, and 25 all live. Replace the addend with 400, add, then adjust.

Multiplication shortcuts on fireground counts

Lengths, crews, and minutes are often multiples of 4, 5, 8, 9, 15, or 25.

Worked example. Six firefighters each carry four 30 m lengths. First 6 × 4 = 24 lengths. Then 24 × 30: 24 × 10 = 240, and 30 = 3 × 10, so 240 × 3 = 720 m. Check: 20 × 30 = 600 and 4 × 30 = 120, sum 720 m.

Worked example. 48 × 25. Because 25 = 100 / 4: 48 × 100 = 4,800, then 4,800 ÷ 4 = 1,200. Check: 50 × 25 = 1,250, minus 2 × 25 = 50, so 1,200.

Worked example. 45 × 18. Split 18 into 20 − 2: 45 × 20 = 900, 45 × 2 = 90, 900 − 90 = 810. Doubling and halving: 45 × 18 = 90 × 9 = 810.

Worked example. 16 × 125. 125 × 8 = 1,000, so 125 × 16 = 2,000. Or 16 × 100 = 1,600 and 16 × 25 = 400, sum 2,000.

Division, remaining time, and a running tank

Division on this paper is usually “how many groups” or “how long until empty.” Prefer factors you can halve.

Worked example. 2,250 L at 25 L/min. Because ÷25 = ÷100 × 4: 2,250 ÷ 100 = 22.5, then 22.5 × 4 = 90 minutes. Check: 25 × 90 = 2,250.

Worked example. 840 ÷ 12. Think 84 ÷ 12 = 7, so 840 ÷ 12 = 70. A 840 L tank split into 12 equal crew shares is 70 L each.

Worked example. 3,600 L at 150 L/min. 150 × 20 = 3,000, remainder 600, 150 × 4 = 600, total 24 minutes. Or 3,600 ÷ 150 = 36 ÷ 1.5, and 36 × 2 / 3 = 72 / 3 = 24.

Multi-step remaining quantity

A 2,000 L tank starts full. The crew uses 350 L, then 125 L, then 225 L. Add uses with pairing: 350 + 125 = 475, then 475 + 225 = 700. Used 700 L. Remaining 2,000 − 700 = 1,300 L. Sense-check: 700 is about one-third of 2,000, so remaining should sit near 1,300, not 300 and not 1,700.

Traps when you cannot write the working

• Restarting the whole sum after each step instead of updating one running total.
• Subtracting 675 as 2,400 − 600 − 65 and dropping the 10.
• Treating ×25 as ×20 + ×6, which is ×26.
• Changing litres to millilitres mid-stem and then treating the answer as litres.
• Hunting for a number-sequence pattern in a remaining-tank stem. Sequences are Chapter 8.

QFD does not publish Core Abilities item counts, time per item, or a percentage pass mark. The 2024 pack says candidates are not expected to answer every question correctly. If a four-factor multiplication is still collapsing after a few seconds, skip if the delivery lets you, and return if review is available.

Read the flowchart as a pacing tool, not as a published QFD rubric. If the numbers are a remaining-tank or crew-count problem, stay on the exact branch. If the options are widely spaced (900 against 1,200 against 1,500), an estimate-first pass can eliminate three choices before you finish the last adjustment. If the stem is a pure next-number pattern, leave this chapter and use Chapter 8.

Independent OpenExamPrep quizzes below use four options. Live OnVUE items may differ in stem length. Practise pairing, breaking, compensation, and sense-checking — not a four-option habit as if it were the live paper’s only shape.