7.2 Fractions, Percentages, and Ratios

Three notations, one quantity

On QFD Core Abilities numeracy, a remaining-tank stem may say three-eighths, 37.5%, or 0.375 of a 960 L appliance tank. Those three strings name the same share. The OnVUE rules still ban a calculator and scrap paper, so you convert by known anchors, not by long division on a pad.

Numeracy is one named Core Abilities element in the April 2024 candidate pack, sitting beside literacy, problem solving, and analytical aptitude. QFD does not publish how many fraction or ratio items you will see or a percentage cut score. Independent OpenExamPrep study material below uses original litre and headcount figures so you can practise the conversions. It does not reproduce official QFD items.

Conversion anchors you should not recompute

If a stem gives 12.5% of a hose lay, switch to 1/8 immediately. If it gives 20% of a tanker load, switch to 1/5. Conversion is cheaper than multiplying by 12.5 in your head.

Taking a fraction of a quantity

The reliable two-step is: divide by the denominator, then multiply by the numerator. If both the quantity and the denominator share a factor, cancel first so the division is a single-digit step.

Worked example. 3/8 of 960 L. 960 ÷ 8 = 120, then 120 × 3 = 360 L. Cancel path: 960 / 8 = 120 still, because 960 and 8 share 8. Sense-check: 1/8 is 120, so 4/8 (a half) would be 480, and 3/8 must sit below 480. 360 is in range.

Worked example. 5/6 of 1,800 L. 1,800 ÷ 6 = 300, then 300 × 5 = 1,500 L. If the stem asked for the remainder after five-sixths is used, that remainder is 1/6 = 300 L, not 1,500 L. Read whether the question wants the taken share or the leftover.

Worked example. 2/5 of 1,250 m of hose in store. Cancel: 1,250 ÷ 5 = 250, then 250 × 2 = 500 m. Check: 1/5 is 250 m, so two-fifths is 500 m.

Worked example. 7/8 of 640 L. 640 ÷ 8 = 80, then 80 × 7 = 560 L. Remainder 1/8 = 80 L. Candidates who invert the fraction compute 8/7 of 640 and land nowhere near a listed option — if options are 80, 560, 480, 320, 560 is the taken share and 80 is the leftover.

Percentages from 10%, 5%, and 1%

Build an awkward percentage from chunks you can take with a decimal shift.

• 10%: move the decimal one place left (2,400 L → 240 L).
• 5%: half of 10% (240 L → 120 L).
• 1%: move the decimal two places left (2,400 L → 24 L).
• 15%: 10% + 5%.
• 12%: 10% + 1% + 1%, or 10% + 2 × 1%.
• 25%: 1/4, or half of 50%.
• 12.5%: 1/8.

Worked example. 12% of 2,500 L. 10% = 250 L. 1% = 25 L, so 2% = 50 L. 250 + 50 = 300 L. Check: 12% = 12/100 = 3/25, and 2,500 ÷ 25 = 100, then 100 × 3 = 300 L.

Worked example. 15% of 1,840 L. 10% = 184 L. 5% = 92 L. 184 + 92 = 276 L. Compensation check: 15% of 1,800 = 270, and 15% of 40 = 6, sum 276 L.

Worked example. Increase 800 L by 25%. 25% of 800 = 200, so the new quantity is 800 + 200 = 1,000 L. A decrease of 25% from 800 L is 800 − 200 = 600 L. Increase and decrease are not symmetric as fractions of the new amount: 600 L increased by 25% is 750 L, not back to 800 L. If the stem says “now 25% higher than 800 L,” land on 1,000 L, not 600 L.

Worked example. A 1,200 L tank is drawn down by 15%. 10% = 120 L, 5% = 60 L, used = 180 L, remaining = 1,200 − 180 = 1,020 L. Remaining as a percentage is 85%, which is 1,200 − 15% of 1,200, not 15% of 1,200.

Ratios as parts

A ratio 5:3 does not mean “5 and 3 litres.” It means 5 parts and 3 parts. Add the parts, divide the total by that sum, then multiply.

Worked example. 48 BA cylinder sets split 5:3 between two stations. Parts = 5 + 3 = 8. One part = 48 ÷ 8 = 6. Larger share = 5 × 6 = 30. Smaller share = 3 × 6 = 18. Check: 30 + 18 = 48.

Worked example. 32 hours of drill time split 4:3:1 across three skills. Parts = 8. One part = 32 ÷ 8 = 4 hours. Shares are 16 hours, 12 hours, and 4 hours. Check: 16 + 12 + 4 = 32.

Worked example. A 840 L foam-and-water mix is 1:6 (1 part foam concentrate to 6 parts water). Parts = 7. One part = 840 ÷ 7 = 120 L. Foam = 120 L. Water = 6 × 120 = 720 L. Check: 120 + 720 = 840. A candidate who treats 1:6 as “one-sixth foam” computes 140 L of foam and fails the check 140 + 700 = 840 — that is a 1:5 mix, not 1:6.

Worked example. Simplify 24:36 before scaling. Divide both by 12: 2:3. If the total is 90 L, parts = 5, one part = 18 L, shares 36 L and 54 L. Cancelling first is the OnVUE-safe move; expanding 24 and 36 against 90 is how working memory overflows.

Which notation to switch into

Use this list as a default, then override if the options are written in another form:

• Stem says “three-eighths of the tank” and options are litres → stay in fractions, divide then multiply.
• Stem says “12.5% remaining” → switch to 1/8 of the start quantity.
• Stem says “in the ratio 5:3” → parts method; do not convert to a percentage unless options are percents.
• Stem says “0.2 of the lay” → 1/5, then divide by 5.
• Options are 0.375, 0.4, 0.5, 0.625 → convert your fraction to a decimal with the anchor table rather than inventing a new long division.

Traps that swap leftover for share

• Computing 3/8 of 960 as 960 × 8 / 3 (inverting).
• Taking 12% as 12 L whenever the tank is 100-something, instead of 10% + 2%.
• Treating 1:6 as 1/6 of the mix rather than 1/7.
• Simplifying 4:3:1 to 4:3 and dropping the last part.
• Reading “remaining after 15% is used” as 15% instead of 85%.

Number sequences such as 1/2, 1/4, 1/8, 1/16 as a pattern are Chapter 8 work. Using 1/8 as a conversion for 12.5% is still this chapter.

The pie is a 5:3:2 split of 2,000 L, not a published QFD water policy. Parts = 10. One part = 2,000 ÷ 10 = 200 L. Attack = 5 × 200 = 1,000 L, backup = 3 × 200 = 600 L, reserve = 2 × 200 = 400 L. If a stem later says the attack line has already used 250 L of its 1,000 L share, remaining on that line is 750 L, and the backup and reserve shares are untouched unless the stem says otherwise. Finish one share’s arithmetic before you mix lines.

QFD does not publish a percentage pass mark for Core Abilities. Practise converting, cancelling, and checking that ratio shares add back to the original total.