Percentages

Percentages appear in roughly 40% of all QR questions - more than any other single skill. They also combine with almost everything else: percentage of a mean, percentage change in a ratio, VAT as a percentage. If you master one thing in QR, make it this.

Memorise these. You'll use them constantly for quick estimation and to avoid unnecessary calculator work.

If a question asks "what fraction of 720?" and you spot the answer is close to 1/8, you know to expect ~90 (720 / 8 = 90). This catches wrong-row or wrong-column errors before they cost marks.

Every percentage question boils down to one of these three. Recognise which one you need before touching the calculator.

Standard formula (three operations): ((Final − Initial) / Initial) × 100

Joyel shortcut (two operations): (Final / Initial) − 1, then read the decimal.

Example: price went from 80 to 100. What's the % change?

Standard: (100 − 80) / 80 = 20 / 80 = 0.25 × 100 = 25% (three calculator ops)
Shortcut: 100 / 80 = 1.25 → subtract 1 → 0.25 → × 100 = 25% (two calculator ops; the subtract-1 is instant in your head)

Why This Is Faster

You often don't even need the final × 100 step - just read the decimal directly:

Reverse percentage questions give you the final value after a percentage has been applied and ask for the original. This is where most students make their biggest QR error.

The VAT trap. "A laptop costs £360 including 20% VAT. What was the price before VAT?"

Wrong: 360 × 0.8 = 288 (subtracting 20% of 360)
Wrong: 360 − (360 × 0.2) = 288 (same mistake, longer)
Right: 360 / 1.2 = 300 (divide by 1 + the percentage)

Check: 300 + 20% of 300 = 300 + 60 = 360 ✓
Check (the wrong answer): 288 + 20% of 288 = 345.60 ✗

The Rule

Think of it this way: the number you divide by is the multiplier that was applied to the original. If the original was increased by 20%, it was multiplied by 1.2. To undo that, divide by 1.2.

When one of the numbers is awkward but the other is easy, flip them. `X% of Y = Y% of X` (multiplication is commutative).

This turns a calculator problem into a 2-second mental calculation. Look for it whenever one number is 25, 50, or small.

These come up constantly. Memorise them to skip calculator use on simple conversions.

If a question says "3/8 of 640 people voted," you instantly know 3/8 = 0.375, type 640 \ 0.375 = 240, done in 5 seconds. Without this, you'd do 640 / 8 \ 3 - an extra step.

Question: The table below shows quarterly smartphone sales. What was the percentage change in sales from Q1 to Q3?

Step 1: identify the data needed → Q1 Total = 4,400, Q3 Total = 5,220.
Step 2: Joyel shortcut → 5220 / 4400 = 1.1864.
Step 3: read the decimal → 1.1864 → 0.1864 → 18.6% increase.

Calculator keystrokes: 5220 / 4400 =. Time: ~10 sec.

Sense check: sales went from 4,400 to 5,220. That's about 800 more on a base of ~4,400 - roughly 1/5.5, so ~18%. Checks out.

Trap 1: Wrong denominator

• % increase → divide by initial (old) value, not final.
• % of total → divide by the total, not a subtotal.

Trap 2: The VAT trap *(covered above)*

"Price includes 20% VAT" → divide by 1.2, not multiply by 0.8.

Trap 3: Successive percentage changes

A 10% increase followed by a 10% decrease does not cancel out.
100 + 10% = 110. 110 − 10% = 99. Net change: −1%.
Always apply each change to the current value, not the original.

Trap 4: Percentage OF vs percentage INCREASE

"120 is 150% of 80" → 120 / 80 = 1.5.
"120 is a 50% increase on 80" → (120 − 80) / 80 = 0.5.
Different questions, same numbers, different answers.

Next lesson: 3.3 Ratios & Rates - ratio problems and speed/distance/time, appearing in about 12% and 5% of questions respectively.