Number and algebra · Lesson 4 of 4

Percentages, discounts and GST

Percent of, percent more, percent less, GST on, GST off. They are all one move: find the base, write the multiplier, multiply.

About 20 min · Interactive · 8 practice questions

Drag the dial. The bar is the base amount, always 100%. A rise adds a gold slice; a discount hatches out a slice. Watch how the multiplier changes with it.

Multiplier×0.85
Change−$36.00
New amount$204.00

Which number is 100%?

Most people learn percentages as three separate skills: find a percentage of something, increase by a percentage, decrease by a percentage. Instead of three skills, there is one move. A percentage is always a percentage of something, and that something is the base. The base is the number that counts as 100%.

Picture the base as a bar. 15% of the bar is a slice 0.15 of its length. A 15% increase is the whole bar plus that slice, so 1.15 of its length. A 15% discount is the whole bar minus that slice, so 0.85 of its length. In every case you scale one bar by one number, and that number is the multiplier.

Reverse questions run the same move backwards. If a sale price is 0.85 of the original, the original is the sale price divided by 0.85. Multiplying stretched the bar, so dividing restores it.

The routine for the test is therefore fixed. Decide which number is 100%. Write the multiplier. Multiply to go forwards or divide to go back.

Three bars built from the same base of $240: 15% of it, 15% more than it, and 15% less than it Base $240 is 100%. Same bar, three questions: 15% of $240 ×0.15 → $36 $240 + 15% ×1.15 → $276 $240 − 15% ×0.85 → $204 100%
The slice is the same $36 each time. What changes is whether you keep only the slice, add it to the bar, or cut it from the bar.

The multiplier recipes

Each recipe below is the same move pointed at a different question. Lesson 1 covered turning a percentage into a decimal, so here the decimal form is used straight away as the multiplier.

1. Percentage of an amount

  1. The amount is the base. Write the percentage as a decimal multiplier.
  2. Multiply.
  3. Without a calculator, build it from anchors: 10% is ÷10, 5% is half of that, 1% is ÷100.

17% of $360: 10% = 36, 5% = 18, 1% = 3.6, so 17% = 36 + 18 + 3.6 + 3.6 = $61.20. Or 0.17 × 360 = 61.2.

2. Increase or decrease

  1. The starting amount is the base.
  2. Increase: multiplier is 1 + the decimal. Decrease: 1 − the decimal.
  3. Multiply once. Do not find the change and then add it unless you need the change itself.

$45 up 8%: 1.08 × 45 = $48.60. $64 down 15%: 0.85 × 64 = $54.40.

3. Finding the percentage change

  1. The original is the base. Always the number you started from, never the new one.
  2. Change = new − original.
  3. Percentage change = change ÷ original × 100.

Enrolment 480 → 432: change = −48. 48 ÷ 480 = 0.1, so a 10% decrease.

4. Reverse: find the original

  1. The original is the base, and it is the unknown.
  2. Write what the known amount is as a percentage of it: after 15% off, the sale price is 85%, multiplier 0.85.
  3. Divide the known amount by the multiplier.

Sale price $102 after 15% off: 102 ÷ 0.85 = $120. Check: 0.85 × 120 = 102.

5. GST in Australia

  1. GST is 10% added to the ex-GST price, so the ex-GST price is the base.
  2. Add GST: × 1.1.
  3. Remove GST: ÷ 1.1 (not minus 10%).
  4. GST inside a GST-inclusive price: ÷ 11.

$1,240 incl. GST: ex-GST = 1240 ÷ 1.1 = $1,127.27; GST = 1240 ÷ 11 = $112.73.

6. Successive changes multiply

  1. Write each change as a multiplier.
  2. Multiply the multipliers together. Never add the percentages.
  3. Order does not matter, because multiplication can be done in any order.

−20% then +20%: 0.8 × 1.2 = 0.96, a 4% net decrease, not zero.

7. Percentage points versus percent

When the quantity being compared is itself a percentage, two different changes are hiding. The gap between 40% and 50% is 10 percentage points: subtract. But 50 is 25% more than 40, because the base for a percent change is the original 40. The test uses "percentage points" when it means subtraction, so read the wording and apply the matching rule.

Points or percent?

  1. "Percentage points": new % − old %.
  2. "Percent increase": (new % − old %) ÷ old % × 100.

72% → 81%: 9 percentage points; 9 ÷ 72 = 0.125, so a 12.5% increase.

Why GST is ÷ 11 and ÷ 1.1

Think of the ex-GST price as 10 equal blocks. GST adds one more block of the same size. The price you pay is 11 blocks. So one block, the GST, is the total ÷ 11, and the ten blocks of price are the total ÷ 1.1. Taking 10% off the total would remove 1.1 blocks, which is too much.

A GST-inclusive price drawn as 11 equal blocks: 10 blocks are the price excluding GST and 1 block is the GST Price excluding GST = 10 blocks (100%) GST = 1 block (10%) 12345 678910 GST Price including GST = 11 blocks (110%) GST = total ÷ 11   ·   ex-GST price = total ÷ 1.1 = total × 10/11
Eleven blocks, not ten. That is why "10% of the total" overstates the GST: 10% of 11 blocks is 1.1 blocks.

Why −20% then +20% is not back to the start

The second change has a different base from the first. After a 20% cut, the base is smaller, so adding 20% of that smaller base puts back less than was taken. Multiply the multipliers and you see it directly.

Three bars: the start at 100 percent, then 80 percent after a 20 percent cut, then 96 percent after adding 20 percent of the smaller bar Start 100% (×1) −20% 80% (×0.8) then +20% 96% (×0.8 × 1.2 = ×0.96) 100%
The gold slice is 20% of 80, which is 16, not 20. The bar ends at 96, a 4% net decrease. Reversing the order (+20% then −20%) gives 1.2 × 0.8, the same 0.96.

Percentage problem solver

Pick the question type, enter the numbers

The solver names the base, writes the multiplier and shows every arithmetic step. Money answers are rounded to the cent only at the end.

    Multipliers to know by heart

    In the no-calculator section you will not have time to work out 1 − 0.15. Know the common multipliers as facts. The fraction column is the mental shortcut: 25% off keeps three quarters, so 25% off $64 is 3 × 16 = $48.

    ChangeMultiplierKeeps / becomesFraction equivalent
    Discounts and decreases
    5% off×0.95keeps 95%keep 19/20
    10% off×0.9keeps 90%keep 9/10
    12.5% off×0.875keeps 87.5%keep 7/8
    15% off×0.85keeps 85%keep 17/20
    20% off×0.8keeps 80%keep 4/5
    25% off×0.75keeps 75%keep 3/4
    1/3 off×0.667 (exactly 2/3)keeps 66.7%keep 2/3
    40% off×0.6keeps 60%keep 3/5
    50% off×0.5keeps 50%keep 1/2
    Increases
    5% more×1.05becomes 105%1 + 1/20
    10% more (GST)×1.1becomes 110%1 + 1/10 = 11/10
    12.5% more×1.125becomes 112.5%1 + 1/8 = 9/8
    20% more×1.2becomes 120%1 + 1/5 = 6/5
    25% more×1.25becomes 125%1 + 1/4 = 5/4
    50% more×1.5becomes 150%1 + 1/2 = 3/2
    100% more×2becomes 200%double

    Hide a column, say each answer aloud, then click a striped cell to check it.

    Shop till drill

    A price and a discount come up on the till. Rounds alternate: find the sale price, then work backwards to the original. Type the dollar amount to the cent.

    Round 1: find the sale price

    Marked price$64.00
    Discount15% off

    Common traps

    • Taking 10% off to remove GST. The GST was 10% of the smaller ex-GST price, not of the total you are looking at. Divide by 1.1 to undo it, or divide by 11 to find the GST. $110 − 10% is $99, but the ex-GST price was $100.
    • Using the new amount as the base. Percentage change is measured against where you started. 480 falling to 432 is 48 ÷ 480 = 10%, not 48 ÷ 432 = 11.1%.
    • Adding successive percentages. A 20% cut and then a 20% rise is 0.8 × 1.2 = 0.96, a 4% net loss. The percentages do not cancel, because each has a different base.
    • Percentage points read as percent. 72% rising to 81% is 9 percentage points but a 12.5% increase. If the question says "percentage points", subtract; if it says "percent increase", divide by the original.
    • "25% more than" treated as "25% of". 25% of $80 is $20. 25% more than $80 is $100. Read whether the question wants the slice or the whole bar with the slice added.
    • Adding a discount and GST instead of multiplying. 15% off then 10% GST is ×0.85 × 1.1 = ×0.935, a 6.5% net cut, not 5%. The order of the two does not change the answer, but adding the percentages does.
    • Rounding money too early. $1,240 ÷ 11 = 112.7272…; keep the full value while you work and round to $112.73 only at the end. Rounding in the middle can shift the final cent and miss the answer the test expects.

    ACER-style practice

    Eight questions in the formats the real test uses: multiple choice, type your answer, and true-or-false tables. Questions marked "no calculator" belong to the test's final section, where you cannot use one. Check each answer for a full explanation.

    0 of 8 checked

    What to remember

    • Every percentage is of a base. Find the base first: it is the number that counts as 100%.
    • One move for everything: answer = base × multiplier. 15% is ×0.15, 15% more is ×1.15, 15% off is ×0.85.
    • Going backwards, divide by the same multiplier: original = sale price ÷ 0.85.
    • Percentage change: (new − original) ÷ original × 100. The original is always the base.
    • GST: add with × 1.1, remove with ÷ 1.1, find the GST inside a total with ÷ 11.
    • Successive changes multiply: 0.8 × 1.2 = 0.96. Never add the percentages.
    • Percentage points subtract; percent increase divides by the original percentage.

    All questions on this page are original and are not reproduced from ACER material. LANTITE is administered by the Australian Council for Educational Research.