Percentages
Percentages as parts per 100; fraction ↔ decimal ↔ percentage equivalence; find a percentage of a quantity; percentages in financial contexts — discounts, interest, budgeting.
A percentage is a fraction whose bottom number is always 100 — a way of measuring any amount against a fixed scale of a hundred equal parts.
What it means
Per cent literally means per hundred. When you say 25%, you mean 25 parts out of 100 — the same as the fraction 25/100 or the decimal 0.25. The % symbol is just shorthand for “divided by 100”.
Percentages are useful because they put everything on the same scale. A test where you scored 18 out of 24 and a test where you scored 30 out of 40 are hard to compare directly — but 75% and 75% are plainly equal. That is the whole point: rescale any part-of-a-whole so the whole is always 100.
Because a percentage, a fraction and a decimal are three faces of one number, you can move between them freely:
- Percentage → decimal: divide by 100 (shift the decimal point two places left).
40% = 0.40. - Decimal → percentage: multiply by 100.
0.7 = 70%. - Fraction → percentage: rewrite so the denominator is 100, or divide top by bottom and multiply by 100.
3/4 = 0.75 = 75%.
To find a percentage of a quantity, turn the percentage into a fraction or decimal and multiply. “20% of 60” means 0.20 × 60 = 12. A handy anchor: 10% is one tenth (÷10), 1% is one hundredth (÷100), and 50% is one half. You can build most everyday percentages from those — 30% is three lots of 10%, and 15% is 10% + 5%.
This matters most with money. A 25% discount on a £40 coat saves 0.25 × 40 = £10, so you pay £30. Interest works the same way: 3% interest on £200 in savings is 0.03 × 200 = £6 added each year.
Worked examples
1. Convert between forms.
2/5 → 0.4 → 40%
0.08 → 8%
65% → 0.65 → 13/202. Find 15% of 80 (build it from parts):
10% of 80 = 8
5% of 80 = 4 (half of 10%)
15% of 80 = 8 + 4 = 123. A £50 game is 30% off. What do you pay?
30% of 50 = 0.30 × 50 = 15 (the discount)
50 − 15 = 35 (the price you pay)4. You save £120 at 4% interest for one year.
4% of 120 = 0.04 × 120 = 4.80
Total after a year = 120 + 4.80 = £124.80The generative-art connection
A percentage is a share of a whole, and the clearest way to see a share is to draw the whole and colour the part. Picture a 10 × 10 grid — 100 cells, one whole. Shade 37 of them and you are literally looking at 37%. Change the number shaded and the picture changes in step: the art is the arithmetic.
This scales up in generative work. Take our dot-multiplier starburst: light up 25% of the dots in a ring and you can see a quarter of the pattern glow — and a child can predict how many that is before the reveal. Colour is percentage too. Saturation and lightness are each a position on a 0–100 scale, so a tool like hue-pulse is quietly moving a percentage every frame: set saturation to 50% and the colour sits exactly halfway between grey and full. Percentages let you dial a picture — brightness at 80%, one colour mixed 30% into another, a shape scaled to 150% of its size.
Common misconceptions
- ”% is a unit like cm.” It is not — it is a hidden “÷100”.
50%of something is not 50 of anything; it is half. - A percentage always needs an “of”.
20%on its own is just a ratio. “20% of 60” is the amount,12. Always ask: a percentage of what whole? - Percentages can pass 100%. 100% is the whole thing; 150% is one-and-a-half times it. “More than the original” is perfectly normal.
- A bigger percentage isn’t always a bigger amount. 50% of £4 (£2) is less than 10% of £40 (£4). The whole matters as much as the percentage.
What a learner should be able to solve
The benchmark for mastering this concept — the problems a child at this stage is expected to be able to work through.
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Fill in the missing forms so each row shows the same number as a fraction, a decimal, and a percentage: (a) `3/4 = ___ = ___%` (b) `___ = 0.6 = ___%` (c) `___ = ___ = 35%`.
Answer
(a) `3/4 = 0.75 = 75%`. (b) `3/5 = 0.6 = 60%`. (c) `7/20 = 0.35 = 35%`. Convert by making the denominator 100 or dividing top by bottom, then shifting the decimal point two places for the %.
Art hook A 3-row strip on Canvas: each row is a 10x10 grid (100 cells). Shade 75, 60, then 35 cells and print the fraction/decimal/percent beside it, so the three faces of one number line up as three pictures.
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Find each amount: (a) `20%` of `45` (b) `75%` of `120` (c) `8%` of `250`.
Answer
(a) `0.20 x 45 = 9`. (b) `0.75 x 120 = 90` (three-quarters). (c) `0.08 x 250 = 20`. Turn each percent into a decimal or fraction and multiply.
Art hook A dial: draw a full circle of 100 dots. Type a percentage and that many dots (from the top, clockwise) light up gold; a label shows 'p% of N = ...' so the glowing arc is the answer made visible.
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A `£60` jacket has `15%` off. How much do you save, and what is the sale price?
Answer
`15%` of `60`: `10%` of `60 = 6`, `5%` of `60 = 3`, so `6 + 3 = £9` saved. Sale price `60 - 9 = £51`.
Art hook A price tag that redraws: a bar of length 60 in blue; a 15% slice (length 9) peels off in red as 'saved', and the remaining 51 stays blue as 'you pay'. Drag a slider to change the discount %.
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Priya has `£40` pocket money for the month. She spends `25%` on a book and `10%` on snacks. How much does she spend in total, and how much is left?
Answer
`25%` of `40 = £10` (book); `10%` of `40 = £4` (snacks). Spent `10 + 4 = £14`. Left `40 - 14 = £26`. (You can also add the percentages: `25% + 10% = 35%`, and `35%` of `40 = £14`.)
Art hook A budget bar of length 40 that splits into coloured segments as you assign percentages: a 25% slice for 'book', a 10% slice for 'snacks', and the remaining 65% (`£26`) stays as 'left over' — a live pie/bar budget where the slices always sum to the whole.
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Which is the better deal on the SAME `£80` item: `25%` off, or a flat `£18` off? Show your reasoning.
Answer
`25%` of `80 = 0.25 x 80 = £20` off. `£20 > £18`, so the `25%` discount saves more (`£20` vs `£18`).
Art hook Side-by-side bars of length 80. On the left, a 20-unit slice peels off; on the right, an 18-unit slice peels off. The longer peeled slice glows to show the better saving.
Training exercises
Practice problems, easier first, that build toward the bar above. Each doubles as a seed for an interactive artwork.
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Say what each percentage means as 'parts per hundred': (a) `40%` (b) `9%` (c) `100%`.
Answer
(a) `40` parts out of `100`. (b) `9` parts out of `100`. (c) `100` out of `100` — the whole thing. The `%` sign just means 'out of 100'.
Art hook A single 10x10 grid where a slider from 0 to 100 shades that many cells, so any percentage becomes a filled-in square you can watch grow.
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Turn each percentage into a decimal: (a) `50%` (b) `7%` (c) `90%`. (Divide by 100 — shift the point two places left.)
Answer
(a) `0.5`. (b) `0.07`. (c) `0.9`. Each is the percentage with the decimal point moved two places left.
Art hook A number line from 0 to 1: type a percentage and a dot slides to its decimal position, labelled with both forms (e.g. 90% lands at 0.9).
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Turn each decimal into a percentage: (a) `0.3` (b) `0.05` (c) `1.2`. (Multiply by 100.)
Answer
(a) `30%`. (b) `5%`. (c) `120%` — yes, over 100% is allowed, it means more than one whole.
Art hook A ring of 100 dots. Feed a decimal and it lights that fraction of the ring; for 1.2 it fills the whole ring once (100%) then starts a brighter second lap (the extra 20%).
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Write each fraction as a percentage: (a) `1/2` (b) `1/4` (c) `1/10` (d) `1/5`.
Answer
(a) `50%`. (b) `25%`. (c) `10%`. (d) `20%`. Rewrite each with denominator 100: `50/100, 25/100, 10/100, 20/100`.
Art hook A pie chart that splits into equal slices: choose the denominator (2, 4, 5, 10) and one slice highlights, printing its percentage — a live 'unit fraction to percent' wheel.
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Find these friendly percentages using the anchors `10%` (÷10), `1%` (÷100) and `50%` (half): (a) `10%` of `70` (b) `1%` of `500` (c) `50%` of `36`.
Answer
(a) `70 ÷ 10 = 7`. (b) `500 ÷ 100 = 5`. (c) half of `36 = 18`.
Art hook A single anchor-strip: a bar of the whole with three buttons '÷10', '÷100', 'half'. Press one and the matching chunk (10%, 1% or 50%) snaps to size and prints its value, showing where each anchor lives on the whole.
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Build these percentages from the `10%` anchor: (a) `30%` of `50` (b) `70%` of `20` (c) `15%` of `40` (use 10% + 5%).
Answer
(a) `10%` of `50 = 5`, so `30% = 3 x 5 = 15`. (b) `10%` of `20 = 2`, so `70% = 7 x 2 = 14`. (c) `10%` of `40 = 4`, `5% = 2`, so `15% = 4 + 2 = 6`.
Art hook A tower of stacked 10%-blocks: set the whole, then stack blocks (each = 10% of it). Stacking 3 blocks shows 30%; a half-block adds the 5%, letting you build any percentage brick by brick.
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Spot the mistake: Sam says '`5%` of `200` is `100`, because you take away the zero.' Is Sam right? Find the correct answer.
Answer
Sam is wrong — that would be `50%`. `5%` means `5/100`, so `5%` of `200 = 0.05 x 200 = 10`. (Check: `1%` of `200 = 2`, and `5 x 2 = 10`.)
Art hook A 'guess vs truth' bar: Sam's claim (100) draws as one bar, the true value (10) as another, over the same whole of 200 — the gap between them makes the error obvious.
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In a class of `25` children, `60%` walk to school. How many children walk?
Answer
`60%` of `25 = 0.6 x 25 = 15` children. (Or `10%` of `25 = 2.5`, and `6 x 2.5 = 15`.)
Art hook A 5x5 grid of 25 child-dots. Turn on 60% of them (15 dots) in green for 'walk', the rest in grey — a pictogram where the coloured share equals the percentage.
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A `£30` book is reduced by `20%`. What is the sale price?
Answer
`20%` of `30 = 0.2 x 30 = £6` off. Sale price `30 - 6 = £24`.
Art hook A price-tag bar of length 30: a 20% slice (6 units) fades out as 'saved' and the remaining 24 stays solid as 'you pay', updating live with a discount slider.
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Two shops sell the same `£50` shoes. Shop A offers `30%` off; Shop B offers `£12` off. Which shop is cheaper?
Answer
Shop A: `30%` of `50 = £15` off, so `£35`. Shop B: `£12` off, so `£38`. Shop A is cheaper (`£35 < £38`).
Art hook Two shop-front bars of length 50 side by side; each peels its own discount slice (15 vs 12). The shorter remaining bar glows to crown the cheaper shop.
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You save `£250` at `2%` interest for a year. How much interest is added, and what is the new total?
Answer
`2%` of `250 = 0.02 x 250 = £5` interest. New total `250 + 5 = £255`.
Art hook A savings jar drawn as a filling bar: the base 250 in blue, then a thin 2% band (5 units) added on top in gold as the yearly interest, with the total labelled.
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Reasoning challenge: is `50%` of `£4` more or less than `10%` of `£40`? Work out both and explain.
Answer
`50%` of `£4 = £2`. `10%` of `£40 = £4`. So `10%` of `£40` is more, even though `50%` is the bigger percentage — the size of the whole matters too.
Art hook Two bars of very different lengths (4 and 40). Each highlights its own percentage chunk (2 vs 4); the smaller-percentage-but-bigger-chunk lights up to show percentages only make sense against their whole.