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Level 8Grade 8skill: proportion· 9 min read

Proportion: Scaling Quantities

In short: Direct proportion keeps the ratio constant (more input, more output — scale by the bigger fraction); inverse proportion keeps the product constant (more workers, fewer days — divide instead). Before scaling, ask whether more input makes the answer rise or fall; the same discipline handles percentages once you name which base each rate acts on.

Two proportions, one question to ask first

Proportion is the machinery for scaling a known relationship up or down. Every proportion problem is one of two kinds, and mixing them up is the single most expensive mistake at this level. So before you touch a calculator, ask one question: when the input goes up, does the answer go up or down?

If the answer rises with the input, it is direct proportion — twice the flour makes twice the muffins, three times the distance costs three times the fuel. Here the ratio stays fixed, so you multiply by a fraction bigger than 1 when you want more.

If the answer falls as the input rises, it is inverse proportion — more workers means fewer days, faster speed means less time. Here the product stays fixed (workers × days is constant), so you multiply by the reciprocal. The numbers look almost the same; the arithmetic runs the opposite way.

Percentages ride on the very same machinery, which is why they turn up in this lesson too. A percent is just a ratio pinned to a particular base — 20% means 20 per 100 of something — so a discount, a tax or a price change is a proportion in disguise. The one extra discipline they demand is naming which base the rate acts on, because successive percentages each land on a different one. Keep that in mind and the worked examples below will move smoothly from scaling recipes to pricing a sale.

  1. Name the two quantities and ask: does more of one give more, or less, of the other?
  2. Direct → the ratio A/B is constant. Inverse → the product A×B is constant.
  3. Find the constant from the known pair (either the ratio or the product).
  4. Use that constant to solve for the asked amount — not the unit amount, the amount you were actually asked for.
  5. For a percentage, name the base it acts on before you multiply — after a discount, that base has already changed.
  6. Sanity-check the direction: your answer should move the way real life moves.

Why the same numbers give two different answers Why it works

Direct
Ratio stays constant: A/B is fixed. Scale by (new input ÷ old input).

350 g flour makes 12 muffins. For 30 muffins: 350 × (30/12) = 875 g. More muffins, more flour — the fraction 30/12 is bigger than 1.

Inverse
Product stays constant: A×B is fixed. Scale by (old input ÷ new input).

6 workers finish a wall in 20 days. For 15 workers: (6×20)/15 = 8 days. More workers, fewer days — you divide by the bigger number, not multiply.

Percent base
A percentage is a rate on a specific base, not a portable number. Successive percentages act on different bases, so they never simply add — not even when the two rates match.

A $1500 item is cut 10%, then marked up 10% again: 1500 × 0.90 × 1.10 = $1485, not back to $1500. The rise acts on 1350, not 1500, so down-then-up still loses money.

Direct proportion holds the ratio still; inverse proportion holds the product still. Whenever you are stuck, compute both the ratio and the product of your known pair, decide which one physically must stay constant, and the correct operation picks itself. Percentages are a special case of the same discipline — a percent is a ratio locked to one base, so the only extra question they add is which base the rate acts on. The trap questions in every CREST paper are built by hoping you scale a ratio when the product is what is really fixed, or that you read a rate off the wrong base.

Worked examples

6 workers build a wall in 20 days, all working at the same rate. How long will 15 workers take? = 8 days (the naive direct answer 20 × 15/6 = 50 days scales the wrong way).
  1. More workers should mean fewer days — this is inverse proportion.
  2. The product workers × days is constant: 6 × 20 = 120 worker-days of effort.
  3. With 15 workers: 120 ÷ 15 = 8.
  4. Check the direction: 15 > 6, so the time should drop below 20 — and 8 does.
A recipe uses 350 g of flour to make 12 muffins. How much flour is needed for 30 muffins? = 875 g of flour.
  1. More muffins need more flour — direct proportion, ratio flour/muffins is fixed.
  2. Scale by 30/12 = 2.5.
  3. 350 × 2.5 = 875.
  4. Check: per muffin is 350/12 ≈ 29.17 g, and 29.17 × 30 = 875 g.
A jacket priced at $2000 goes up by 25% for the season, then sells in a festival sale at 20% off. What is the final price? = $2000 (a 25% rise followed by a 20% cut cancels out).
  1. Apply the 25% rise first: 2000 × 1.25 = 2500.
  2. Apply the 20% discount to the NEW price, not the old one: 2500 × 0.80 = 2000.
  3. So the final price is $2000 — exactly the start.
  4. Insight: +25% then −20% is not +5%, because the 20% acts on the larger base 2500. On different bases, percentages never simply add.
A $1200 backpack is marked down 20%, and loyalty members get a further 10% off at the till. Is that the same as a flat 30% off? = $864 (a 20% then 10% discount is not 30% off — it is 28%).
  1. First discount: 1200 × 0.80 = 960.
  2. The second 10% comes off the already-reduced price, not the original: 960 × 0.90 = 864.
  3. A flat 30% off would be 1200 × 0.70 = 840 — actually cheaper for the customer.
  4. The gap appears because the 10% acts on 960, not 1200: stacked discounts land on shrinking bases, so 20% then 10% is only 28% off overall.

Four ways this goes wrong

Scaling a ratio when the product is fixed

✗ 6 workers take 20 days, so 15 workers take 20 × 15/6 = 50 days.

This treats a workers-and-days problem as direct proportion. But adding workers cannot make a job take longer — the physical quantity that stays constant is the product (worker-days), not the ratio. Using direct proportion where inverse is needed flips the answer the wrong way.

Fix: Ask whether more input gives more or less output. Fewer days from more workers means inverse: keep 6 × 20 = 120 constant, so 120 ÷ 15 = 8 days.

wrong_operation

Finding the unit and forgetting to scale

✗ 5 pens cost $90, so 8 pens cost $18.

$18 is the price of ONE pen (90 ÷ 5), not eight. The unitary method has two steps — find one, then multiply up — and stopping after the first step answers a question nobody asked.

Fix: Always scale the unit to the amount requested: one pen is $18, so 8 pens cost 18 × 8 = $144.

partial_computation

Applying a percentage to the wrong base

✗ A $1500 shirt has 20% off, then 5% sales tax: sales tax = 5% of 1500 = $75, so total = 1200 + 75 = $1275.

sales tax is charged on the price the customer actually pays, which is the discounted $1200 — not the original $1500. Taking the percentage from the wrong base quietly overcharges by $15 here, and much more on bigger numbers.

Fix: Settle the base first. Discounted price = 1500 × 0.80 = 1200; sales tax = 5% of 1200 = $60; total = $1260.

base_confusion

Treating a rate as a raw number

✗ Sales tax is 8% on a $250 bill, so tax = $8 and the total is $258.

8% is a rate, not eight dollars. Reading the percent as an absolute amount ignores the base it is supposed to act on, and the error grows with the bill.

Fix: A percentage always multiplies its base: tax = 8% of 250 = 0.08 × 250 = $20, so the total is $270.

percent_as_whole

Practice: scale it the right way

Work through a mixed set of direct and inverse proportion problems with discounts, sales tax and successive percentages. For each one, decide first whether the ratio or the product stays constant, then confirm the percentage is charged on the correct base before you commit to an answer.

Keep going

First, make sure of

Frequently asked questions

How do I quickly tell direct from inverse proportion?

Ask what happens to the answer when the input grows. If it grows too (more muffins need more flour), it is direct and the ratio stays constant. If it shrinks (more workers means fewer days), it is inverse and the product stays constant. Computing both the ratio and the product of your known pair, then deciding which one physically must stay fixed, settles it every time.

Can I just add successive percentages, like +25% then −20% giving +5%?

No. Percentages act on their base, and successive percentages act on different bases. A $2000 price rising 25% becomes 2500, and the 20% cut applies to 2500, not 2000: 2500 × 0.80 = 2000. The two moves cancel exactly, not to +5%. Always multiply the factors (1.25 × 0.80 = 1.00) rather than adding the rates.

When there is a discount and sales tax, which comes first and on what base?

Apply the discount first, then charge sales tax on the discounted price, because that is the amount the customer actually pays. For a $8000 phone with 15% off and 18% sales tax: 8000 × 0.85 = 6800, then 6800 × 1.18 = $8024. Charging sales tax on the original $8000 uses the wrong base and overstates the total.

Is the unitary method direct or inverse proportion?

The classic 'find one, then multiply up' unitary method is direct proportion — the per-unit ratio is constant. But the unitary idea also works for inverse problems if you find the total product first (like worker-days) and then divide. The key is to keep the correct quantity constant: the ratio for direct, the product for inverse.

What is the most common trap in proportion questions?

Scaling a ratio when the product is what is actually fixed — the direct-versus-inverse mix-up. A problem like '6 workers take 20 days, how long for 15?' tempts you into 20 × 15/6 = 50 days, but more workers cannot make a job longer. The product 6 × 20 = 120 is constant, so the answer is 120 ÷ 15 = 8 days.