Ratio and percentages

Q: 5 people out of 20. What percent is that?

25% — 5 ÷ 20 = 0.25 → 25%. The whole here is the 20 people.

Q: What is ¥1000 with 30% off?

¥700 — 1000 × 0.3 = ¥300 off. 1000 − 300 = ¥700.

Q: Take another 10% off ¥700. What do you get?

¥630 — 700 × 0.1 = ¥70 off, which leaves ¥630. That is not the same as 40% off the original ¥1000 (¥600).

Q: A pre-tax price of ¥1200 plus 10% sales tax. What is the total?

¥1320 — 1200 × 1.1 = ¥1320.

Q: Which formula gives you the ratio?

the part you're comparing ÷ the whole — Ratio = the part you're comparing ÷ the whole. Whatever comes after the ÷ is the whole.

Where this is used

"30% off." "10% cash back." "45% approval rating." Most of the numbers you meet in real life are percentages. If you can't spot what the percentage is being taken of, you can walk away thinking you saved money when you actually lost it. Percentages are the math adults use more than any other.

Key points

Ratio = the part you're comparing ÷ the whole you're comparing it to

If 5 people out of 20 own a dog, then 5 ÷ 20 = 0.25. Multiply by 100 and you get 25%. The key question is always: what is the whole (the number after the ÷)? Get that wrong and your answer will be wrong every time.

5 people out of 20 = 1/4 = 25%

A percent is how many parts out of 100

Percent means "how many out of 100." Once you agree that the whole is always 100, you can compare things of completely different sizes side by side. That is what makes percentages so useful.

Favourite sport (we asked 40 people)

Always ask: a percent of what?

Take ¥1000 and knock off 30%: you get ¥700. Now take "another 10% off" and you get ¥630 — not ¥600, which is what a straight 40% off would give you. That's because the second 10% is taken from ¥700, not from the original ¥1000. The whole changed partway through.

What 1000 yen turns into

Percentages that go up

Add 10% sales tax to a pre-tax price of ¥1000 and you get ¥1100. You're multiplying the original by 0.1 and adding it on — in other words, 1000 × 1.1. Multiplying by 1.1 gets you there in one step. For 10% off, multiply by 0.9 instead.

Check yourself 5questions

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Q1 5 people out of 20. What percent is that?

Q2 What is ¥1000 with 30% off?

Q3 Take another 10% off ¥700. What do you get?

Q4 A pre-tax price of ¥1200 plus 10% sales tax. What is the total?

Q5 Which formula gives you the ratio?

Jobs that use this

Every job uses this. The ones below are just where it shows up most.

Banker / financial planner $75k

Retail sales planner $60k

Data analyst $95k

Banker / financial planner$75k

Interest rates, fees, rates of increase — all of it is percentages. If you can't explain what a percentage actually means, you can't protect your clients' money.

Maths used: RatiosPercentagesCompound interest

Retail sales planner$60k

How big a discount can you offer and still make money? You decide by working out both the discount rate and the profit margin.

Maths used: RatiosPercentagesProfit margin

Data analyst$95k

"5% click rate." "3% churn rate." Almost every metric in this job is a ratio. The mistake analysts watch for hardest is comparing two numbers that were measured against different wholes.

Maths used: RatiosPercentagesStatistics