I look for the whole (the base) · 跟马丁学编程

You know the part and the percent, but not the whole. The way back goes via one percent.

Twenty percent is 90 Kč. How much was the whole?

Last time you knew the whole and looked for the part. Today the other way: you know the part and you know how many percent of the whole it is. You look for the whole. The whole in questions is called the base.

A life example: the deposit for camp is 90 Kč and the leader says that is 20% of the price. How much does the whole camp cost?

The path is the same as last time, just backwards. If 20% is 90 Kč, then 1% is 90 : 20 = 4.50 Kč. And the whole, that is 100%, is a hundred times that: 4.50 × 100 = 450 Kč.

Remember it as a ladder: from the percent you know, climb down to one percent (divide), and from one percent climb up to a hundred (times a hundred). Down by dividing, up by multiplying. This ladder will hold you even in questions where you are not sure what to divide by and what to multiply by.

Ema found a bike at a jumble sale with a tag: “After the discount only 30% of the original price — 1 800 Kč.” Jonáš whistles: “That must have been expensive. But how much did it cost then?” Ema takes the receipt and writes on the back: 30% … 1 800 Kč. One percent: 1 800 : 30 = 60. A hundred percent: 60 × 100 = 6 000 Kč. “Six thousand,” she says. Jonáš tries a check: 30% of 6 000 is 0.3 × 6 000 = 1 800. That checks out. “So a discount of 4 200. That already sounds like a reason it is hanging here.”

💡

Always write the pair you know: “20% … 90 Kč”. Then climb down to 1% by dividing and climb up to 100% by multiplying. The ladder will not let you fall.

The ladder via one percent

The method for finding the base:

Step 1: write what you know as a pair: number of percent … value. For example 20% … 90 Kč.

Step 2: climb down to one percent — divide the value by the number of percent: 90 : 20 = 4.5. Now you know: 1% … 4.50 Kč.

Step 3: climb up to a hundred percent — multiply by a hundred: 4.5 × 100 = 450. The base is 450 Kč.

Step 4: check the other way: 20% of 450 = 0.2 × 450 = 90. That checks out.

How do you know you are looking for the base? The question has a sentence like “that is 20% of the price” and the price itself is missing. The thing you look for is the whole, not a piece.

A faster writing for the confident: base = part : (percent as a decimal). So 90 : 0.2 = 450. It is the same ladder climbed in one jump — use it when it is clear.

The base is usually bigger than the part (if the percent is less than 100). If the base comes out smaller, something is wrong — most likely you multiplied where you should divide.

15% is 45. What is the base?

Step 1: pair: 15% … 45.

Step 2: one percent: 45 : 15 = 3.

Step 3: a hundred percent: 3 × 100 = 300.

Result: the base is 300. Check: 15% of 300 = 0.15 × 300 = 45. That checks out. Notice the base (300) is bigger than the part (45) — with percents under 100 that must be so.

A discount of 240 Kč was 30% of the price. How much did the goods cost?

Watch: 240 Kč here is not the new price, but the discount itself — so 30% of the base.

Step 1: pair: 30% … 240 Kč.

Step 2: one percent: 240 : 30 = 8.

Step 3: a hundred percent: 8 × 100 = 800.

Result: the original price was 800 Kč. After the discount the goods cost 800 − 240 = 560 Kč. Read questions carefully: the discount, the new price and the original price are three different numbers.

There are 6 left-handers in the class, which is 25% of the class. How many children are in the class?

Path via the ladder: 25% … 6 children. One percent: 6 : 25 = 0.24. A hundred percent: 0.24 × 100 = 24.

A faster path via an anchor: 25% is a quarter. If a quarter of the class are 4 left-handers… wait: a quarter is 6 children, so the whole is 4 × 6 = 24.

Result: there are 24 children in the class. With nice percents (25%, 50%, 20%, 10%) the path via a fraction is fastest: a quarter → times four.

⚠️

Do not calculate a percent of the part when you are looking for the whole. The mistake looks like this: “20% is 90, so I take 20% of 90.” No — 90 already is those 20%. The direction is the other way: from 90 you get to 1% by dividing and to the whole by multiplying.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. 开始 with ten. When it goes well, add more.

Look for the whole via the ladder: down to 1% by dividing, up to 100% by multiplying. Write only the number.

Practise the questions

On paper: the ladder there and back

For each question write the pair “percent … value”, then both ladder steps and a check.

Ten percent of an amount is 32 Kč. What is the whole amount? Check by calculating 10% of the result.
A deposit of 360 Kč is 40% of the trip price. How much does the trip cost and how much is left to pay?
Adam read 84 pages, which is 35% of the book. How many pages does the book have? Write a check too.
Invent your own question for finding the base with a nice percent (25% or 50%) and solve it via a fraction anchor.

Now you 💪

I recognise a base question: I know the part and the percent, I look for the whole.
I know the ladder: value divided by percent = 1%, then times 100.
I check every result by calculating the other way.

✅

Done when: All four questions have a written pair, both ladder steps and a check that fits.

What to take from this lesson

The base = the whole the percent is calculated from.
The ladder: divide by the number of percent → 1%, times 100 → the base.
Discount, new price and original price are three different numbers — read the question carefully.
With anchors (25%, 50%) the faster path is via a fraction.
The base must come out bigger than the part when the percent is less than 100.