Every percent question asks for one of three numbers
You know all three percent calculations: a part of a whole, finding the base and the number of percent. Word problems only wrap them in words — sometimes on purpose tangled. Today training in unwrapping.
The strategy has four steps:
1. Find the whole (the base). Of what are the percents calculated? The original price, the whole class, a full tank. It often sits after the word “of”.
2. Find what else you know. The part (in korunas, pupils, litres), or the percent (a number with a % sign)?
3. Decide what is missing. The part is missing → base times percent. The base is missing → the ladder via 1%. The percent is missing → part : whole × 100.
4. Calculate and check with an estimate. Did a discount come out bigger than the price? Percents over a thousand? There is a mistake somewhere.
Tangled questions have a favourite trick: they give you a number after a change and want the state before it. Watch for that — you spot them by the words “after a discount”, “after a rise”, “is left”.
Adam and Filip solve homework: “After 20% off a game costs 480 Kč. How much did it cost originally?” Filip fires: “20% of 480 is 96, so 576!” Adam shakes his head: “Check: 576 minus 20% of 576… that is 576 − 115.20 = 460.80. It does not fit.” Filip stops: “Ah, those 480 are already reduced. That is 80% of the original price.” The ladder: 480 : 80 = 6, times 100 = 600 Kč. Check: 600 − 120 = 480. That checks out. Adam: “The words ‘after a discount’ are a trap for fast shooters.”
For every question write three lines: whole = ?, part = ?, percent = ? Fill in what you know and you see at once which of the three calculations to use.
Unwrapping questions by the strategy
Type 1 — the part is missing. “There are 640 pupils in the school, 45% are boys. How many boys?” Whole 640, percent 45, part missing: 0.45 × 640 = 288.
Type 2 — the base is missing. “I saved 210 Kč, which is 30% of the price. How much does the thing cost?” Part 210, percent 30, whole missing: 210 : 30 × 100 = 700.
Type 3 — the percent is missing. “Of 25 tries, 19 worked. What is the success rate?” Part 19, whole 25: 19 : 25 × 100 = 76%.
Tangled variants and how to handle them:
“After p% off it costs…” → the given price is (100 − p)% of the base. The ladder from that, not from 100.
“The price rose by 15% to 460 Kč” → 460 is 115% of the original price.
“By how many percent…” → count the change, always divide by the original value.
Two changes one after another → multiply the coefficients (0.8 × 1.1), never add the percents.
A check is part of the solution: plug the result back into the story of the question and recalculate the forward way. With word problems the mistakes are in reading, not in multiplying — a check catches reading mistakes.
A tank is 35% full and contains 42 litres. How much does the whole hold?
Three lines: whole = ? (I look for it), part = 42 l, percent = 35.
The base is missing → the ladder via 1%.
Step 1: one percent: 42 : 35 = 1.2 litres.
Step 2: a hundred percent: 1.2 × 100 = 120 litres.
Check: 35% of 120 = 0.35 × 120 = 42. That checks out.
Result: the tank holds 120 litres. The word “of” pointed to the whole: full of 35 percent — the percents are calculated from the volume of the whole tank.
A hoodie rose from 800 Kč by 15%, then fell by 15%. How much does it cost?
A trap for adding: +15% and −15% do not cancel.
Step 1: after the rise: 800 × 1.15 = 920 Kč.
Step 2: the discount is calculated from the new price: 920 × 0.85 = 782 Kč.
Result: 782 Kč — less than the original 800! Why: 15% of 920 (= 138) is more than 15% of 800 (= 120). The discount took from a higher base than the rise added. The coefficients show the same faster: 1.15 × 0.85 = 0.9775, so 97.75% of the original price.
A test had 40 questions. Ema managed 34, Filip 80%. Who was better?
Convert to a common language — percents.
Ema: part 34, whole 40: 34 : 40 = 0.85 = 85%.
Filip: 80% given straight. (For interest: 0.8 × 40 = 32 questions.)
Result: Ema (85% against 80%). Word problems like mixing units of answers — pieces and percents. Before comparing, convert both values to the same: either both to percents, or both to questions. Mixing makes an argument, not a result.
A number in a question need not be the base. The words “after a discount”, “after a rise”, “is left” mean you are getting an already changed value. Who calculates percents from it directly calculates from the wrong whole. Always make it clear: is this number before the change, or after it?
Practice: now you do it 🎲
Now try it for real. You will see all the questions at once, like on paper. Mulai with ten. When it goes well, add more.
A quick warm-up: a percent of a number. Write only the number without %.
On paper: four tangled questions
For each write three lines (whole, part, percent), the type of question, the calculation and a check.
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A book costs 240 Kč after 25% off. How much did it cost before the discount and how much did you save?
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Of 720 cinema viewers, 288 were children. How many percent of the viewers were adults? (Watch, I ask about adults.)
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The price of a ticket rose by 10% to 66 Kč. What was the original price?
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A backpack at 1 500 Kč was first reduced by 20%, then raised by 20%. Calculate the final price and write in one sentence why 1 500 Kč did not come out.
Now you 💪
- For every question I fill the trio whole–part–percent and I decide what is missing.
- I recognise a value after a change and I calculate from it with the ladder, not directly.
- I check every result with a forward calculation.
Done when: Four questions have a trio, a type, a calculation and a check: 320 Kč, 60%, 60 Kč and 1 440 Kč with the right reason.
What to take from this lesson
- Three lines (whole, part, percent) decide the type of question.
- The words “after a discount / after a rise” = you get a changed value, go by the ladder.
- “By how many percent” = the change divided by the original value.
- Changes one after another multiply as coefficients, they do not add.
- A check by a forward calculation is part of the solution.
© 2026 Ing. Martin Polak / AlgoRhino · Ketentuan penggunaan konten