MISSION · Number sense
Percents as Parts of 100
Why “per cent” literally means “out of 100”
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LEVEL 2 · LEARN
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The idea in one sentence
A percent is a fraction that always has 100 on the bottom, so \(35\%\) means \(\frac{35}{100}\) — nothing more mysterious than that.
Why 100, every time
Percents exist to make things comparable. If one class got 17 out of 20 questions right and another got 42 out of 50, which did better? The numbers don’t line up. Turn both into “out of 100” and the answer is immediate: 85 and 84.
That’s the whole job of a percent. It’s a common denominator everyone agreed on in advance.
Three ways to find a percent of a number
All three give the same answer. Use whichever fits the numbers in front of you.
As a fraction. \(35\%\) of 80 is \(\frac{35}{100} \times 80 = 28\).
As a decimal. Move the decimal point two places left, then multiply: \(0.35 \times 80 = 28\).
From benchmarks, which is the one to do in your head. \(10\%\) of 80 is 8, so \(30\%\) is 24. \(5\%\) is half of \(10\%\), so 4. Add them: \(24 + 4 = 28\).
The benchmark method is worth practicing until it’s automatic. It’s how you check a tip, a discount, or a test score without reaching for anything.
Percent of what
This is the part that trips people, and it isn’t really about arithmetic.
A percent on its own is meaningless — it’s always a percent of some whole. Ten percent of your allowance and ten percent of a country’s population are the same fraction and wildly different amounts. Whenever a percent appears, find the whole it belongs to before you do anything else.
Common mix-ups
- Reading \(0.5\) as \(5\%\). It’s \(50\%\). Two decimal places, not one: \(0.05\) is the one that means \(5\%\).
- Adding percents of different wholes. \(20\%\) off a $50 shirt and \(20\%\) off a $200 coat don’t combine into \(40\%\) off anything.
- Assuming an increase and a decrease cancel. Take 100, add \(10\%\) to get 110, then subtract \(10\%\) — that \(10\%\) is now of 110, so you land on 99, not 100. The second percent measures a different whole than the first.
- Thinking a percent can’t pass 100. A price that triples rose by \(200\%\). Anything that can more than double can exceed \(100\%\).
Check yourself
What is \(25\%\) of 60? Then find \(15\%\) of 60 using benchmarks only.
\(25\%\) of 60 is 15, because \(25\%\) is one quarter. For \(15\%\): \(10\%\) of 60 is 6, half of that is 3, and \(6 + 3 = 9\).
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