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مرکز یادگیری / ریاضی روزمره

How to calculate percentages: the three types of problem

How to solve percentage problems of all three kinds — a percentage of an amount, one number as a percentage of another, and percentage change — with checked examples.

2 دقیقه مطالعهبه‌روزرسانی ۱۱ آبان ۱۴۰۵نوشته Sumzo teamبازبینی: Sumzo team

“Per cent” means “out of 100”, so p%p\% is the fraction p100\frac{p}{100}. To solve a percentage problem, first decide which of three types it is — the method follows from that. This lesson shows how to solve percentage questions of each type, with a quick way to check every answer.

Which type of percentage problem is it?

  • “What is 15% of 80?” — a percentage of an amount. You know the percentage and the whole.
  • “37 is what percent of 40?” — one number as a percentage of another. You know the part and the whole.
  • “A price rises from 50 to 65. By what percent?” — percentage change. You know the old and new amounts.

1. A percentage of an amount

p% of N=p100×Np\% \text{ of } N = \frac{p}{100} \times N

مثال حل‌شده

Easy: 15% of 80

What is 15% of 80?

نمایش راه‌حل
  1. Convert 15% to a fraction and multiply.
    15100×80=12\frac{15}{100} \times 80 = 12
  2. Check with 10% and 5%: 10% of 80 is 8 and 5% is 4, and 8+4=128 + 4 = 12.

پاسخ: 12

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Large numbers work the same way: 20% of 200,000 is 0.2×200,000=40,0000.2 \times 200{,}000 = 40{,}000. A shortcut for 20% is to find 10% (move the decimal point one place) and double it.

2. One number as a percentage of another

percentage=partwhole×100%\text{percentage} = \frac{\text{part}}{\text{whole}} \times 100\%

مثال حل‌شده

Medium: a test score out of 40

You score 37 out of 40 on a test. What percentage is that?

نمایش راه‌حل
  1. Divide the part by the whole.
    3740=0.925\frac{37}{40} = 0.925
  2. Multiply by 100.
    0.925×100%=92.5%0.925 \times 100\% = 92.5\%
  3. Check: 92.5% of 40 is 0.925×40=370.925 \times 40 = 37 ✓.

پاسخ: 92.5%

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3. Percentage change

Always divide the change by the original amount, not the new one.

change=new−originaloriginal×100%\text{change} = \frac{\text{new} - \text{original}}{\text{original}} \times 100\%

مثال حل‌شده

Medium: from 50 to 65

A price rises from 50 to 65. What is the percentage increase?

نمایش راه‌حل
  1. Find the change.
    65−50=1565 - 50 = 15
  2. Divide by the original and multiply by 100.
    1550×100%=30%\frac{15}{50} \times 100\% = 30\%

پاسخ: A 30% increase

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مثال حل‌شده

Hard: working backwards from a sale price

After a 25% discount a coat costs 66. What was the original price?

نمایش راه‌حل
  1. After 25% off you pay 75% of the original price pp.
    0.75p=660.75p = 66
  2. Divide by 0.75.
    p=660.75=88p = \frac{66}{0.75} = 88
  3. Check: 25% of 88 is 22, and 88−22=6688 - 22 = 66 ✓.

پاسخ: The original price was 88.

بررسی‌شده با موتور ریاضی Sumzo

Common mistakes

  • Dividing by the new amount in a percentage change — always use the original.
  • Taking 25% off the sale price to find the original: 25% of 66 is not the discount that was taken.
  • Adding percentages of different amounts as if they were the same whole.
  • Forgetting to multiply by 100 when turning a decimal such as 0.925 into a percentage.

پرسش‌های متداول

How do you solve a percentage problem?

Decide which type it is: a percentage of an amount (multiply by p100\frac{p}{100}), a part as a percentage of a whole (divide, then multiply by 100), or a percentage change (divide the change by the original).

How do I turn a test score into a percentage?

Divide the score by the total and multiply by 100: 3740×100=92.5%\frac{37}{40} \times 100 = 92.5\%. The percentage calculator does this instantly.

Why does a 20% increase and a 20% decrease not cancel out?

The second percentage is taken of a bigger amount. 100 rises to 120, and 20% of 120 is 24, so it falls to 96.

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