Percentages

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Quick recall

• Always use the correct whole or original value as the denominator.
• For successive changes, increase = positive; decrease = negative.

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Formula Cards

1. Percentage of a whole

$$\text{Percentage}=\frac{\text{Part}}{\text{Whole}}\times100$$

2. Percentage change

$$\%\Delta=\frac{\text{New}-\text{Old}}{\text{Old}}\times100$$

Use the original value as the denominator.

3. Find X% of N

$$\text{Value}=\frac{X}{100}N$$

4. Recover the whole

$$N=\frac{\text{Value}\times100}{X}$$

5. Two successive changes

$$\text{Net change (\%)}=a+b+\frac{ab}{100}$$

Use positive numbers for increases and negative numbers for decreases.

6. Three or more changes

$$\text{Final}=\text{Initial}\prod_{i=1}^{n}(1+p_i)$$

Here each p is a signed decimal: +20% becomes +0.20; −5% becomes −0.05.

7. Net change from growth factors

$$\text{Net change (\%)}=\left[\prod_{i=1}^{n}(1+p_i)-1\right]100$$

8. Constant spending after a price rise

$$\text{Consumption reduction (\%)}=\frac{X}{100+X}\times100$$

Price rises by X%.

9. Constant spending after a price fall

$$\text{Consumption increase (\%)}=\frac{X}{100-X}\times100$$

Price falls by X%, with X less than 100.

Successive Changes — Cases

Example 1

Question: Apply successive increases of 20% and 10%.

Step 1: Use positive signed changes.

$$a=20,\qquad b=10$$

Step 2: Apply the shortcut.

$$20+10+\frac{20(10)}{100}=32\%$$

Answer: 32% increase. Growth-factor check: 1.20 × 1.10 = 1.32.

Example 2

Question: Apply a 25% rise followed by a 20% fall.

Step 1: Use a negative value for the fall.

$$a=25,\qquad b=-20$$

Step 2: Apply the shortcut.

$$25-20+\frac{25(-20)}{100}=0\%$$

Answer: No net change. Successive percentages generally do not cancel; calculate the factors.

Example 3

Question: Apply successive decreases of 10% and 20%.

Step 1: Use negative signed changes.

$$a=-10,\qquad b=-20$$

Step 2: Apply the shortcut.

$$-10-20+\frac{(-10)(-20)}{100}=-28\%$$

Answer: 28% decrease.

Example 4

Question: A value rises 20%, rises 10%, then falls 5%. Find net change.

Step 1: Multiply the three growth factors.

$$1.20(1.10)(0.95)=1.254$$

Step 2: Subtract 1, then convert to percentage.

$$(1.254-1)100=25.4\%$$

Answer: 25.4% increase.

Worked Examples

Example 1

Question: A student answers 45 out of 60 questions correctly. Find the percentage.

Step 1: Identify part and whole.

$$\text{Part}=45,\qquad\text{Whole}=60$$

Step 2: Divide and multiply by 100.

$$\frac{45}{60}\times100=75\%$$

Answer: 75%.

Example 2

Question: A price rises from Rs. 200 to Rs. 250. Find the percentage increase.

Step 1: Find the increase.

$$250-200=50$$

Step 2: Divide by the original price.

$$\frac{50}{200}\times100=25\%$$

Answer: 25% increase.

Example 3

Question: Find the overall discount after successive discounts of 10% and 20%.

Step 1: Convert discounts into remaining-price factors.

$$1-0.10=0.90,\qquad1-0.20=0.80$$

Step 2: Multiply the factors.

$$0.90\times0.80=0.72$$

Step 3: Subtract from the original whole.

$$(1-0.72)\times100=28\%$$

Answer: 28% discount.

Example 4

Question: Find 15% of 800.

Step 1: Convert the percentage into a fraction.

$$\frac{15}{100}$$

Step 2: Multiply by 800.

$$\frac{15}{100}\times800=120$$

Answer: 120.

Example 5

Question: 30% of a number is 90. Find the number.

Step 1: Set up the equation.

$$0.30N=90$$

Step 2: Divide by 0.30.

$$N=\frac{90}{0.30}=300$$

Answer: 300.

Example 6

Question: 30% of a number is 8 more than 40% of 100. Find the number.

Step 1: Calculate the amount on the right.

$$0.40\times100+8=48$$

Step 2: Solve for the number.

$$0.30N=48\quad\Rightarrow\quad N=160$$

Answer: 160.

Drill

Try each question before opening its answer.

Question 1

What is 20% of 150?

• Show answer

  30

Question 2

30 is what percent of 120?

• Show answer

  25%

Question 3

A number increased from 80 to 100. Find % increase.

• Show answer

  25%

Question 4

A price fell from 500 to 400. Find % decrease.

• Show answer

  20%

Question 5

If 40% of a number is 60, find the number.

• Show answer

  150

Question 6

Two successive increases of 10% and 10%

• Show answer

  net %? — 21%

Question 7

A student scored 68 out of 80. Find percentage.

• Show answer

  85%

Question 8

If the price of an item increases by 25%, by what % must it be reduced to return to original?

• Show answer

  20%

Question 9

15% of 15% of 2000 = ?

• Show answer

  45

Question 10

A salary increased by 20% then decreased by 20%. Net change?

• Show answer

  -4%

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