Percentage Increases and Decreases

Worked example

1. 1

   $2400,\text{up }15\%$ — The question: 2400 goes up by 15%. We want the new amount, not just the increase.

2. 2

   $100\%+15\%=115\%$ — The new amount is the 100% you started with plus the 15% added on.

3. 3

   $115÷100=1.15$ — Divide by 100 to turn 115% into a decimal. This is the multiplier.

4. 4

   $2400\times 1.15=2760$ — Multiply the original by the multiplier. One multiplication gives the new amount.

Worked example

1. 1

   $\$60,20\%\text{ off}$ — The question. “Off” means the price goes down, so this is a decrease.

2. 2

   $100\%-20\%=80\%$ — After 20% is taken off, you keep 80% of the price.

3. 3

   $80÷100=0.8$ — Divide by 100 to get the multiplier. It is less than 1 because the price goes down.

4. 4

   $\$60\times 0.8=\$48$ — Multiply the original price by the multiplier. The sale price is $48.

Worked example

1. 1

   $\$80\text{to}\$92$ — The question: the price went up from $80 to $92. Find the percentage increase.

2. 2

   $\text{change}=92-80=12$ — Find the change first: new minus original. The price went up by $12.

3. 3

   $\frac{12}{80}=0.15$ — Divide the change by the original, $80. A percentage change is always compared with where you started.

4. 4

   $0.15\times 100=15\%$ — Multiply by 100 to turn the decimal into a percentage. So it is a 15% increase.

Worked example

1. 1

   $\text{change}=7.40-6.85=0.55$ — For a decrease, take the new price from the original, so the change is positive.

2. 2

   $\frac{0.55}{7.40}=0.074324\dots$ — Divide by the original, $7.40. Keep every digit on your calculator for now.

3. 3

   $0.074324\dots \times 100=7.4324\dots \%$ — Multiply by 100 to make it a percentage.

4. 4

   $7.43\%\text{ decrease}$ — Round only at the end. The first 3 significant figures are 7, 4, 3, and the next digit, 2, rounds down.

Worked example

1. 1

   $\text{up }20\%:100\%+20\%=120\%=1.2$ — The first change is an increase, so its multiplier is 1.2.

2. 2

   $\text{down }25\%:100\%-25\%=75\%=0.75$ — The second change is a decrease. It acts on the new amount, so it gets its own multiplier.

3. 3

   $1.2\times 0.75=0.9$ — One change after the other means multiplying by both. Multiply the two multipliers to get one multiplier for the whole journey.

4. 4

   $0.9=90\%$ — You end with 90% of what you started with.

5. 5

   $100\%-90\%=10\%\text{ decrease}$ — 10% has been lost, so the overall change is a 10% decrease.

6. 6

   $\$400\times 0.9=\$360$ — If the question gives a starting amount, say $400, multiply it by 0.9 to get the final amount.

Worked example

1. 1

   $2500\times 1.06=2650$ — First find Ali's new pay. A 6% rise means multiply by 1.06.

2. 2

   $\text{Sara’s new pay}=2650$ — They now earn the same, so Sara's new pay is also $2650.

3. 3

   $2650-2400=250$ — Sara's change: her new pay minus her original pay.

4. 4

   $\frac{250}{2400}\times 100=10.416\dots$ — Divide by Sara's original pay, $2400, not Ali's, because it is her rise.

5. 5

   $10.4\%$ — Round to 3 significant figures. Sara got a 10.4% rise.

Worked example

1. 1

   $20000\times 0.85\times 0.9$ — Apply the two known changes: down 15% is × 0.85, down 10% is × 0.9.

2. 2

   $=15300$ — This is the value after two years, just before the unknown change.

3. 3

   $\frac{16218}{15300}=1.06$ — Divide the final value by the value before the change. This gives the third multiplier.

4. 4

   $1.06=106\%$ — Multiply by 100: the value became 106% of what it was.

5. 5

   $106\%-100\%=6\%$ — Take away the 100% it started with. So x = 6.