Reverse Percentages

Worked example

1. 1

   $100\%-25\%=75\%$ — The question: after 25% off, a price is $51. The original is always 100%. 25% was taken off, so $51 is 75% of the original.

2. 2

   $\text{multiplier}=0.75$ — 75% as a decimal is 75 ÷ 100 = 0.75. This is the number the original was multiplied by.

3. 3

   $0.75\times \text{original}=51$ — Write the equation: the original times the multiplier gave the sale price. This line alone earns a mark for working.

4. 4

   $\text{original}=51÷0.75$ — Multiplying by 0.75 got you to $51, so divide by 0.75 to go back. Dividing undoes multiplying.

5. 5

   $=\$68$ — The normal price. It is bigger than $51, which makes sense: the price went down to reach $51.

6. 6

   $\text{check: }68\times 0.75=51$ — Multiply forwards. You get the sale price back, so $68 is right.

Worked example

1. 1

   $100\%+20\%=120\%$ — The question: a salary went up 20% to $31 200. The old salary is 100%. 20% was added, so $31 200 is 120% of it.

2. 2

   $\text{multiplier}=1.2$ — 120% as a decimal is 120 ÷ 100 = 1.2. A multiplier bigger than 1 means an increase.

3. 3

   $1.2\times \text{old}=31200$ — Write the equation: the old salary times 1.2 gave the new salary.

4. 4

   $\text{old}=31200÷1.2$ — Divide by 1.2 to undo the multiply and go back to the start.

5. 5

   $=\$26000$ — The old salary. It is smaller than $31 200, which makes sense before a rise.

6. 6

   $\text{check: }26000\times 1.2=31200$ — Multiply forwards. You get the new salary back, so the answer is right.

Worked example

1. 1

   $100\%-30\%=70\%$ — The question: after 30% off, a coat costs $63. So $63 is 70% of the normal price.

2. 2

   $70\%=\$63$ — Write down what you know. This is the line that earns the first mark.

3. 3

   $10\%=63÷7=\$9$ — 70% is 7 lots of 10%, so divide by 7 to find 10%.

4. 4

   $100\%=9\times 10=\$90$ — 100% is 10 lots of 10%, so multiply by 10. This is the normal price.

5. 5

   $\text{check: }90-27=63$ — 30% of $90 is $27, and $90 − $27 = $63. It works.

Worked example

1. 1

   $35\%\text{ of a number is }91$ — The question. 91 is only part of the number, so the number must be bigger than 91.

2. 2

   $0.35\times \text{number}=91$ — 35% as a decimal is 0.35. “35% of” means “0.35 times”.

3. 3

   $\text{number}=91÷0.35$ — Divide by 0.35 to undo the multiply.

4. 4

   $=260$ — The whole number. Check: 0.35 × 260 = 91.

Worked example

1. 1

   $\$900=12\%\text{ of the old salary}$ — The question: a pay rise of $900 was a 12% rise. The $900 is the 12% itself, not the new salary.

2. 2

   $0.12\times \text{old}=900$ — 12% as a decimal is 0.12. 12% of the old salary is $900.

3. 3

   $\text{old}=900÷0.12$ — Divide by 0.12 to undo the multiply.

4. 4

   $=\$7500$ — The old salary. Check: 0.12 × 7500 = 900.

5. 5

   $\text{new}=7500+900=\$8400$ — Only if the question asks for the new salary: add the rise to the old salary.

Worked example

1. 1

   $\text{up }10\%:\times 1.1$ — The question: a price went up 10%, then down 20%, and is now $264. Up 10% means 110%, so the first multiplier is 1.1.

2. 2

   $\text{down }20\%:\times 0.8$ — Down 20% leaves 80% of the new price, so the second multiplier is 0.8.

3. 3

   $1.1\times 0.8=0.88$ — Multiply the two multipliers. This gives one multiplier that does both changes at once.

4. 4

   $0.88\times \text{start}=264$ — Write the equation: the start price times 0.88 gave $264.

5. 5

   $\text{start}=264÷0.88$ — Divide by 0.88 to undo both changes in one go.

6. 6

   $=\$300$ — The start price. Check: 300 × 1.1 = 330, then 330 × 0.8 = 264.

Worked example

1. 1

   $\text{Ali}=A$ — The question: Ali, Bo and Cy share $620. Bo gets 80% of Ali's share. Cy gets 30% more than Ali. Everyone is compared with Ali, so call Ali's share A.

2. 2

   $\text{Bo}=0.8A$ — 80% of Ali's share is 0.8 times A.

3. 3

   $\text{Cy}=1.3A$ — 30% more than Ali means 100% + 30% = 130%, which is 1.3 times A.

4. 4

   $A+0.8A+1.3A=3.1A$ — Add the three shares. Together they are 3.1 lots of Ali's share.

5. 5

   $3.1A=620$ — The three shares make the total, $620. This equation earns a mark for working.

6. 6

   $A=620÷3.1=\$200$ — Divide by 3.1 to find Ali's share.

7. 7

   $\text{check: }200+160+260=620$ — Bo gets 0.8 × 200 = $160 and Cy gets 1.3 × 200 = $260. They add up to $620.