Reverse Percentages

The sale price is half the original price.

What percentage of the original price remains?

The new wage is 110% of the original.

Reverse the multiplier for a 20% increase.

The reduced price is 80% of the original.

The new population is 105% of the original.

1. 1260\div1.05=1200.

Find the percentage still being paid.

1. 100\%-15\%=85\%.
2. 119\div0.85=140 pounds.

Use a multiplier of 1.125.

1. After a 12.5% rise, 90 cm is 112.5% of the original length, or 1.125 times the original. Divide by 1.125 to find 100%: 90\div1.125=80.
2. The original length was 80 cm.

£27 is the reduction itself.

1. £27 represents 30\% of the original.
2. 27\div0.30=90 pounds.

The given price represents 115% of the original.

1. The new price equals the old price multiplied by 1.15.
2. Reverse this by calculating 92\div1.15.

Recover the original price before finding the saving.

1. After 35% off, 65% of the original price remains. So £156 is 0.65 of the original price. Divide by 0.65 to find 100%: 156\div0.65=240. The original price was £240.
2. The saving was 240-156=84, so £84 was saved.

Recover the original weekly pay, then use the hours.

1. After an 8% rise, £453.60 is 108% of Milo’s original weekly pay, or 1.08 times as much. Divide by 1.08 to find the original weekly pay: 453.60\div1.08=420.
2. He works 35 hours, so his hourly pay before the rise was 420\div35=12, or £12.

Remove delivery before reversing the reduction.

1. Remove the £12 delivery charge first: 108-12=96. The reduced chair price was £96.
2. After a 20% reduction, 80% of the original price remains. So £96 is 0.80 of the original. Divide by 0.80 to find 100%: 96\div0.80=120. The original price was £120.

Convert the remaining volume and identify its percentage.

1. Convert the remaining water to litres: 1.02\times1000=1020 litres.
2. After a 15% loss, 85% of the original amount remains. So 1020 litres is 0.85 of the original. Divide by 0.85 to find 100%: 1020\div0.85=1200. The tank started with 1200 litres.

Divide by the increase multiplier, then round once.

1. Including a 12% charge, £47 is 112% of the price before the charge, or 1.12 times that price. Divide by 1.12 to find 100%: 47\div1.12=41.964285\ldots.
2. Round once, at the end: the price before the charge was £41.96 to the nearest penny.

What fraction of the normal price is the sale price?

1. The sale price is 1-\frac{1}{4}=\frac{3}{4} of the normal price.
2. \frac{1}{4} of the normal price is 255\div3=85 pounds.
3. The saving is \frac{1}{4} of the normal price, so £85 is saved. (The normal price is 4\times85=340 pounds.)

£63 is 15% of the old fee, not of the new fee.

1. £63 is 15\% of the old fee.
2. Old fee: 63\div0.15=420 pounds.
3. New fee: 420+63=483 pounds.

Equal final prices need not mean equal original prices.

1. After a 20% reduction, 80% remains. So £72 is 80% = 0.80 of the radio’s original price. Divide by 0.80 to find 100%: 72\div0.80=90. The original price was £90.
2. After a 40% reduction, 60% remains. So £72 is 60% = 0.60 of the speaker’s original price. Its original price was 72\div0.60=120, or £120.
3. The difference between the original prices is 120-90=30, or £30.

Undo the departures before undoing the percentage increase.

1. Add back the 23 members who left: 138+23=161. This is the membership after the increase, before anyone left.
2. A 15% increase makes the new membership 115% of the original, so 161 is 1.15 times the original. Divide by 1.15 to find 100%: 161\div1.15=140.
3. The club had 140 members before the increase.

Recover the correct original using the percentage left.

1. £96 is 60\% of the original price.
2. Correct original: 96\div0.60=160 pounds.
3. Jo is 160-134.40=25.60 pounds below the correct price.