Percentage Change and Multipliers

The multiplier must include all of the original amount as well as the increase.

1. After an 18\% increase, the new amount is 118\% of the original.
2. Divide by 100 to get the multiplier 1.18.

Subtract the percentage decrease from the original hundred per cent.

1. A 7.5\% decrease leaves 100\%-7.5\%=92.5\% of the amount.
2. Divide by 100 to turn the percentage into a multiplier: 92.5\div100=0.925.

Use a multiplier that includes the original amount.

1. The multiplier for a 15\% increase is 1.15.
2. The new amount is 240\times1.15= £276.

Find the percentage remaining before multiplying.

1. A 12\% decrease leaves 88\%, so use multiplier 0.88.
2. The new mass is 80\times0.88=70.4\text{ kg}.

Divide the change by the original value, not the final value.

1. The increase is 80-64=16.
2. The percentage increase is \frac{16}{64}\times100\%=25\%.

Express the fall as a fraction of the original number of visitors.

1. The decrease is 150-126=24.
2. The percentage decrease is \frac{24}{150}\times100\%=16\%.

The sale price represents the percentage remaining after the discount.

1. After a 20\% discount, the sale price is 80\% of the original price.
2. So original price \times0.8=72.
3. Undo the multiplication by dividing: 72\div0.8=90.
4. The original price was £90. (Adding 20\% of £72 is wrong: the 20\% was taken off the original price, not the sale price.)

Divide the final amount by the original amount.

1. The multiplier is \frac{322}{280}=1.15.
2. It represents a 15\% increase because the new amount is 115\% of the original.

You need 35\% of 60\% of the members, so multiply the two multipliers.

1. Adults make up 60\% of the members, a multiplier of 0.6.
2. Adult coaches are 35\% of the adults, so they are 0.35\times0.6=0.21 of all the members.
3. As a percentage, 0.21 is 21\% of the club members.

The second percentage is applied to the increased price, so the two changes do not cancel.

1. The combined multiplier is 1.2\times0.8=0.96.
2. The final price is 96\% of the original.
3. The overall decrease is 100\%-96\%=4\%.

The selling price is more than one hundred per cent of the cost price.

1. A profit of 35\% of the cost price means the selling price is 135\% of the cost price, a multiplier of 1.35.
2. So cost price \times1.35=81.
3. Divide to undo the multiplier: 81\div1.35=60.
4. The cost price is £60. (Taking 35\% off £81 gives the wrong answer, because the 35\% is a percentage of the cost price, not of £81.)

Calculate each final price in the stated order before comparing them.

1. Shop A charges 250\times0.7= £175.
2. Shop B charges (250-60)\times0.9= £171.
3. Shop B is cheaper by 175-171= £4.

Compare the tax part with the total after tax, rather than with the untaxed amount.

1. Imagine the price before VAT is £100. The VAT is 20\% of this, which is £20.
2. The price including VAT is 100+20= £120.
3. The VAT as a fraction of the price including VAT is \frac{20}{120}=\frac16.
4. \frac15 is the VAT as a fraction of the price before VAT, which is not what the question asks.

Choose a convenient value for the smaller amount, then use the larger amount as the comparison base.

1. Take B=100, so A=120. The difference is 20.
2. As a percentage of A, this is \frac{20}{120}\times100\%=16.666\ldots\%.
3. To one decimal place, B is 16.7\% less than A.

Divide the combined multiplier by the known first multiplier to find the second.

1. The first multiplier is 1.08. The second must be 0.864\div1.08=0.8.
2. A multiplier of 0.8 means 80\% remains.
3. The reduction is therefore 20\%, so p=20.

Find total income from both selling prices, then compare the total profit with the purchase cost.

1. The purchase cost is 40\times3= £120.
2. The remaining 10 mugs sell for 6\times0.75= £4.50 each, so total income is 30\times6+10\times4.5= £225.
3. The profit is 225-120= £105.
4. The percentage profit is \frac{105}{120}\times100\%=87.5\%.

Apply each percentage change only to its own share of the original budget.

1. Take the original budget as £100, with £60 for wages and £40 for equipment.
2. The new spending is 60\times1.08= £64.80 and 40\times0.95= £38.
3. The new total is £102.80, an increase of £2.80 on £100.
4. The overall percentage increase is 2.8\%.

Area is the product of the two dimensions, so multiply their change multipliers.

1. The length multiplier is 1.25 and the width multiplier is 0.88.
2. The area multiplier is 1.25\times0.88=1.10.
3. The area is 110\% of its original value, so the increase is 10\%.

The restoring multiplier is the reciprocal of the reduction multiplier.

1. The reduction multiplies the price by 0.82.
2. To get back to the original price we must undo this, so multiply by \frac{1}{0.82}=1.219512\ldots
3. A multiplier of 1.219512\ldots is an increase of 21.9512\ldots\%.
4. To one decimal place the increase needed is 22.0\%. (An 18\% increase is not enough: it is 18\% of the lower price, which is less than the amount taken off.)

Remove the unaffected delivery charge before reversing the percentage change.

1. The discounted merchandise costs 216-12= £204.
2. This is 85\% of its original price, so the original merchandise costs 204\div0.85= £240.
3. Adding the delivery charge gives the original order total 240+12= £252.