Percentage Power

Every percentage is a multiplier

Percentage questions look like a dozen different topics: finding a share, adding a rise, taking off a discount, interest, working backwards. They are one idea wearing different clothes. A percentage change is a single number you multiply by. An increase of 10 per cent is a multiply by 1.1. A decrease of 10 per cent is a multiply by 0.9. To find the multiplier you turn the percentage into a decimal, then for a rise you add it to 1, and for a fall you take it away from 1. Once a change is just a multiply, the hard parts stop being hard. Two changes one after another are found by multiplying their multipliers, which is why they do not simply add. Working backwards to a starting value is just dividing by the multiplier, because dividing undoes multiplying. Carry one question through everything that follows: what is the multiplier, and which value is it acting on?

The words percentages run on

Five terms, each defined by what it is. The last one is where most marks are lost.

Tap the two multipliers above 1

A multiplier above 1 makes a value bigger. Tap the TWO changes that use a multiplier above 1.

• A rise of 15 per cent, which is a multiply by 1.15
• An increase of 100 per cent, which is a multiply by 2
• Taking 20 per cent off, which is a multiply by 0.8
• A fall of 50 per cent, which is a multiply by 0.5

Find a share of an amount

A jacket costs 60 pounds. To find 15 per cent of it, multiply 60 by 0.15. How much is 15 per cent of 60 pounds, in pounds?

Adding on against taking off

Both an increase and a decrease are one multiply. The only difference is whether the multiplier is above or below 1.

Match the change to its multiplier

Two things a multiplier makes easy

Select the TWO statements that are true once you treat a percentage change as a single multiplier.

• Two changes one after another can be done by multiplying their multipliers together
• A rise then a fall of the same per cent does not bring you back to the start
• Two percentage changes can always be added together to get the total change
• A bigger percentage always means a bigger actual amount

Answering a percentage question

A shape that works on almost any percentage question. Find the multiplier first. Turn the percentage into a decimal. For a rise, add it to 1. For a fall, take it from 1. This one number does the whole job. Decide which value it acts on. A percentage is always a share of something, so before you multiply, be clear which amount is the whole. To go forwards, multiply. The starting value times the multiplier gives the value after the change. To go backwards, divide. If you know the value after a change and want the original, divide by the multiplier, because dividing undoes the multiply. This is the reverse percentage, and it is where most marks are won or lost. Two habits cost marks. The first is adding percentages that should be multiplied, so two changes in a row are treated as one combined percentage, which is wrong. The second is the reverse trap: taking the percentage off the final value instead of dividing, which finds a share of the wrong amount. Always ask which value is the whole before you start.

Work back to the original

After a rise of 20 per cent a phone costs 240 pounds. The rise was a multiply by 1.2, so to undo it divide 240 by 1.2. What was the price before the rise, in pounds?

Up then down

A coat priced at 50 pounds goes up by 10 per cent, and then the new price is reduced by 10 per cent. Is the coat back to 50 pounds?

• No. Multiplying by 1.1 and then by 0.9 gives 0.99, so it ends at 49 pounds and 50 pence, just below the start
• Yes, because the same percentage went on and then came off
• No, it ends higher than 50 pounds
• It cannot be worked out without more information

Order the reverse percentage method

Put the steps of a reverse percentage problem into the right order, for a value that was increased by 20 per cent.

• Write down the multiplier that was applied, which for a rise of 20 per cent is 1.2
• See that the final value equals the original multiplied by that number
• Divide the final value by the multiplier to undo the change
• Read off the original value from that division
• Check by multiplying the original back up and seeing it matches the final value

Build the reverse rule

This is the rule for a reverse percentage. Assemble it.

One shop, a few price changes

Here is an invented example, written for practice. A shirt is priced at 20 pounds. In a sale it is reduced by 10 per cent. The number you multiply by is 0.9, and 20 times 0.9 is 18, so the sale price is 18 pounds. For a special event the shop then adds 10 per cent onto the sale price. The number to multiply by is 1.1, and 18 times 1.1 is 19 pounds and 80 pence. Notice it did not go back to 20 pounds. The rise added a share of 18 pounds, which is smaller than the 20 pounds the fall was taken from, so the two do not cancel. That is why changes are multiplied and not added. Now the other direction. A customer sees a jacket at 30 pounds on a rail marked as 20 per cent off, and wants to know what it cost before the sale. The 20 per cent was a share of the original, not of the 30 pounds, so you cannot just add 20 per cent onto 30. The sale was a multiply by 0.8, so to work backwards you divide: 30 divided by 0.8 is 37 pounds and 50 pence. That was the original price, and multiplying it by 0.8 gives 30 again, which checks out.

Complete the percentage facts

The single number you multiply by to carry out a percentage change is the _____. A change that makes a value larger, using a number above 1, is a percentage _____. A change that makes it smaller, using a number below 1, is a percentage _____. Finding the original value before a change, given the value after it, is a _____ percentage problem.

Three answers to fix

Three student answers about percentages. The arithmetic is not the problem; the reasoning is.

• A student works out a rise of 20 per cent then a fall of 20 per cent and says the price is back where it started. What is the flaw?
• To find the price before a rise of 25 per cent, a student takes 25 per cent off the final price. Why is that wrong?
• A student says 30 per cent of a number is always a bigger amount than 10 per cent of a number. How would you correct it?

Explain how percentages really work

A friend can find a simple percentage of an amount but keeps getting percentage change and reverse percentage questions wrong. Explain the single idea that ties all of these together.

• Explain what it means to treat a percentage change as one multiplier
• Explain how to find the multiplier for an increase and for a decrease
• Explain why two percentage changes one after another do not simply add together
• Explain how to work backwards to an original value using the multiplier
• Finish with the one question worth asking on any percentage problem, about the multiplier and which value it acts on