Working with Percentages

A visual representation

Calculating 10% amounts

• Here is a block representing 100% of an amount.

• Each block is worth 10%. We calculate this by taking the full amount and dividing by 10.

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Increasing by 20%

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•When we add 20% onto an amount, it is like adding 2 more blocks. Now there are 12 in total.•Each block is still worth 10% so the total length of the blocks is worth 120% or 100% +20%.•This is the same as multiplying by 1.2.

Example-increasing %

Task: Increase $30 by 20%.30 x

•First mark the 30 at the end of the first 10 blocks.•30/10 = 3 so each block (worth 10%) =3.•Increasing by 20% means add 2 blocks.•So, x = 30 + 2x3 = 30 + 6 = 36.

Decreasing by 20%

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•When we decrease an amount by 20%, it is like taking 2 blocks away from 10. So now there are 8 blocks left.•Each block is still worth 10%, so the length of the remaining blocks is worth 80% or 100%-20%.•This is the same as multiplying by 0.8.

Example-decreasing %

Task: Decrease $40 by 30%.

• First mark 40 at the 10th block.• 40/10 = 4 so each block (worth 10%) = 4.• A decrease of 30% means takeaway 3 blocks. • So, x= 40 – 3x4 = 40 -12 = 28.

40x

Calculating 1% amounts1 2 3 4 5 6 7 8 9 10

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100

•Here is another way of representing 100% of an amount. It is made up of 100 blocks. •Each block is worth 1%. This is calculated by dividing the full amount by 100.

Increasing by 37%

•When we increase by 37%, it is like adding 37 little blocks. Now there are 137 in total.•Each one is still worth 1% so the total amount of blocks is 137% or 100%+37%.•This is the same as multiplying by 1.37.

Example 2 – increasing %

Task: Increase $20 by 32%.•

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y

•First mark 20 at the 100th block.•20/100 = 0.2 so each block (worth 1%) = 0.2.•Increasing by 32% means adding 32 blocks.•So y = 20 + 32x0.2 = 26.2.

Decreasing by 37%

• When we decrease by 37%, it is like taking away 37 little blocks from 100. Now there are 63 left.

• Each block is still worth 1% so the total amount remaining is 63% or 100%-37%.

• This is the same as multiplying by 0.63.

Example 2 – decreasing by %

Task: Decrease $60 by 37%.

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y

•First mark 60 in the 100th block.•60/100=0.6 so each block (worth 1%) = 0.6.•Decreasing by 37% means takeaway 37 blocks.•So y = 60 – 37x0.6 =37.8.

Finding the Original AmountBefore, We knew: which number went with 100% (or the last block).We had to find out: what number went with y (which was the

result of some % increase or decrease).

Now,We know: y which is the result of some % increase or decrease.We have to find out: what number goes with x (which

represents 100%).x is used to stand for the original amount, and y for the amount

after % increase/decrease because x comes before y in the alphabet.

Finding the original amount before a 21% increase was added on.

• Recall the earlier example which showed that % increase meant blocks added onto the original 100.

•Here, the x is placed at the 100th block. The end amount, y, (after the % increase) would go in the 121st block.•To work out what 1% is worth, we would divide the end amount by 121.•Then x would be 100 x that number (which goes with 1%).•Dividing by 121 then multiplying by 100 is just the same as dividing by 1.21.

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y

Example 1 – finding original amount

Task: Find the original amount for a t-shirt which was increased by 21% to sell for $30.

x

30•First mark x in the 100th block and 30 in the last block.•30/121 = 0.247933......•So, x = 100 x 0.247933.... = 24.7933... Or $24.79 (2d.p.)

Finding original amount before a 37% decrease was taken off.

x

• Again, the x is placed in the 100th block, and the end amount , y, (after the % decrease) goes in the 63rd block.•This is because 100 blocks minus 37 blocks, leaves 63 blocks.•To find out what 1% is worth, take the end amount, y, and divide it by 63.•Then the original amount, x, is 100 x that number.•Dividing by 63 and then multiplying by 100 is just the same as dividing by 0.63.

y

Example 2 – finding original amount.

Task: Find the original amount for a t-shirt which was discounted by 45% to sell for $30.

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30

•First mark x in the 100th block.•45% decrease means 45 blocks are taken away. This leaves 55 blocks. Mark 30 in the 55th block.•30/55 = 0.545454.... (which is 1%)•x = 100 x 0.545454.... = 54.5454...or $54.55(2 d.p.)

Key IdeasWhen finding the end amount, y, (after a % increase or decrease) we

multiply by a number.When finding the original amount, x, (before a % increase or decrease)

we divide by a number.

It is helpful if you remember the tricks for multiplying and dividing by 10 or 100.

Multiplying a number by 10: move the decimal point 1 place to the right.

Dividing a number by 10: move the decimal point 1 place to the left.Multiplying a number by 100: move the decimal point 2 places to the

right.Dividing a number by 100: move the decimal point 2 places to the left.

Number Line alternativeYou can use a number line like this, which is divided into 10’s

and 1’s.For example, calculating the end amount when $40 is

increased by 15%.1% is equal to 40/100 = 0.4y = 40 + 15x0.4 = 46 (which is the same as 40x1.15)

100%

$40

115%

y