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

Increasing by 20% 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%. 30x 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 = x3 = = 36.

Decreasing by 20% 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 10 th block. 40/10 = 4 so each block (worth 10%) = 4. A decrease of 30% means takeaway 3 blocks. So, x= 40 – 3x4 = = x

Calculating 1% amounts 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%. 20 y First mark 20 at the 100 th block. 20/100 = 0.2 so each block (worth 1%) = 0.2. Increasing by 32% means adding 32 blocks. So y = x0.2 = 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%. 60 y First mark 60 in the 100 th 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 Amount Before, 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 100 th block. The end amount, y, (after the % increase) would go in the 121 st 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 x 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 100 th block and 30 in the last block. 30/121 = So, x = 100 x = Or $24.79 (2d.p.)

Finding original amount before a 37% decrease was taken off. x Again, the x is placed in the 100 th block, and the end amount, y, (after the % decrease) goes in the 63 rd 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 y

Example 2 – finding original amount. Task: Find the original amount for a t-shirt which was discounted by 45% to sell for $30. x 30 First mark x in the 100 th block. 45% decrease means 45 blocks are taken away. This leaves 55 blocks. Mark 30 in the 55 th block. 30/55 = (which is 1%) x = 100 x = or $54.55(2 d.p.)

Key Ideas When 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 alternative You 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 = 0.4 y = x0.4 = 46 (which is the same as 40x1.15) 100% $40 115% y