 Comparing Percentages Revision | KS3 Maths Resources

## What you need to know

Things to remember:

• It doesn’t matter which number you choose as the percentage

$x\% \text{ of } y = y\% \text{ of } x$

We have two methods for finding percentage amounts:

To do use this method, you need to remember how to find some simple percentages.

Percentage Division

100% $\div1$

50% $\div2$

25% $\div4$

20% $\div5$

10% $\div10$

5% $\div20$

4% $\div25$

2% $\div50$

1% $\div100$

To do this method, we need to remember that we can add and subtract percentages to make other ones. This method is best when you don’t have a calculator.

$$9\%=5\%+4\%$$

$$28\%=20\%+10\%-2\%$$

Find 35% of 250

$$35\%=25\%+10\%$$

Step 2: Find the value of these percentages

$$25\%\text{ of } 250 = 250\div4=62.5$$

$$10\%\text{ of } 250 = 250\div10=25$$

$$35\%=25\%+10\%$$

$$35\%\text{ of } 250=62.5+25=87.5$$

Multiplication works too.

$$35\%=7\times5\%$$

Step 2: Find the value of these percentages

$$5\%\text{ of } 250 = 250\div20=12.5$$

$$35\%=7\times5\%$$

$$35\%\text{ of } 250=7\times12.5=87.5$$

Our second method is the most powerful can we can find any percentage amount in two steps. This is useful for decimal amounts and you have a calculator.

Method 2: Multiplying from 1%

Find 12.56% of 300.

Step 1: Divide the number you’re finding the percentage of by 100 to find 1%.

$$300\div100=3$$

$$1\%=3$$

Step 2: Multiply the value of 1% found in Step 1 by your starting percentage.

$$12.56\%=12.56\times1\%$$

$$12.56\%\text{ of }300=12.56\times3=37.68$$

We can now use these two methods to compare two percentage amounts.

Now that we have the strategies, we can look at some questions.

Which is bigger, 55% of 20 or 40% of 30?

Find 55% of 20

Method 1

$$55\%=50\%+5\%$$

Step 2: Find the value of these percentages

$$50\%\text{ of }20=20\div2=10$$

$$5\%\text{ of }20=20\div20=1$$

$$55\%=50\%+5\%$$

$$55\%\text{ of }20=10 + 1 =11$$

Find 40% of 30

Method 2

Step 1: Divide the number you’re finding the percentage of by 100 to find 1%.

$$30\div100=0.3$$

$$1\%=0.3$$

Step 2: Multiply the value of 1% found in Step 1 by your starting percentage.

$$40\%=40\times1\%$$

$$40\%\text{ of }30=40\times0.3=12$$

Which is bigger, 70% of 55 or 55% of 70?

Find 70% of 55

Method 1

$$70\%=50\%+20\%$$

Step 2: Find the value of these percentages

$$50\%\text{ of }55=55\div2=27.5$$

$$20\%\text{ of }55=55\div5= 11$$

$$70\%=50\%+20\%$$

$$70\%\text{ of }55= 27.5+11=38.5$$

Find 55% of 70

Method 2

Step 1: Divide the number you’re finding the percentage of by 100 to find 1%.

$$70\div100=0.7$$

$$1\%=0.7$$

Step 2: Multiply the value of 1% found in Step 1 by your starting percentage.

$$55\%=55\times1\%$$

$$55\%\text{ of }70=55\times0.7=38.5$$

So, these two percentages are the same. It is worth remembering that whenever you find a percentage of something, it is the same as using the number as the percent and the percent as the number.

55% of 20 = 20% of 55

78.5% of 24 = 24% of 78.5

13.35% of 72.8 = 72.8% of 13.35

## Example Questions

28% of 110

Method 1

$$28\%=25\%+2\%+1\%$$

Step 2: Find the value of these percentages

$$25\%\text{ of }110=110\div4=27.5$$

$$2\%\text{ of }110=110\div50=2.2$$

$$1\%\text{ of }110=110\div100=1.1$$

$$28\%=25\%+2\%+1\%$$

$$28\%\text{ of }110=27.5+2.2+1.1=30.8$$

27% of 120

Method 2

Step 1: Divide the number you’re finding the percentage of by 100 to find 1%.

$$120\div100=1.2$$

$$1\%=1.2$$

Step 2: Multiply the value of 1% found in Step 1 by your starting percentage.

$$27\%=27\times1\%$$

$$27\%\text{ of }120=27\times1.2=32.4$$

So, 27% of 120 is bigger than 28% of 110.

Neither is smaller, they’re equal! We have just swapped the numbers around.

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