Mixing decimals, fractions, percentages and ratios

As we can convert decimals to fractions or to percentages or to ratios, we can do operations (add, subtract, divide, multiply) on these different ways of representing numbers and proportions by converting between them.

Here are some examples:

$0.5 +\frac{3}{4}$

$=\frac{5}{10} + \frac{3}{4}$

$= \frac{1}{2} + \frac{3}{4}$

$= \frac{2}{4} + \frac{3}{4}$

$= \frac{5}{4}$

$= 1 \frac{1}{4}$

Here is an example question involving mixing calculations with fractions and percentages in which we introduce a diagram called a frequency tree.

A survey was carried out in which 750 people were asked if they exercise. $\frac{6}{10}$ people say “Yes”.
30% of the people who say “Yes” exercise twice a week or more.

Complete the frequency tree

To calculate how many exercise:

$\frac{6}{10} \times 750 = \frac{750\times 6}{10} = \frac{750}{10} \times 6 = 75 \times 6$

We can do the multiplication in the way we learned earlier in the course.

This means that $750 - 450 = 300$ don’t exercise.

Another way to do this is to first work out how many don’t exercise. $\frac{4}{10}$ don’t exercise.

$\frac{4}{10} \times 750 = \frac{750\times 4}{10} = \frac{750}{10} \times 4 = 75 \times 4$

This calculation can be used as a check on the 300 that we worked out above.

We can fill in some of the frequency tree now.

Now we need to calculate how many of the people who say that they exercise, do so twice a week or more. This is 30% of 450.

$\frac{30}{100} \times 450 = \frac{3}{10} \times 450 = 3 \times \frac{450}{10} = 3 \times 45$

We can calculate what 3 times 45 is

So 135 people exercise twice or more a week. This means that $450 - 135 = 315$ people don’t.

We can do a check of this number as we know that 70% of people don’t exercise twice a week.

$\frac{70}{100} \times 450 = \frac{7}{10} \times 450 = 7 \times \frac{450}{10} = 7 \times 45$

This confirms that we have done the calculations correctly.

Here is another question. Now we have completed the frequency tree, what percentage of people surveyed exercise twice or more a week?

The fraction of people who exercise twice a week or more is $\frac{135}{750}$

To convert the fraction in to a percentage, the denominator needs to be converted into 100

To change 750 into 100, we need to divide it by 7.5

To transform the fraction $\frac{135}{750}$ into a percentage, we need to divide 135 by 7.5

It is difficult dividing by a number with a decimal in in. We can change it into a number without a decimal. To do this we could multiply 7.5 by 2, for example. Or by 10. Whatever we choose to multiply the divisor by, we need to do the same to the dividend.

Let’s multiply the divisor and dividend by 10 as multiplying by 10 is easy.

75 doesn’t divide into 1 or 13 but it goes once into 135

Then we multiply the 1 by 75 and take it away from 135 to leave 60

Bring down the zero and then we need to find out how many 75s there are in 600

One way to find out how many times a number goes into another number is the “doubling technique.” You just double the number and then keep doubling it until you get to or above the dividend.

By doubling 15 to 30 to 60 to 120, we can see there that 8 x 15 = 120

$\frac{1350}{75} = 18$

$\frac{135}{750} = \frac{18}{100} = 18%$

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