Fractions, Decimals and Percentages

\(\frac{1}{2}, \frac{2}{3}, \frac{3}{8}\) are all proper fractions.

\(\frac{3}{2}, \frac{3}{3}, \frac{8}{3}\) are all improper fractions.

Convert the following to mixed fractions.

a. \(\frac{5}{3}\)

b. \(\frac{27}{6}\)

Solution

a. Converting \(\frac{5}{3}\) to mixed fraction.

Step 1: Divide the numerator by the denominator.

\[5\div 3=1\, r\, 2\]

where \(r\) signifies 'remainder'.

Step 2: since your whole number is \(1\) and your remainder is \(2\), you can place \(1\) at the left side. Then \(2\) becomes the new numerator while \(3\) is retained as the denominator, so that you have

\[\frac{5}{3}=1\frac{2}{3}\]

Step 3: Verify that no factor is common between the new numerator and the denominator. \(2\) and \(3\) are both prime numbers hence without a common factor.

Therefore, the improper fraction, \(\frac{5}{3}\), has the mixed fraction, \(1\frac{2}{3}\).

b. Converting \(\frac{27}{6}\) to mixed fraction.

Step 1: Divide the numerator by the denominator.

\[27\div 6=4\, r\, 3\]

where \(r\) signifies 'remainder'.

Step 2: since your whole number is \(4\) and your remainder is \(3\), you can place \(4\) at the left side. Then \(3\) becomes the new numerator while \(6\) is retained as the denominator, so that you have

\[\frac{27}{6}=4\frac{3}{6}\]

Step 3: Verify that no factor is common between the new numerator and the denominator. Between the numerator, \(3\), and the denominator, \(6\), there is common factor which is \(3\). Hence, simplify \(\frac{3}{6}\) by dividing by \(3\) to arrive at

\[4+\frac{3\div 3}{6\div 3}=4\frac{1}{2}\]

Since the current numerator and denominator cannot be further divided, therefore, the improper fraction, \(\frac{27}{6}\), has the mixed fraction, \(4\frac{1}{2}\).

0.5, 2.46, and 0.0057 are decimal numbers.

In the decimal 2.46, 2 is a whole number and is an integer while 0.4 (4 tenths) and 0.06 (6 hundredths) are non-integers because they are just parts of a whole number.

\(50\%\), \(40\%\), \(13\%\) are examples of percentages.

Actually \(50\%\) means

\[50\%=\frac{50}{100}=\frac{1}{2}\]

Likewise, \(40\%\) means

\[40\%=\frac{40}{100}=\frac{2}{5}\]

and \(13%\) means

\[13\%=\frac{13}{100}\]

Convert the following fractions to percentages:

a) \(\frac{1}{4}\)

b) \(2 \cdot \frac{1}{5}\)

Solution

a) Step 1: Multiply the fraction by 100%.

\[\frac{1}{4} \cdot 100\%\]

b) Step 2: Simplify the previous result, by dividing it by 4

\[\frac{1}{4} \cdot 100\% = \frac{100}{4} \% = \frac{\cancel{100}}{\cancel{4}} \% = 25\%\]

b) When you are to convert a mixed fraction to a percentage, you have an extra step. Firstly, convert the mixed fraction to an improper fraction:

\[2 \cdot \frac{1}{5} = \frac{(2 \cdot 5) +1}{5} = \frac{11}{5}\]

Now to the steps.

Step 1: Multiply the improper fraction by 100%; \(\frac{11}{5} \cdot 100\%\)

Step 2: Simplify by dividing by 5;

\(\frac{11}{\cancel{5}}\cdot \cancel{100}\% = 11 \cdot 20\%= 220\%\)

Convert the following percentages to fractions:

a) 40%

b) 120%

Solution

a) Step 1: Divide the percentage by 100%.

\(\frac{40\%}{100\%}\)

Step 2: Simplify until you can divide no further.

\(\frac{\cancel{40\%}}{\cancel{100\%}} = \frac{4}{10} = \frac{\cancel{4}}{\cancel{10}} = \frac{2}{5}\)

b) Step 1: Divide the percentage by 100%.

\(\frac{120\%}{100\%}\)

Step 2: Simplify until you can divide no further.

\(\frac{\cancel{120\%}}{\cancel{100\%}} = \frac{12}{10} = \frac{\cancel{12}} {\cancel{10}} = \frac{6}{5} = 1 \cdot \frac{1}{5}\)

Convert the following to decimal:

a) \(\frac{1}{5}\)

b) \(\frac{1}{8}\)

Solution

a)\(\frac{1}{5}\)

The numerator 1 is less than the denominator 5. So you add a 0 in front of 1 making it 10 but place a 0 and decimal point after it above the fraction as seen below,

\[0.\frac{10}{5}\]

Now the numerator is large enough to be divided by the denominator, hence you can divide;

\[0.\frac{10}{5} = \frac{\cancel{10}}{\cancel{5}}\]

Place your answer after the decimal point. Continue dividing if there is a remainder; but in this case, there is no remainder. Therefore:

\[0.2 \frac{1}{5}\]

So our answer is 0.2.

b) \(\frac{1}{8}\)

The numerator 1 is less than the denominator 5. So you add a 0 in front of 1 making it 10 but place a 0 and decimal point after it above the fraction as seen below;

\[0.\frac{10}{8}\]

10 divided by 8 is 1 remainder 2; write the 1 after the decimal point and leave the remainder on top of 10.

\[0.1 \frac{\cancel{10^2}}{\cancel{8}}\]

Next, you divide the remainder 2 by 8, 2 is less than 8 so you add another 0 beside it making it 20 and divide by 8;

\[0.1 \frac{\cancel{20^4}}{\cancel{8}}\]

Next, you divide the remainder 4 by 8, 4 is less than 8, so you add another 0 beside it making it 40 and divide by 8;

\(0.12 \frac{\cancel{40}}{\cancel{8}}\)

There is no remainder anymore, thus:

\[\frac{1}{8} = 0.125\]

So our answer is 0.125.

Convert the following to fraction:

a) 0.2

b) 0.125

Solution

a) Step 1: Determine how many decimal places; the d.p. of 0.2 is 1 because there is only one number that comes after the decimal point.

Step 2: Since it has 1 d.p, it means 10 is what you divide with.

Step 3: Remove the decimal point and divide the number by 10, 100, 1000 etc. depending on the d.p; when you remove the decimal point, the number you have is 02, but the 0 is actually omitted so you have 2. Now divide 2 by 10:

\(\frac{2}{10}= \frac{\cancel{2}}{\cancel{10}} = \frac{1}{5}\)

b) Step 1: Determine how many decimal places; the d.p of 0.125 is 3 because there are three numbers that come after the decimal point.

Step 2: Since it has 3 d.p, it means 1000 is the divisor.

Step 3: Remove the decimal point and divide the number by 10, 100, 1000, etc. depending on the d.p; when you remove the decimal point, the number you have is 0125, but the 0 is actually omitted so you have 125. Now divide 125 by 1000;

\(\frac{125}{1000}\)

Divide by 5:

\(\frac{\cancel{125}}{\cancel{1000}}=\frac{25}{200}\)

Continue dividing until you can no longer divide:

\(\frac{\cancel{125}}{\cancel{1000}}=\frac{25}{200} = \frac{5}{40} = \frac{1}{8}\)

Convert the following to percentage:

a) 0.7

b) 1.6

Solution

a) Multiply the decimal by 100%:

\(0.7 \cdot 100\% = 70\%\)

b) Multiply the decimal by 100%:

\(1.6 \cdot 100\% = 160\%\)

Convert the following to decimals:

a) 70%

b) 160%

Solution

a) Divide by 100%:

\(\frac{70\%}{100\%} = \frac{\cancel{70\%}}{\cancel{100\%}} = \frac{7}{10}\)

Convert the fraction to decimal using the steps explained earlier;

\(\frac{7}{10} = 0.70.7\)

b) Divide by 100%:

\(\frac{160\%}{100\%} = \frac{\cancel{160\%}}{\cancel{100\%}} = \frac{16}{10} = \frac{8}{5}\)

Convert the fraction to decimal using the steps explained earlier:

\(\frac{8}{5} = 1.6\)

A man earns £1000, and he spends £400 on accommodation.

a) Determine the fraction he spends on accommodation.

b) Express in decimal the portion spent on groceries if he spends £100 on that.

c) Determine the percentage of his income donates if he gives alms worth £50.

Solution

a) The fraction he spends on accommodation is calculated as

\(\frac{400}{1000} = \frac{\cancel{400}}{\cancel{1000}} = \frac{4}{10}=\frac{2}{5}\)

b) The portion spent on groceries in decimal is

\(\frac{100}{1000} = \frac{\cancel{100}}{\cancel{1000}} = \frac{1}{10}\)

Using the steps explained earlier, the fraction is converted to

\(\frac{1}{10} = 0.10.1\)

c) the percentage he gives to the poor is

\(\frac{50}{1000} \cdot 100\% = \frac{50}{10\cancel{00}}\cdot \cancel{100}\% = \frac{5}{10} \% = \frac{1}{2} \% = \frac{1}{5} \%\)

or this can be expressed as a decimal percentage by converting the fraction part to a decimal

\(\frac{1}{5} \% = 0.2\%\)