Fractions, Decimals & Percentages

From GCSE(WIKI)

Fractions, Decimals and Percentages are three different forms for representing numbers. Sometimes, it may be easier to work in a specific form, so it useful to understand how to convert between them.

1 Fractions to Decimals
2 Decimal to Fractions
3 Fraction to Percentages

[edit] Fractions to Decimals

If you already understand some of the basics of algebra, you will understand that a horizontal line between two numbers means divide. Fractions use the exact same syntax so we can just divide to get a decimal number.

On a calculator, this is quite easy to do - just enter the numerator divided by the denominator:

$\frac{5}{8} = 5 \div 8 = 0.625$

If we do not have a calculator, we can use short or long division

[edit] Decimal to Fractions

[edit] Non-recurring decimals

If a decimal doesn't recur (keep repeating the same digits), we can follow this method:

Place the decimal over 1. For example, 0.21 becomes $\frac{0.21}{1}$
Multiply both the numerator and denominator by 10 until we get whole numbers and no decimal places:

$\frac{0.21}{1} = \frac{0.21 \times 10}{1 \times 10} = \frac{2.1}{10} = \frac{2.1 \times 10}{10 \times 10} = \frac{21}{100}$

[edit] Recurring decimals

If a decimal does recur, such as $0.\dot{3}$ and $0.\dot{2}\dot{7}$ , follow this method:

Shift the number left by the number of decimal places the recurring decimals occupy. $0.\dot{3}$ uses 1 decimal place. Shifting one to the left gives us $3.\dot{3}$ . Remember the number of places we shift by.
Subtract your second number by the original number: $3.\dot{3} - 0.\dot{3} = 3$ . This is our numerator: 3.
Our denominator begins with 9. Its total length is the number of places we shifted by (1). We add zeros as necessary. We add no zeros because the length is already 1 - our denominator is therefore 9. Our final answer then is $\frac{3}{9} = \frac{1}{3}$

We can also use some basic algebra to help us. This is how we would convert $0.\dot{2}\dot{7}$

$\begin{array}{rcr}x & = & 0.\dot{2}\dot{7} \\100x & = & 27.\dot{2}\dot{7} \\100x-x & = & 27.\dot{2}\dot{7} - 0.\dot{2}\dot{7} \\99x & = & 27 \\x & = & \frac{27}{99}{\ }& = & \frac{3}{11}\end{array}$

[edit] Advanced Decimals to Fractions

Sometimes, not all digits recur. We may get a number like $0.1\dot{3}$ . Before we start, we must get the recurring digits on their own. In this case, shifting to the left by 1 will place the 1 to the left of the decimal point, while leaving the recurring 3 to the right: $1.\dot{3}$ . We can then follow the first method:

Shift the number left by the number of decimal places the recurring decimals occupy. $1.\dot{3}$ uses 1 decimal place. Shifting one to the left gives us $13.\dot{3}$ . Remember the number of places we shift by - this is 1 plus the 1 we shifted by before we started - so, in total, we have shifted by 2.
Subtract your second number by the original number: $13.\dot{3} - 1.\dot{3} = 12$ . This is our numerator: 10.
Our denominator begins with 9. Its total length is the number of places we shifted by (2). We add zeros as necessary. We add 1 zero - our denominator is therefore 90. Our final answer then is $\frac{12}{90} = \frac{2}{15}$

Using Algebra:

$\begin{array}{rcr}x & = & 0.1\dot{3} \\10x & = & 1.\dot{3} \\100x & = & 13.\dot{3} \\100x-10x & = & 13.\dot{3} - 1.\dot{3} \\90x & = & 12 \\x & = & \frac{12}{90} \\{\ } & = & \frac{2}{15}\end{array}$