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Fractions can be equivalently represented as decimals and vice versa. One form may be more useful in a context than another and it is useful to practise how to change between them.

\n

Have a go at these questions involving fractions and decimals, remembering to write your answer in its simplest form.  

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Identify well-known fractional equivalents of decimals. Convert obscure decimals and recurring decimals into fractions.

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Express these common decimals as their fraction equivalent.

\n

i)

\n

$\\var{a}=$  [[0]] [[1]]

\n

ii)

\n

$\\var{b}=$  [[2]] [[3]]

\n

iii)

\n

$\\var{d}=$  [[6]] [[7]]

\n

iv)

\n

$0.\\dot{\\var{c}}=$  [[4]] [[5]]

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Convert this decimal to a fraction, giving your answer in its simplest form. 

\n

$\\displaystyle\\var{f} = $  [[0]] [[1]]

\n

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a)

\n

To convert a decimal into a fraction, firstly place the decimal over $1$ as a fraction, and then multiply both the numerator and denominator by $10$ for however many decimal places the decimal has. For example, if the decimal was $0.1$, you would multiply the fraction by $10$ as there is one decimal place. If the decimal was $0.01$, you would multiply it by $100$, as there are two decimal places.  

\n

i)

\n

$\\var{a}$

\n

\\[
\\frac{\\var{a}}{1}\\times\\frac{10}{10}=\\frac{\\var{10a}}{10}\\text{.}
\\]

\n

ii)

\n

$\\var{b}$

\n

\\[
\\frac{\\var{b}}{1}\\times\\frac{100}{100}=\\frac{\\var{100b}}{100}=\\simplify{{100b}/{100}}\\text{.}
\\]

\n

iii)

\n

\n

$\\var{d}$

\n

\\[
\\frac{\\var{d}}{1}\\times\\frac{10}{10}=\\frac{\\var{10d}}{10}=\\simplify{{10d}/{10}}\\text{.}
\\]

\n

iv)

\n

\n

$0.\\dot{\\var{c}}$

\n

To convert a recurring decimal to a fraction, the first step is to set up a simple equation where

\n

\\[
x=0.\\dot{\\var{c}}\\text{.}
\\]

\n

By multiplying both sides by $10$, we can gain another simple equation where

\n

\\[
10x=\\var{c}.\\dot{\\var{c}}\\text{.}
\\]

\n

By subtracting one equation from the other, we can find the fraction equivalent of the recurring decimal. 

\n

\\[
\\begin{align}
&&\\var{c}.\\dot{\\var{c}}&={10}x\\\\
-&&{0.\\dot{\\var{c}}}&=x\\\\
&&\\overline{\\qquad} & \\overline{\\qquad}\\\\
&&{\\var{c}}&=9x\\\\
\\\\
&&\\frac{\\var{c}}{9}&=x
\\end{align}
\\]

\n

$\\displaystyle\\frac{\\var{c}}{9}$ simplifies to $\\simplify{{{c}}/{9}}$ by dividing by $3$ and therefore, $0.\\dot{\\var{c}}=\\simplify{{{c}}/{9}}$ in its fractional form.}

\n

\n

b)

\n

$\\displaystyle\\var{f}$

\n

\\[
\\var{f}\\times\\frac{\\var{f1000}}{\\var{f1000}}=\\frac{\\var{f2}}{\\var{f1000}}\\text{.}
\\]

\n

From this, we can look to see if we can cancel down the fraction by finding the highest common divisor between the numerator and denominator. This is $\\var{mygcd}$.

\n

Therefore, it is not possible to simplify the answer any further and the final answer is

\n

Simplifying by this amount gives the final answer

\n

\\[\\frac{\\var{f3}}{\\var{f4}}.\\]

\n

c)

\n

$\\var{h}.\\dot{\\var{j}}\\dot{\\var{k}}.$

\n

To convert a recurring decimal to a fraction, the first step is to set up a simple equation where,

\n

$x=\\var{h}.\\dot{\\var{j}}\\dot{\\var{k}}.$

\n


By multiplying both sides by $100$ to isolate the recurring section on the left hand side of the decimal point, we can gain another simple equation

\n

$100x=\\var{h}\\var{j}\\var{k}.\\dot{\\var{j}}\\dot{\\var{k}}.$

\n

\n

Now that we have two equations in terms of $x$, we can subtract one from the other and solve to get a value of $x$.

\n

\\[
\\begin{align}
&&\\var{h}\\var{j}\\var{k}.\\dot{\\var{j}}\\dot{\\var{k}}&=100x\\\\
-&&\\var{h}.\\dot{\\var{j}}\\dot{\\var{k}}&=x\\\\
&&\\overline{\\qquad} & \\overline{\\qquad} 
\\\\
&&{{\\var{h}}\\var{j}\\var{k-h}}&=99x\\\\
\\\\
&&\\frac{\\var{numerator}}{\\var{g}}&=x\\text{.}\\\\
\\end{align}
\\]

\n

From this, we should look to see if it is possible to simplify by finding the greatest common divisor of the numerator and the denominator. The greatest common divisor is $\\var{gcd1 }.$

\n

Therefore, it is not possible to simplify and so

\n

Simplifying by this value gives the fraction $\\displaystyle\\simplify{{{numerator}}/{g}}$ and so  

\n

\\[
\\begin{align}
\\var{h}.\\dot{\\var{j}}\\dot{\\var{k}}=\\simplify{{{numerator}}/{g}}\\text{ in its fractional form.}\\\\
\\end{align}
\\]

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