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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.

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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.

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i)

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$\\var{a}=$  [[0]] [[1]]

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ii)

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$\\var{b}=$  [[2]] [[3]]

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iii)

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$\\var{d}=$  [[6]] [[7]]

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iv)

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$0.\\dot{\\var{c}}=$  [[4]] [[5]]

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

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

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#### a)

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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.

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i)

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$\\var{a}$

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

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ii)

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$\\var{b}$

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

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iii)

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$\\var{d}$

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

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iv)

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

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

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\$x=0.\\dot{\\var{c}}\\text{.} \$

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

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\$10x=\\var{c}.\\dot{\\var{c}}\\text{.} \$

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

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\\\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} \

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

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#### b)

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$\\displaystyle\\var{f}$

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

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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 $\\var{mygcd}$.

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

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Simplifying by this amount gives the final answer

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\$\\frac{\\var{f3}}{\\var{f4}}.\$

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#### c)

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

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

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

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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

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

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

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\\\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} \

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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 $\\var{gcd1 }.$

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Therefore, it is not possible to simplify and so

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

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

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