Decimal to Fraction | Fraction Proper and Improper | Multiplying Fractions

  • decimal to fraction
  • fraction to decimal
  • improper fraction
  • equivalent fractions
  • multiplying fractions
  • proper fraction

In this chapter we read about fractions and its related topic like decimal to fraction, fraction to decimal, Fraction Proper and Improper, equivalent fractions, multiplying fractions, proper fraction, adding fractions, dividing fractions etc. We teach about fraction concepts and solve questions related to fraction.

Each fraction topic we understand with examples and question. Our expert team choose many questions about fractions, which help you to get highest marks in your exam.

Fraction

a fraction is a number representing part of a whole. The whole may be a single object or a group of objects.
Example – \(\displaystyle \frac{3}{4},\frac{7}{4}\)
\(\displaystyle \frac{5}{{12}}\) is a fraction. We read it as “five-twelveth”.
in this fraction 12 stand for the number of equal parts into which the whole has bees divided.
5 stand for the number of equal parts which have been taken out.

Numerator and Denominator

Take a fraction \(\displaystyle \frac{5}{{7}}\) . In this fractions 5 is called the numerator and 12 is called the denominator.
Example – \(\displaystyle \frac{15}{{13}}\) in this fraction numerator is 15 and denominator is 13.
\(\displaystyle \frac{4}{{13}}\) in this fraction numerator is 4 and denominator is 13.

Proper Fraction and Improper Fraction

Proper Fraction

A proper fraction, is a number representing part of a whole. In a proper fraction the denominator shows the number of parts into which the whole is divided and the numerator shows the number of parts which have been considered.
Therefore, in a proper fraction the numerator is always less than the denominator.

Example – \(\displaystyle \frac{5}{7},\,\frac{{11}}{{13}},\,\frac{{19}}{{60}},\,\frac{5}{9}\)

Improper Fraction

The fractions, where the numerator is bigger than the denominator are called improper fractions.

Example – \(\displaystyle \frac{7}{5},\,\frac{{11}}{{3}},\,\frac{{9}}{{5}},\,\frac{7}{3}\)

Mixed Fraction

A mixed fraction has a combination of a whole and a part.

Example – \(\displaystyle 1\frac{5}{7},\,2\frac{{11}}{{13}},\,5\frac{{19}}{{60}},\,11\frac{5}{9}\)

How to Change Mixed Fraction in to Improper Fraction

First of all we multiple denominators with a whole number
\(\displaystyle 2\frac{5}{7}\)
Here we multiple 7 with 2 = 7 × 2 = 14
Then we add a numerator to the product.
14 + 5 = 19
This is our numerator of improper fractions and we put the same denominators in the fraction.
Now improper fraction is = \(\displaystyle \frac{{19}}{7}\)

Thus, we can express a mixed fraction as an improper fraction as
\(\displaystyle \frac{{(Whole\,\times \,Denominator)\,+\,Numerator}}{{Denominator}}\)

Example – Express the following as improper fractions :
\(\displaystyle 4\frac{5}{7}\,=\,\frac{{33}}{7}\)
\(\displaystyle 4\frac{1}{5}\,=\,\frac{{21}}{5}\)
\(\displaystyle 1\frac{8}{3}\,=\,\frac{{11}}{3}\)

Changing Improper Fraction in to Mixed Fraction

We can express an improper fraction as a mixed fraction by dividing the numerator by denominator to obtain the quotient and the remainder. Then the mixed fraction will be written as \(\displaystyle \text{Quotient}\frac{{\text{Remainder}}}{{\text{Divisor}}}\)

Example –
Express the following as mixed fractions :
\(\displaystyle \frac{{17}}{3}\)
\(\displaystyle \begin{array}{l}3\overset{5}{\overline{\left){{17}}\right.}}\\\,-\underline{{15}}\\\,\,\,\,\,\,\,2\end{array}\)5 whole and \(\displaystyle \frac{2}{3}\) more
So, mixed fraction is \(\displaystyle 5\frac{2}{3}\)

\(\displaystyle \frac{{12}}{7}\)
\(\displaystyle \begin{array}{l}7\overset{1}{\overline{\left){{12}}\right.}}\\\,\,\,-\underline{7}\\\,\,\,\,\,\,\,5\end{array}\)
1 whole and \(\displaystyle \frac{5}{7}\) more
So, mixed fraction is \(\displaystyle 1\frac{5}{7}\)

Equivalent Fractions

Each proper or improper fraction has many equivalent fractions. To find an equivalent fraction of a given fraction, we may multiply or divide both the numerator and the denominator of the given fraction by the same number.

To find an equivalent fraction of a given fraction, you may multiply both the numerator and the denominator of the given fraction by the same number.
Example –
Find the equivalent fractions of \(\displaystyle \frac{2}{3}\)
\(\displaystyle \frac{{2\times 2}}{{3\times 2}}\,=\,\frac{4}{6}\) , \(\displaystyle \frac{{2\times 3}}{{3\times 3}}\,=\,\frac{6}{9}\) ,\(\displaystyle \frac{{2\times 4}}{{3\times 4}}\,=\,\frac{8}{{12}}\)

To find an equivalent fraction, we may divide both the numerator and the denominator by the same number.
Example –
Find the equivalent fractions of \(\displaystyle \frac{12}{15}\)
\(\displaystyle \frac{{12\div 3}}{{15\div 3}}\,=\,\frac{4}{5}\)

Simplest Form of a Fraction

A fraction is said to be in the simplest (or lowest) form if its numerator and denominator have no common factor except 1.
Example – Find the simplest form of \(\displaystyle \frac{36}{54}\)
\(\displaystyle \frac{{36\div 18}}{{54\div 18}}\,=\,\frac{2}{3}\)
Now 2 and 3 have no common factor except 1; we say that the fraction \(\displaystyle \frac{2}{3}\) is in the simplest form.

Like Fractions and Unlike Fractions

Like Fractions

Fractions with same denominators are called like fractions.
Example –
\(\displaystyle \frac{2}{7},\,\frac{8}{7},\,\frac{3}{7},\,\frac{9}{7}\)

Unlike Fractions

Fractions with different denominators are called unlike fractions.
Example –
\(\displaystyle \frac{{12}}{7},\,\frac{8}{{17}}\)

Fraction to Decimal

Convert Fraction to Decimal Value

\(\displaystyle \frac{8}{{10}},\,\frac{9}{{10}},\,\frac{{18}}{{10}},\,\frac{{21}}{{10}}\)
The denominator of all the fractions given above is 10 i.e all the factors have tenth value. In a decimal number the place of tenth is immediately right of the decimal point. On this basis the above given factors can be expressed in decimal numbers as follows:
\(\displaystyle \begin{array}{l}\frac{8}{{10}}\,=\,0.8\\\frac{9}{{10}}\,=\,0.9\\\frac{{18}}{{10}}\,=\,1.8\\\frac{{21}}{{10}}\,=\,2.1\end{array}\)

It is clear the above examples – “Every fraction with the denominator 10 can be converted to decimal number by putting the decimal point after one leaving one digit from the right in the numerator.”
Example – 
\(\displaystyle \begin{array}{l}\frac{3}{{10}}\,=\,0.3\\\frac{{91}}{{10}}\,=\,9.1\\\frac{{118}}{{10}}\,=\,11.8\\\frac{{2251}}{{10}}\,=\,225.1\end{array}\)

Similarly “even fraction with 100 as denominator can be converted to decimal number by putting a decimal point after leaving two digits from the right of the numerator.”
Example –
\(\displaystyle \begin{array}{l}\frac{2}{{100}}\,=\,0.02\\\frac{{86}}{{100}}\,=\,0.86\\\frac{{218}}{{100}}\,=\,2.18\\\frac{{2251}}{{100}}\,=\,22.51\end{array}\)

If a fraction with denominator 1000 is to be changed in a decimal value then pit a decimal point leaving three digits from the right of the numerator.”
Example – 
\(\displaystyle \begin{array}{l}\frac{7}{{1000}}\,=\,0.007\\\frac{{28}}{{1000}}\,=\,0.028\\\frac{{725}}{{1000}}\,=\,0.725\\\frac{{7859}}{{1000}}\,=\,7.859\end{array}\)
It is clear from the above example that while converting the fractions into decimal numbers with denominators 100 and 1000, put as many zeros which are less while counting the digits from the right of the numerator to put a decimal point.”

Let’s convert the following fractions into decimal numbers –
\(\displaystyle \frac{4}{5},\,\frac{9}{2},\,\frac{7}{{25}},\,\frac{9}{4},\,\frac{5}{8}\)
If the denominators of the fractions are 2, 5 or their multiples, then they can be easily converted to equivalent fractions with denominator 10, 100 or 1000. Then these fractions can be converted to decimal numbers by the above mentioned methods.
\(\displaystyle \begin{array}{l}\frac{4}{5}\,=\,\frac{{4\times 2}}{{5\times 2}}\,=\,\frac{8}{{10}}\,=\,0.8\\\frac{9}{2}\,=\,\frac{{9\times 5}}{{2\times 5}}\,=\,\frac{{45}}{{10}}\,=\,4.5\\\frac{7}{{25}}\,=\,\frac{{7\times 4}}{{25\times 4}}\,=\,\frac{{28}}{{100}}\,=\,0.28\\\frac{9}{4}\,=\,\frac{{9\times 25}}{{4\times 25}}\,=\,\frac{{225}}{{100}}\,=\,2.25\\\frac{5}{8}\,=\,\frac{{5\times 125}}{{8\times 125}}\,=\,\frac{{625}}{{1000}}\,=\,0.625\end{array}\)

Converting Decimal Numbers into Fractions

We have seen that to convert a fraction with denominator as 10 or 100 into decimal number we count two or three digits respectively from the right of the numerator and put a decimal point. On the other hand to convert a decimal number into a fraction we count the number of digits at the tenth, hundredth, or the thousandth place after decimal point and accordingly put the 10, 100 or 1000 respectively in the denominator and remove the decimal point.
Example –
\(\displaystyle \begin{array}{l}\frac{4}{5},\,\frac{9}{2},\,\frac{7}{{25}},\,\frac{9}{4},\,\frac{5}{8}\\4.9\,=\,\frac{{45}}{{10}}\,=\,\frac{9}{2}\\6.45\,=\,\frac{{645}}{{100}}\,=\,\frac{{129}}{{20}}\\0.05\,=\,\frac{5}{{100}}\,=\,\frac{1}{{20}}\\5.25\,=\,\frac{{525}}{{100}}\,=\,\frac{{105}}{{20}}\,=\,\frac{{21}}{4}\end{array}\)

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