Question

How do you write \[\dfrac{280}{100}\] as a mixed fraction?

Answer 1

Hint: For converting the given equation \[\dfrac{280}{100}\] into a mixed fraction, we have to convert the given equation into a proper fraction. By using the greatest common factor we have to lower the terms of our fraction. Now we have to convert this to a mixed fraction by writing the divisor as denominator, remainder as numerator and quotient as the whole number part.

Complete step-by-step solution:
For converting a given fraction as a mixed fraction. Let us assume and consider the given equation as ‘A’ and equation (1) respectively.
Let us consider
\[A=\dfrac{280}{100}...........\left( 1 \right)\]
By observing equation (1) we can say that this fraction consists of two numbers and a fraction bar. The number above the bar is 280 (which is called numerator). The number below the bar is 280 (which is called denominator).
Now, we have to reduce the fraction in equation (1), we have to divide the numerator and denominator by their greatest common factor, GCF.
Let us build the prime factorizations of the numerator and denominator in the equation (1), we get
\[\Rightarrow A=\dfrac{2\times 2\times 2\times 5\times 7}{2\times 2\times 5\times 5}\]
Let us rewrite the above equation in exponential form.
\[\Rightarrow A=\dfrac{{{2}^{3}}\times 5\times 7}{{{2}^{2}}\times {{5}^{2}}}\]
Let us consider the above equation as equation (2).
\[\Rightarrow A=\dfrac{{{2}^{3}}\times 5\times 7}{{{2}^{2}}\times {{5}^{2}}}............\left( 2 \right)\]
Calculating the GCF, we get
Multiplying all the common prime factors, by the lowest exponents.
GCF\[\left( {{2}^{3}}\times 5\times 7;{{2}^{2}}\times {{5}^{2}} \right)={{2}^{2}}\times 5\].
Dividing both numerator and denominator with GCF to convert given improper function to proper function, we get
\[\Rightarrow A=\dfrac{\dfrac{{{2}^{3}}\times 5\times 7}{{{2}^{2}}\times 5}}{\dfrac{{{2}^{2}}\times {{5}^{2}}}{{{2}^{2}}\times 5}}\]
\[\Rightarrow A=\dfrac{2\times 7}{5}\]
\[\Rightarrow A=\dfrac{14}{5}\]
Therefore the fraction is reduced in lower terms, let us consider it as equation (3).
\[A=\dfrac{14}{5}............\left( 3 \right)\]
Now we have to write this improper fraction to mixed fraction. For that we have to write a whole number and a proper fraction, of the same sign.
\[14\div 5=2\] and remainder =4, we can write it as
\[\Rightarrow A=14\div 5\Rightarrow 2\times 5+4\]
\[\Rightarrow 2+4/5\]
\[\Rightarrow A=2\dfrac{4}{5}\]
Let us consider the above equation as equation (4).
\[\Rightarrow A=2\dfrac{4}{5}...............\left( 4 \right)\]
Therefore a mixed fraction of equation (1) is equation (4).

Note: In some problems we cannot convert the fraction into lower terms because numerator and denominator may be not divisible with the same numbers. Students may get confused in writing a mixed fraction i.e. \[A=4\dfrac{2}{5}\] which is wrong because in mixed fraction always quotients should be at outside the fraction, remainder at the numerator and divisor at the denominator.