Question

If four-fifths of a number is greater than three-fourths of the number by 4. Find the number.

Answer 1

Hint: Let the required number be ‘x’. The terminology four-fifth means the fraction $\dfrac{4}{5}$ and four-fifths of a number means that a number is multiplied by the fraction $\dfrac{4}{5}$ and similarly three-fourths of the number means that a number is multiplied by the fraction $\dfrac{3}{4}$. We have to multiply the number ‘x’ by the fractions and apply the condition given in the question to get the value of ‘x’.

Complete step-by-step answer:
Let the number be ‘x’. We have to apply the steps mentioned in the question one by one.
Consider the phrase four-fifths of a number. It means that the assumed number ‘x’ is multiplied by the fraction $\dfrac{4}{5}$. Let the four-fifths of the number be ‘a’. Numerically, we can write
$\begin{align}
  & a=\left( \dfrac{4}{5} \right)\times \left( x \right) \\
 & a=\dfrac{4x}{5}\to \left( 1 \right) \\
\end{align}$
Consider the phrase three-fourths of a number. It means that the assumed number ‘x’ is multiplied by the fraction $\dfrac{3}{4}$. Let the four-fifths of the number be ‘b’. Numerically, we can write
$\begin{align}
  & b=\left( \dfrac{3}{4} \right)\times \left( x \right) \\
 & b=\dfrac{3x}{4}\to \left( 2 \right) \\
\end{align}$
In the question it is given that four-fifths of a number is greater than three-fourths of the number by 4. This means that ‘a’ is greater than ‘b’ by 4. Numerically, we can write it as
$\begin{align}
  & a=b+4 \\
 & a-b=4 \\
 & \\
\end{align}$
Substituting the values of ‘a’ and ‘b’ from equations (1) and (2), we get
$\dfrac{4x}{5}-\dfrac{3x}{4}=4$
Taking ‘x’ outside in L.H.S gives us
$\left( \dfrac{4}{5}-\dfrac{3}{4} \right)\left( x \right)=4$
Taking LCM of the denominator and solving the fractions gives us
$\begin{align}
  & \left( \dfrac{\left( 4\times \dfrac{20}{5} \right)-\left( 3\times \dfrac{20}{4} \right)}{20} \right)\left( x \right)=4 \\
 & \left( \dfrac{\left( 4\times 4 \right)-\left( 3\times 5 \right)}{20} \right)\left( x \right)=4 \\
 & \left( \dfrac{16-15}{20} \right)\left( x \right)=4 \\
 & \left( \dfrac{1}{20} \right)\left( x \right)=4 \\
\end{align}$
Multiplying by 20 on both sides, we get
$\begin{align}
  & 20\times \left( \dfrac{1}{20} \right)\left( x \right)=4\times 20 \\
 & x=80 \\
\end{align}$
$\therefore $ The required number is equal to 80.

Note: There is a chance of mistake by students by considering the statement in the question as two fractions multiplied to the number ‘x’ is equal to 4. That is $\dfrac{4}{5}\times \dfrac{3}{4}\times \left( x \right)=4$ , this assumption leads to a wrong answer. So, to solve questions of this kind, we have to understand the statement properly and we can check backwards if the solution we got is right or not. In this question, with the answer 80, we can check backwards.
Four-fifths of 80 is $\dfrac{4}{5}\times 80=4\times 16=64$
Three-fourths of 80 is $\dfrac{3}{4}\times 80=3\times 20=60$
We can see that $64-60=4$ satisfies the condition given in the question.