Question

If a number is decreased by 10%, it becomes 54. By what per cent should it be increased to make it 72.

Answer 1

Hint: In order to solve this problem, we need to find the original number which gives 54 when decrease by 10%. After this, by using that number, we need to add another unknown percentage and equate to 72. We can see that the original number will lie between 54 and 72 as according to the first condition and less than 72 according to the second condition.

Complete step by step answer:
There is a number which when decrease by 10% gives 54.
So firstly we need to find that number.
Let the original number by x.
Therefore, by calculating the 10% of x we get,
10% of x = $\dfrac{10}{100}\times x=\dfrac{x}{10}$ .
We need to subtract this quantity as it is said that the quantity when decreased gives 54.
Therefore, the equation says that,
$x-\dfrac{x}{10}=54$ .
Solving this equation, we get,
$\begin{align}
  & \dfrac{10x-x}{10}=54 \\
 & 9x=540 \\
 & x=60 \\
\end{align}$
Therefore, the value of the number is 60.
Now we have said that by what per cent when increased it will make 72.
So let the percentage by which it will increase be $y$ .
The percentage increase of 60 is given by $\dfrac{y}{100}\times 60=\dfrac{3y}{5}$ .
We need to add this number to the original number and equate to 72 as it is said that when increased.
The equation forms as follows,
$60+\dfrac{3y}{5}=72$
By solving we get,
$\begin{align}
  & \dfrac{3y}{5}=72-60 \\
 & \dfrac{3y}{5}=12 \\
 & 3y=60 \\
 & y=20 \\
\end{align}$

Hence, when we increase the number 60 by 20% then the value will be increased by 72.

Note: We need to understand while assuming $y$ we are assuming the percentage therefore, we need to divide the number by 100 before multiplying by the original number. Similarly, in the first condition, we are given 10% so we need to divide the equation by 100 before applying. The same number which we assume as x is used in the second condition.