Question

If a family spends on food, housing and clothing in the rate of 5:3:2 and experiences the rise in prices of these heads by 40, 30 and 20 percent respectively, the final budget will be increased by:
(a) $33\%$
(b) $40\%$
(c) $37\%$
(d) None of the above

Answer 1

Hint: First of all, we will convert the ratio of spending on food, housing and clothing i.e. 5:3:2 into numbers by multiplying each ratio by x so the amount spent on food, housing and clothing is 5x, 3x and 2x respectively. Then, we are going to find the increase in food, housing and clothing expenditure by using the formula $a\left( 1+\dfrac{b}{100} \right)$. In this formula, $a$ is the expenditure and $b$ is the percentage increase. Then add all the food, housing and clothing increased expenditure and then find the percentage increase.

Complete step by step solution:
The ratio of expenditure on food, housing and clothing is given in the above problem as follows:
5:3:2
Now, converting the above ratio into number, we are going to multiply the ratio by x we get,
$\Rightarrow 5x,3x,2x$
Adding all the three expenditures on food, housing and clothing we get,
$\begin{align}
  & \Rightarrow 5x+3x+2x \\
 & =10x \\
\end{align}$
Now, the budget experiences rise in food, housing and clothing by 40, 30 and 20 percent respectively. So, the increase in budget is calculated as follows:
$a\left( 1+\dfrac{b}{100} \right)$
In the above formula, $a$ is the expenditure and $b$ is the percentage increase so finding the expenditure on food by substituting $a$ as 5x and $b$ as 40 in the above equation we get,
$\begin{align}
  & \Rightarrow 5x\left( 1+\dfrac{40}{100} \right) \\
 & =5x\left( 1+\dfrac{2}{5} \right) \\
 & =5x\left( \dfrac{5+2}{5} \right) \\
 & =5x\left( \dfrac{7}{5} \right) \\
\end{align}$
In the above, 5 will be cancelled out from the numerator and the denominator and we get,
$\begin{align}
  & \Rightarrow x\left( 7 \right) \\
 & =7x \\
\end{align}$
Now, the expenditure for housing is calculated by substituting $a=3x$ and $b=30$ we get,
$\begin{align}
  & \Rightarrow 3x\left( 1+\dfrac{30}{100} \right) \\
 & =3x\left( 1+\dfrac{3}{10} \right) \\
 & =3x\left( \dfrac{10+3}{10} \right) \\
 & =\dfrac{39x}{10} \\
\end{align}$
The expenditure for clothing is calculated by substituting $a=2x$ and $b=20$ we get,
$\begin{align}
  & \Rightarrow 2x\left( 1+\dfrac{20}{100} \right) \\
 & =2x\left( 1+\dfrac{1}{5} \right) \\
 & =2x\left( \dfrac{5+1}{5} \right) \\
 & =\dfrac{12x}{5} \\
\end{align}$
Adding the new expenditure on food, housing and clothing we get,
$\Rightarrow 7x+\dfrac{39x}{10}+\dfrac{12x}{5}$
Taking 10 as L.C.M in the above expression we get,
$\begin{align}
  & \Rightarrow \dfrac{70x+39x+24x}{10} \\
 & =\dfrac{133x}{10} \\
 & =13.3x \\
\end{align}$
The new total expenditure is 13.3x and the old total expenditure is 10x so finding the percentage increase by subtracting 10x from 13.3 x we get,
$\begin{align}
  & \Rightarrow 13.3x-10x \\
 & =3.3x \\
\end{align}$
Dividing 10x in the above expression we get,
$\begin{align}
  & \Rightarrow \dfrac{3.3x}{10x} \\
 & =0.33 \\
\end{align}$
Multiplying 100 to the above number we get,
$\begin{align}
  & \Rightarrow 0.33\times 100 \\
 & =33\% \\
\end{align}$
Hence, the correct option is (a).

Note: The mistake that could be possible in the above problem is that while finding the increase in expenditure you will make this mistake in the formula which is equal to:
$a\left( \dfrac{b}{100} \right)$
In the above formula, $a$ is the expenditure and $b$ is the percentage increase. Now, the problem with the formula is that you missed to add 1 in the bracket so make sure you won’t make this mistake.