Question

If the labour cost 20% of the cost of production and raw material cost 10% of the cost of production and the price on which article is sold is 20% above the cost of production. If the price of the labour is increased by 40% and the price of raw material increases by 20% and the rest other expenditure of cost remains the same. The industry thus decided to increase the selling price by 10%. Find the new profit percent?

Answer 1

Hint:Let us assume that the cost of production is x then labour cost is $ \dfrac{20}{100}x $ , raw material cost is $ \dfrac{10}{100}x $ and S.P. (Selling Price) is $ x\left( 1+\dfrac{20}{100} \right) $ . After that when price of labour is increased by 40% and the price of raw material increased by 20% and rest expenditures are remain constant then cost of production is increased to $ x+\dfrac{20}{100}x\left( \dfrac{40}{100} \right)+\dfrac{10}{100}x\left( \dfrac{20}{100} \right) $ so this is the new cost price of the production. Selling price is also increased by 10% i.e. $ \left( \dfrac{20}{100}+1 \right)x\left( 1+\dfrac{10}{100} \right) $ . To find the new profit percent subtract the new cost price from the new selling price and divide this result by new cost price and hence, multiply this division by 100.

Complete step-by-step answer:
Let us assume that the cost of production is x.
It is given that:
The labour cost 20% of the cost of production so the cost of the labour is:
  $ \dfrac{20}{100}x=\dfrac{1}{5}x $
The raw material cost 10% of the cost of production so the cost of the raw material is:
  $ \dfrac{10}{100}x=\dfrac{1}{10}x $
Selling price of the article is given by:
  $ \dfrac{20}{100}x+x $
  $ \begin{align}
  & =x\left( \dfrac{20}{100}+1 \right) \\
 & =\dfrac{120}{100}x=\dfrac{12}{10}x \\
\end{align} $
Thus, the selling price of the article is $ \dfrac{12}{10}x $ .
After that the price of the labour is increased by 40% and the price of raw material increases by 20% and the other expenditure of cost remains the same. This means that the cost of production is increased and we have to add the percentage increase of cost of labour and cost of raw material.
Percentage increase of the cost of labour is $ \dfrac{40}{100}\left( \dfrac{x}{5} \right) $ and percentage increase of the cost of raw material is $ \dfrac{20}{100}\left( \dfrac{x}{10} \right) $ . Now adding these percentages to the earlier cost of production we get,
  $ \begin{align}
  & x+\dfrac{40}{100}\left( \dfrac{x}{5} \right)+\dfrac{20}{100}\left( \dfrac{x}{10} \right) \\
 & =x+\dfrac{20}{100}x\left( \dfrac{2}{5}+\dfrac{1}{10} \right) \\
 & =x+\dfrac{20}{100}x\left( \dfrac{4+1}{10} \right) \\
 & =x+\dfrac{2}{10}x\left( \dfrac{5}{10} \right) \\
 & =x+\dfrac{1}{10}x=\dfrac{11}{10}x \\
\end{align} $
Hence, the new cost price of the production is equal to $ \dfrac{11}{10}x $ .
It is given that the selling price is increased by 10% of the earlier selling price. The earlier selling price that we have calculated is $ \dfrac{12}{10}x $ so adding this earlier selling price in 10% of the earlier selling price we get,
  $ \begin{align}
  & \dfrac{12}{10}x+\dfrac{10}{100}\left( \dfrac{12}{10}x \right) \\
 & =\dfrac{12}{10}x\left( 1+\dfrac{10}{100} \right) \\
\end{align} $
  $ \begin{align}
  & =\dfrac{12}{10}x\left( 1+\dfrac{1}{10} \right) \\
 & =\dfrac{12}{10}x\left( \dfrac{11}{10} \right) \\
\end{align} $
Hence, the new selling price is equal to $ \dfrac{12}{10}x\left( \dfrac{11}{10} \right) $ .
New profit percent is the subtraction of cost price of the production (C.P.) from the selling price of the article (S.P.) and then divide this result of subtraction by new cost of production (C.P.) and multiply this result of division by 100.
  $ \text{Profit Percent}=\left( \dfrac{\text{S}\text{.P - C}\text{.P}\text{.}}{\text{C}\text{.P}\text{.}} \right)\times 100 $
Substituting the new selling price and cost price we get,
  $ \text{Profit Percent}=\left( \dfrac{\dfrac{12}{10}\text{x}\left( \dfrac{11}{10} \right)-\dfrac{11}{10}x}{\dfrac{11}{10}x} \right)\times 100 $
Taking $ \dfrac{11}{10}x $ as common from the numerator and the denominator we get,
  $ \text{Profit Percent}=\left( \dfrac{\dfrac{11}{10}x\left( \dfrac{12}{10}-1 \right)}{\dfrac{11}{10}x} \right)\times 100 $
As you can see that $ \dfrac{11}{10}x $ is common in both the numerator and the denominator so it will be cancelled out.
  $ \begin{align}
  & \text{Profit Percent}=\left( \dfrac{12}{10}-1 \right)\times 100 \\
 & \Rightarrow \text{Profit Percent}=\left( \dfrac{12-10}{10} \right)\times 100 \\
 & \Rightarrow \text{Profit Percent}=\left( \dfrac{2}{10} \right)\times 100 \\
 & \Rightarrow \text{Profit Percent}=20\% \\
\end{align} $
From the above solution, the new profit percentage that we have got is equal to 20%.

Note: Kindly avoid the silly mistakes in calculations. And while writing the formula for the profit percentage make sure you have applied the correct signs in the formula as the formula for the profit percentage is:
  $ \text{Profit Percent}=\left( \dfrac{\text{S}\text{.P - C}\text{.P}\text{.}}{\text{C}\text{.P}\text{.}} \right)\times 100 $
You might mistakenly put positive sign instead of negative sign in the numerator.