Question

The cost of manufacturing a product including labour, material, and overhead are in the ratio \[5:7:3\]. If the profits are calculated as \[20\% \] on costs, what is the ratio of material costs to profit?
(a) \[5:2\]
(b) \[10:3\]
(c) \[11:9\]
(d) \[7:3\]

Answer 1

Hint:
Here, we need to find the ratio of material costs to profit. Let the labour costs be \[5x\], material costs be \[7x\], and overhead costs be \[3x\]. We will find the total costs by adding the labour, material and overhead costs. Then, we will calculate the profit on the total cost. Finally, we will use this to find the ratio of the material cost to profit.

Complete step by step solution:
It is given that the labour costs, material costs, and overhead costs are in the ratio \[5:7:3\].
Let the labour costs be \[5x\], material costs be \[7x\], and overhead costs be \[3x\].
We will add the three costs to get the total cost.
Therefore, we get
\[ \Rightarrow \]Total cost \[ = 5x + 7x + 3x\]
Adding the like terms, we get
\[ \Rightarrow \]Total cost \[ = 15x\]
Now, the profits are \[20\% \] on the total cost.
Therefore, we get
\[ \Rightarrow \]Profit \[ = 20\% \] of total cost
Substituting \[15x\] as total cost, we get
\[ \Rightarrow \]Profit \[ = 20\% {\text{ of }}15x\]
Rewriting the expression, we get
\[ \Rightarrow \]Profit \[ = \dfrac{{20}}{{100}} \times 15x\]
Simplifying the expression, we get
\[ \Rightarrow \]Profit \[ = \dfrac{1}{5} \times 15x = 3x\]
We need to find the ratio of material costs to profit.
The material costs are \[7x\].
The profits are \[3x\].
Therefore, we get
\[\dfrac{{{\text{Material costs}}}}{{{\text{Profit}}}} = \dfrac{{7x}}{{3x}}\]
Simplifying the expression, we get
\[ \Rightarrow \dfrac{{{\text{Material costs}}}}{{{\text{Profit}}}} = \dfrac{7}{3}\]
Thus, we get
Material costs : Profit = \[7:3\]
Therefore, the ratio of material costs to profit is \[7:3\].
The correct option is option (d).


Note:
Note: We needed to find the ratio of material costs to profit. We calculated the profits on the total costs because ‘profits’ are assumed to be on the total costs, unless specified. A common mistake is to assume ‘ratio of material cost to profits’ as ‘ratio of material cost to profits on material cost’. This will result in an incorrect answer.