Question

The cost of making an umbrella is divided between material, labour and overheads in the ratio 6:4:1. If the material costs  RS.132, find the cost of production of an umbrella.

Answer 1

Hint: We assume the cost of overhead as $x$ rupees, then the cost of material is  $6x$ and the cost of labour is $4x$ rupees. We find the sum of the terms in an expression of  $x$. We are given that the cost of material is 132 which means $6x=132$. We solve for $x$ and put the value in the expression.

Complete step by step answer:

A ratio is a fraction with both numerator and denominator as positive integers expressed in standard form.  The standard of a fraction is $\dfrac{p}{q}$ where both $p$ and $q$ are positive integers and the highest common factor of $p$ and $q$ is 1. It means  $p$ and $q$ are co-prime or relatively prime. 

The  ratio between two positive integers numbers $A$ and $B$ is written as $a:b$ and is given as 

\[A:B=\dfrac{\dfrac{A}{n}}{\dfrac{B}{n}}=\dfrac{a}{b}=a:b\]

where $n$ is the greatest common divisor of $A$ and $B$.  

If two numbers $A$ and $B$ are in a ratio $a:b$  and then for some positive integer $k$ if multiplied, $kA$ and $kB$ will so have the same ratio $a:b$.

If two numbers $A$ and $B$ are in a ratio $a:b$  and then for some positive integer $k$, if divided $\dfrac{A}{k}$ and $\dfrac{B}{k}$ will so have the same ratio $a:b$ where k is a factor of both $A$ and $B$. If there is more than one ratio involved we call it continued proportion.

The given ratio is a continued proportion 6:4:1 which represents the proportion in which the cost is spent to make an umbrella for material, labour and overheads respectively. Let us assume the cost of the overhead is in rupees is $x$. So the cost of the material is $6x$ rupees and the cost of the labour is $4x$ rupees. The costs $6x,4x,x$ will be in the same ratio as 6:4:1. The sum of the cost is the cost of making an umbrella which is 

\[6x+4x+x=11x\]

We are given in the question that the material cost is 132 rupees. So we have,

\[6x=132\]

\[ \Rightarrow x=\dfrac{132}{6}=22 \] 

We are asked to find the total cost. The total cost in rupees is 

\[11x=11\times 22=242\]

Therefore, the total cost is Rs.242.

 

Note:
We note that we can also find the cost of overhead as $x=22$ rupees and the cost of labour as $4x=88$ rupees. The lengths, breadth, height $\left( l:b:h \right)$ of an cuboid, the angles $\left( A:B:C \right)$ or sides $\left( a:b:c \right)$ of a triangle are also in continued proportion with three numbers.