The average salary of the workers in a workshop is RS.8500. If the average monthly salary of 7 technicians is RS 10000 and the average monthly salary of the rest is RS 7800. Then the total number of the workers in the workshop is?
An averages problem
3 Answers
What do we know?
- We know that $\dfrac{\text{Total salary earned}}{\text{number of workers}} = 8500$.
- And we know that $\dfrac{\text{salary earned by technicians}}{7 = \{\text{the number of tecnhicias} \}} = 10000$.
- Finally, we know that $\dfrac{\text{salary earned by workers besides the technicians}}{\text{number of workers}-\text{number of technicians}} = 7800$.
From 2. we know the total salary earned by the technicians. We can use this to combine 1. and 3. into a system of equations with two unknowns: the total salary earned and the total number of workers.
Can you handle it from there?
Let $n$ be the number of workers among the "rest." Then the total number of workers is $n+7$.
The overall average wage is is $8500$. So the total wage bill for all the workers is $(8500)(n+7).\tag{$1$}$
We find the total wage bill in another way. The $n$ "other" workers between them earn $(7800)(n)$. The $7$ technicians between them earn $(10000)(7)$. So the total wage bill is $(7800)(n)+(10000)(7).\tag{$2$}$
The two expressions $(1)$ and $(2)$ for the total wage bill must give equal results. From this you should be able to get a nice linear equation, which is not difficult to solve for $n$.
Remark: There is a more informal way to solve the problem. It substitutes thinking for the algebra. The $7$ technicians each earn $1500$ more than the average salary of $8500$. So between them, they get a total of $(1500)(7)$ more than they would if they got average salary.
This $(1500)(7)$ must come out of the hides of the poor "other workers," whose individual earnings are $700$ below average. So each of the poor other workers contributes $700$ to the wealth of the technicians. Since the total contribution is $(1500)(7)$, the number of "other workers" must be $\frac{(1500)(7)}{700}.$ That is more or less how this type of problem was solved before "algebra" came into widespread use. People learned "rules" to partly automate the process, and bypass thinking.
Let $n$ represents the total number of workers. Total salary of $n$ workers=$8500n$.$$ Total Salary of $7$ technicians $=70000$ and total salary of the rest $n-7$ workers $=7800(n-7)$ \implies 70000+7800(n-7)=8500n$ $\implies 70000-7800*7=(8500-7800)n$ $\implies 15400=700n$ $\implies n=22$ Thus, total number of workers =22.$