# An averages problem

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?

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What have you tried? –  Dan W Jul 11 '12 at 16:39

What do we know?

1. We know that $\dfrac{\text{Total salary earned}}{\text{number of workers}} = 8500$.
2. And we know that $\dfrac{\text{salary earned by technicians}}{7 = \{\text{the number of tecnhicias} \}} = 10000$.
3. 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?

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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.

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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 $=22.$

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