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I am having difficulty with the following problem:

Mark was formerly paid a salary of $400 per week and 8% commission on his total sales.Later Mark was given a 20% salary cut and his commission rate was increased to 10%. What is the smallest amount of sales Mark has to make to earn as much under the new plan as he did with the old. (Ans=4000)

Here is how i am attempting it.

Marks new salary is $320 and his commission rate is 10%.

So He needs 80 dollars more to be back to his original salary so he needs a sale of $800

Now further compensation requires that the previous sales income must match the onward sales income . I don't know how to do that. I guess that's why I am not getting the right answer. Any suggestions that could help me out.

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up vote 1 down vote accepted

Your statement that

He needs 80 dollars more to be back to his original salary

is incorrect; recall that his original salary wasn't just \$400, but also included a commission of 8%. If $d$ is the number of dollars in sales he needs to make so that his total salary would be the same under both his new plan and his old plan, instead of solving $$320+(0.1\times d)=400$$ (which is what you proposed), we need to solve $$320+(0.1\times d)=400+(0.08\times d)$$ This equation becomes $$(0.1\times d)-(0.08\times d)= 400-320$$ $$0.02\times d = 80$$ $$d=80\times\frac{1}{0.02}=80\times 50=4000,$$ which is the correct answer.

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There is not enough information to solve the problem, since we do not know how much in the way of sales Mark used to make, so we don't know what his total income used to be.

So we will have to make an assumption. Let us assume that Mark's monthly sales total will remain unchanged. This is perhaps unreasonable, since Mark, if he wants to continue eating, has additional incentive to sell harder.

Let this unchanging total sales amount be $x$. Then Mark's previous commission was $(0.08)x$ and his new commission is $(0.10)x$. The difference $(0.02)x$ will have to be $80$ to make up for the drop in salary.

This gives us the equation $(0.02)x=80$. Thus $x=\frac{80}{0.02}=4000$.

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Let total sales be of $x$.Then previous income = $400 + 0.08x$. New income = $320+0.1x$. Equate them, you will get $x=4000$.

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