How to calculate the minimum amount of time to reach sought value with multiple variables?

Question

A person has two jobs and gets payed $500 per week from one job, and gets payed $1100 every two weeks from the other job. How many weeks does it at least take for the person to gain $10,000?

Answer 1

The simplest and most efficient way to solve this problem actually does not require multiple variables. If we read the problem closely, we see the only variable we are being asked to solve for is time. Therefore, if we put everything in terms of time then we only need a single equation and to solve for x. In this case x is going to represent a time period of 1 week. As such, the equation is:

500(x) + 1100(x/2) = 10,000

because we are putting the money earned in terms of time worked in total, not time worked at either job. This now becomes an equation to solve for x.

We simplify:

500x + 550x = 10000

1050x = 10000

x = 9.52380....

Now, this answer is useful but not the final answer, as the question wants to know the minimum number of weeks it would take to gain $10000, so giving a decimal answer is giving "partial weeks" and thus incorrect as it wants a whole number. As a final step we have to apply our decimal number to the context of the problem and an easy way to do that is to try both whole numbers closest to our decimal, in this case 9 and 10, to see which will get us a true inequality.

First we plug in 9, with the goal that the sum is greater than 10000:

500(9) + 1100 (9/2) > 10000

4500 + 4950 > 10000

9450 > 10000

This is FALSE.

So now we try x = 10.

500(10) + 1100(10/2) > 10000

5000 + 5500 > 10000

10500 > 10000

This is TRUE.

Now we see 10 is the lowest possible whole number we can use to make a true inequality, and we can correctly say it would take at least 10 weeks to earn 10000$.