# GMAT Math Example Problem – Word Problems & System of Equations

A junior investment banker is paid hourly for her work at Sisyphus, Inc. If her current hourly rate were to be increased by \$300, 30 fewer hours of work could be bought for \$10,000, excluding income tax. What is her current hourly rate?

This is a particularly tricky problem that requires us to not only figure out just what exactly is trying to be said, but also requires us to set up a system of equations for our two unknown variables. In this case the two unknowns we have are the investment banker’s current hourly wage and how many hours she is working. We will assign the variable R to her hourly rate, and the variable H to her hours worked.

We calculate earned income by multiplying the hourly rate by hours worked. Since she currently works H hours at a rate of R dollars per hour to earn \$10,000, we get:

$R*H=10000$

We are then told that were we to increase her rate R by \$300, it would take her 30 fewer hours H to earn \$10,000, yielding the following equation:

$(R+300)*(H-30)=10000$

We expand this equation and get:

$R*H-30R+300H-9000=10000$

We can then substitute directly from our first equation to get the following:

$10000-30R+300H-9000=10000$
$1000-30R+300H=10000$
$-30R+300H=9000$

Let’s take a second to tie everything together. After the work we’ve done, we now have the following system of equations:

$R*H=10000$
$-30R+300H=9000$

We will now solve this by using the substitution method. From the first equation, we can say:

$H=\frac{10000}{R}$

Substituting for H in our second equation, we get this single variable equation:

$-30R +300(\frac{10000}{R})=9000$
$-30R +\frac{3000000}{R}=9000$

To simplify the large numbers we have, let’s divide both sides by -30:

$R-\frac{100000}{R}=-300$

Now we need to get rid of that R in the denominator, which we will accomplish by multiplying everything by R, yielding a quadratic equation:

$R^{2}-100000=-300R$
$R^{2}+300R-100000=0$

While unwieldy, this expression factors quite nicely:

$(R+500)(R-200)=0$

$R+500=0$
$R=-500$

OR

$R-200=0$
$R=200$

Since it would not make a whole lot of sense to say our investment banker is earning -\$500 each hour, it stands to reason that she currently earns a tidy \$200 per hour.

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