Avoiding Calculations with Variables in Denominator

We have discussed before how GMAT is not a calculation intensive exam. Whenever you land on an equation which looks something like this: $\frac{60}{(n - 5)} - \frac{60}{n} = 2$ , you probably think that we don’t know what we are talking about! You obviously need to cross multiply, make a quadratic and finally, solve the quadratic to get the value of $n$ . Actually, you usually don’t need to do any of that for GMAT questions. You have an important leverage – the options. Even if the options don’t directly give you the values of $n$ or $n-5$ , you can use the knowledge that every GMAT question is doable in 2 mins and that the numbers fit in beautifully well.

Let’ see whether we can get a value of n which satisfies this equation without going the whole nine yards. We will not use any options and will try to rely on our knowledge that GMAT questions don’t take much time.

$\frac{60}{(n - 5)} - \frac{60}{n} = 2$

So, the difference between the two terms of the left-hand side is 2. Try to look for values of $n$ which give us simple numbers i.e. try to plug in values which are factors of the numerator.

Say, if $n = 10$ , you get $\frac{60}{5} - \frac{60}{10} = 12 - 6 = 6$ . The difference between them is much more than 2

$\frac{60}{n}$ and $\frac{60}{(n - 5)}$ need to be much closer to each other so that the difference between them is 2. The two terms should be smaller to bring them closer together. So increase the value of $n$

Put $n = 15$ since it is the next number such that ( $15 - 5 = 10$ ) as well as 15 divides 60 completely. You get $\frac{60}{10} - \frac{60}{15} = 6 - 4 = 2$ . It satisfies and you know that a value that $n$ can take is 15. Usually, you will get a solution within 2-3 iterations. This is enough for a PS question. Notice that this equation gives us a quadratic so be careful while working on DS questions.

You might need to manipulate the equation a little to figure out whether the other root is a possible solution as well. Anyway, today we will focus on the application of such equations in PS questions only. Let’s take a question now to understand the concept properly:

Question: Machine ${A}$ takes 2 more hours than machine ${B}$ to make 20 widgets. If working together, the machines can make 25 widgets in 3 hours, how long will it take machine ${A}$ to make 40 widgets?
(A) 5
(B) 6
(C) 8
(D) 10
(E) 12

Solution:

We need to find the time taken by machine $A$ to make 40 widgets. It will be best to take the time taken by machine $A$ to make 40 widgets as the variable $x$ . Then, when we get the value of $x$ , we will not need to perform any other calculations on it and hence the scope of making an error will reduce. Also, value of $x$ will be one of the options and hence plugging in to check will be easy.
Machine $A$ takes $x$ hrs to make 40 widgets.

Rate of work done by machine $A = \frac{Work\ done}{Time\ taken} = \frac{40}{x}$

Machine $B$ take 2 hrs less than machine $A$ to make 20 widgets hence it will take 4 hrs less than machine $B$ to make 40 widgets. Think of it this way: Break down the 40 widgets job into two 20 widget jobs. For each job, machine $B$ will take 2 hrs less than machine $A$ so it will take 4 hrs less than machine $A$ for both the jobs together.

Time taken by machine $B$ to make 40 widgets $= x - 4$

Rate of work done by machine $B = \frac{Work\ done}{Time\ taken} = \frac{40}{x - 4}$

We know the combined rate of the machines is $\frac{25}{3}$

So here is the equation:

$\frac{40}{x} + \frac{40}{x - 4} = \frac{25}{3}$

The steps till here are not complicated. Getting the value of $x$ poses a bit of a problem.

Notice here that that the right-hand side is not an integer. This will make the question a little harder for us, right? Wrong! Everything has its pros and cons. The 3 of the denominator gives us ideas for the values of $x$ (as do the options). To get a 3 in the denominator, we need a 3 in the denominator on the left-hand side too.

$x$ cannot be 3 but it can be 6. If $x = 6$ , $\frac{40}{6 - 4} = 20$ i.e. the sum will certainly not be 20 or more since we have $\frac{25}{3} = 8.33$ on the right hand side.

The only other option that makes sense is $x = 12$ since it has 3 in it.

$\frac{40}{12} + \frac{40}{12 - 4} = \frac{10}{3} + 5 = \frac{25}{3}$

Answer (E)

If we did not have the options, we might have tried $x = 9$ too before landing on $x = 12$ . Nevertheless, these calculations are not time consuming at all since you can get rid of the incorrect numbers orally. Making a quadratic and solving it is certainly much more time consuming.

Another method could be to bring 3 to the left-hand side to get the following equation:

$\frac{120}{x} + \frac{120}{x - 4} = 25$

This step doesn’t change anything but it helps if you face a mental block while working with fractions. Try to practice such questions using these techniques – they will save you a lot of time.

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