kamalakarthi wrote:Hi Experts,

Thanks all for helping by answering the doubts. The below question is from Manhattan CAT.

Even though, I got to the right answer by back solving the answers, I could not get to the right choice immediately by solving thru equations.

My equations considering R is the rate and T is the time :

(R-4). (T+16)= 96

1.5 R * T = 96

so I tried to equate

Why cant I equate like the below

(R-4). (T+16) = 1.5 R * T

You have to eliminate a variable in order to solve the equation. When you equated that way, you left yourself without a way to eliminate a variable.

But, when I tried 96/R-4 = 96/1.5r +16, I was able to solve. It took time to come to this way so how should I approach these questions during setting up the equation.

The key here is to see that you have to eliminate a variable in order to solve.

It is often the case that in doing an algebraic translation in the process of answering a GMAT quant question, you are best off eliminating a variable in creating the setup if possible.

For instance, in an age question involving the ages of three people, you may be able to express the ages of two people in terms of the age of the third person. So, if it were the case that you could do so, you could set up the equations for answering the question by using just one variable.

In this case, the key to getting the right answer is to see that you can eliminate T in the process of setting up. When you do so, you are left with only one variable, R.

You CAN use this equation:

(R-4) * (T+16) = 1.5 R * T

You're just missing a crucial piece: the distance of 96.

With that, we have

(r - 4) * (t + 16) = 1.5rt

and

1.5rt = 96

and we can solve.

Just in case anyone reading wants to actually solve, there are two approaches:

I:: The Lazy Testwriter Principle

The testwriter will probably make the answer an integer. (Here, in fact, the answers are all integers, so we should be OK.) With that in mind, once we have

1.5rt = 96

or

rt = 64

We know that r and t should each be a factor of 64.

Looking at the answers, that's either r = 8, t = 8, or r = 16, t = 4.

Plugging those into our other equation, we have

(r - 4) * (t + 16) = 96

Only r = 8 and t = 8 work here, so r = 8 is our answer.

II: Algebra

We've got two equations:

1.5rt = 96, or rt = 64

and

(r - 4) * (t + 16) = 1.5rt

The second one gives us

rt - 4t + 16r - 64 = 1.5rt

16r - 4t - 64 = 0.5rt

We know rt = 64, so we can replace 0.5rt with 0.5*64, or 32:

16r - 4t - 64 = 32

16r - 4t = 96

4r - t = 24

4r - 24 = t

This gives us t in terms of r. Now we can replace t with 4r - 24 in our first equation to solve for r:

rt = 64

r * (4r - 24) = 64

4r*r - 24r - 64 = 0

r*r - 6r - 16 = 0

(r - 8) * (r + 2) = 0

Our solutions are r = 8 and r = -2. The rate can't be negative (at least in this reality!), meaning r = 8 is our only solution.