# Math Basics – Distance, Rate and Time

The 4^{th} of July weekend is upon us, and many of us will be taking road trips, or even taking a plane somewhere. To commemorate this collective movement, let’s learn the most fundamental formula when dealing with movement over time. First, let’s meet Bob…

Q1. Bob drives at an average rate of 50 mph from Berkeley to Los Angeles, a distance of 350 miles. How long does it take him to complete the trip?

(A) 4 hrs

(B) 5 hrs

(C) 7 hrs

(D) 10 hrs

(E) 12 hrs

When dealing with distance, rate and time, we always want to remember the nifty little formula, D = R \times T, in which D stands for the distance, R stands for the rate (or speed), and T stands for the time.

With the problem above, the distance between Berkeley and Los Angeles is 350 miles. So D = 350. Bob is traveling at 50 mph, so that is his rate. The question is how long will it take him to complete this trip. Therefore, we have to solve for T. Let’s set up the equation, plugging in the values for D and R:

350 = 50T.

Solving for T, we get 7. Because we are dealing with miles per hours, the 7 corresponds to hours. So T = 7.

Now let’s try another problem:

Q2. Charlie takes 2.5 hours to fly from Los Angeles to Mexico City, a distance of 1200 miles. What is the average speed of his plane in miles per hour?

(A) 200 mph

(B) 240 mph

(C) 410 mph

(D) 480 mph

(E) 533 mph

Setting up the equation, we get 1200 (Distance) = 2.5 (Time) \times R; 1200 = 2.5R. Before solving (assuming this is on the current GRE, in which you do not have a calculator), let’s change 2.5 to \dfrac{5}{2}, as it is much easier to do the math with fractions than with decimals. We get 1200 = \dfrac{5}{2R}. Solving for R, we multiply both sides by the reciprocal, \dfrac{5}{2}. This gives us R on the right-hand side of the equation, and 1200 \times \dfrac{5}{2} on the left-hand side. \dfrac{2400}{5} = 480 (D).

Note you could have solved this problem back-solving, in which you put the answer choices back into the question. Or, even better, we could just plug-in 500 mph. Of course that is not one of the answer choices, but using 500 will make the math very easy. If the number is a little too high when we plug-in 500, then the answer must be (D). If it is a little low, then the answer must be (E) 533. Again, you never try to back solve with an ugly number like 533 — it will take too long.

Using 500 mph, we get 1200 (Distance) = 500 (Rate) \times 2.5 (Time). You can see that multiplying these two numbers gives us 1250. Meaning, we flew too fast and missed Mexico City by 50 miles. Therefore, we have to slow down the plane a little — the answer is 480.