Quarter Wit, Quarter Wisdom: And Now, Evading Formulas!

Quarter Wit, Quarter WisdomToday, we again pay homage to the lazy bum within each one of us in our QWQW series. If you are wondering what we mean by ‘again’, check out our last two posts of the QWQW series. We have been discussing how to avoid calculations. Today let’s learn why it is advisable to avoid learning formulas too!

You really don’t need to know many formulas for GMAT – just the basic ones e.g. Distance = Speed*Time, Work = Rate*Time (which are actually the same if you look at them closely) etc. If a Time-Distance-Speed question pertains to GMAT, rest assured it can be solved using just the formula given above and that too, within 1-2 mins. Then, do you need to learn the many formulas that people claim speed up question solving? No! In fact, the more specific the formula, the more constraints it has. It can be used in only particular circumstances and hence when the situation differs even a little bit from the ideal, you could end up using the formula incorrectly. Therefore, we recommend our students to stay away from the umpteen, less generic formulas until and unless they have already used them extensively. Let’s discuss this point with an example:

Question: A and B start from Opladen and Cologne respectively at the same time and travel towards each other at constant speeds along the same route. After meeting at a point between Opladen and Cologne, A and B proceed to their destinations of Cologne and Opladen respectively. A reaches Cologne 40 minutes after the two meet and B reaches Opladen 90 minutes after their meeting. How long did A take to cover the distance between Opladen and Cologne?

(A) 1 hour
(B) 1 hour 10 minutes
(C) 2 hours 30 minutes
(D) 1 hour 40 minutes
(E) 2 hours 10 minutes

Solution: People often like to use a formula for this situation. Let’s quickly discuss that first.

If two objects A and B start simultaneously from opposite points and, after meeting, reach their destinations in ‘a’ and ‘b’ hours respectively (i.e. A takes ‘a hrs’ to travel from the meeting point to his destination and B takes ‘b hrs’ to travel from the meeting point to his destination), then the ratio of their speeds is given by:

Sa/Sb = sqrt(b/a)

i.e. Ratio of speeds is given by the square root of the inverse ratio of time taken.

Sa/Sb = sqrt(90/40) = 3/2

This gives us that the ratio of the speed of A : speed of B as 3:2. We know that time taken is inversely proportional to speed. If ratio of speed of A and B is 3:2, the time taken to travel the same distance will be in the ratio 2:3. Therefore, since B takes 90 mins to travel from the meeting point to Opladen, A must have taken 60 (= 90*2/3) mins to travel from Opladen to the meeting point

So time taken by A to travel from Opladen to Cologne must be 60 + 40 mins = 1 hr 40 mins

Now let’s see how we can solve the question without using the formula.

Think of the point in time when they meet:

A starts from Opladen and B from Cologne simultaneously. After some time, say t mins of travel, they meet. Since A covers the entire distance of Opladen to Cologne in (t + 40) mins and B covers it in (t + 90) mins, A is certainly faster than B and hence the Meeting point is closer to Cologne.

Now think, what information do we have? We know the time taken by A and B to reach their respective destinations from the meeting point. We also know that they both traveled the same distance i.e. the distance between Opladen and Cologne. So let’s try to link distance with time taken. We know that ‘Distance’ varies directly with ‘Time taken’. (Check out this post if you don’t know what we are talking about here.)

Distance between Opladen and Meeting point /Distance between Meeting point and Cologne = Time taken to go from Opladen to Meeting point/Time taken to go from Meeting point to Cologne = t/40 (in case of A) = 90/t (in case of B)
t = 60 mins

So A takes 60 mins + 40 mins = 1 hr 40 mins to cover the entire distance.

Answer (D)

We could easily solve the question without using any specific formula. So stick to your basics and kick those little grey cells to get to the answer!

Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!