I thought I will take up “Roots” today but the rules of exponents and roots can get pretty monotonous. So let’s take a break today and do something more interesting.  I keep telling my students that GMAT questions do not involve long calculations. If you find yourself dividing a three digit number with a two digit number, it means you have missed the point of the question. Today, I will depict what I mean with the help of a couple of examples. I start with one my favorite questions from the Veritas book.

Question 1: A certain portfolio consisted of 5 stocks, priced at \$20, \$35, \$40, \$45 and \$70, respectively. On a given day, the price of one stock increased by 15%, while the price of another decreased by 35% and the prices of the remaining three remained constant. If the average price of a stock in the portfolio rose by approximately 2%, which of the following could be the prices of the shares that remained constant?

(A) 20, 35, 70
(B) 20, 45, 70
(C) 20, 35, 40
(D) 35, 40, 70
(E) 35, 40, 45

What we notice here is that one stock price increases by 15% and another decreases by 35% but still, there is an overall INCREASE of 2%. Hence, the amount of increase should be greater than the amount of decrease. Say, the stock whose price increases by 15% is A and the stock whose price decreases by 35% is B. We can infer that
15% of A > 35% of B

Now an interesting point is that 15% of A will be equal to 30% of B if A is twice of B. But 15% of A is greater than 30% of B so A must be greater than twice of B. In fact 15% of A is greater than 35% of B so A must be substantially greater than twice of B. Can we infer something about the value of B from this? Can we say that A, B is a pair of numbers such that A is more than twice of B? Out of 20, 35, 40, 45 and 70, what can be the value of B? B has to be 20 because we have values more than twice of 20 (which are 45 and 70). But we don’t have any values which are more than twice of 35, 40, 45 and 70.

If B is 20, A must be either 45 or 70 (since A must be more than twice of B). I would bet on 70 since A has to be substantially greater than twice of 20. But say, there is this little voice inside you that is constantly telling you to be extra careful before marking the answer. After all, there is no partial credit. Only the final answer is what matters and if you slip up in the last step, it’s as bad (actually worse since you used up the time allotted to this question) as being totally bowled over by the question (in such a case, you should move on within 30 secs and save the time for another, more do-able problem).

Let’s do some quick calculations to confirm that the correct answer is indeed 70 and not 45:

10% of 20 is 2.

30% of 20 is 6.

5% of 20 is 1.

Therefore, 35% of 20 is 6+1 = 7.

15% of A has to be greater than 7.

70 satisfies this condition – 10% of 70 is 7 so 15% of 70 must be greater than 7.

45 does not satisfy this condition – 10% of 45 is 4.5 and 5% of 45 is 2.25, therefore, 15% of 45 is 6.75 which is less than 7

We see that the stock prices that did not change are 35, 40 and 45. Hence the answer is (E).

The question might have seemed intimidating at first but wouldn’t you say that it looks much more reasonable now? Let’s try another question (from some outside source) which looks calculation intensive, but in fact, involves no calculations at all.

Question 2: Which of the following, when squared, will yield a value greater than 3/4

(A) 2/7
(B) (.75)^2
(C) 2/3
(D) 6/7
(E) 7/8

You might think that you need to square each option and divide to find out the answer but don’t get alarmed just yet. Let’s give ‘logic’ a try first. We know that when a positive number less than 1 is squared, it becomes even smaller.

For example:

(1/2)^2 = 1/4. 1/4 is smaller than 1/2

(2/3)^2 = 4/9. 4/9 is smaller than 2/3. If you are wondering how to figure this out, 4/9 is less than ½ and 2/3 is greater than ½. Or multiply  and divide 2/3 by 3 to get 6/9. 4/9 is less than 6/9

All the given options are less than 1. When you square them, they will become even smaller.

So the answer must be greater than 3/4 to begin with and must be much greater than 3/4 so that even after squaring, it remains greater than 3/4. Let’s consider the individual options.

Option (A) 2/7 < 3/4 – Ignore

Option (B) (.75)^2 < .75 (which is 3/4) – Ignore

Option (C) 2/3 < 3/4 – Ignore

6/7 and 7/8 are both greater than 3/4. If 6/7 were the answer and the square of 6/7 were greater than 3/4, since 7/8 is even greater than 6/7, its square would be greater than 3/4 too. But we cannot have multiple answers. Hence, only the square of 7/8 must be greater than 3/4. Answer must be 7/8.

Or you can just consider that each option is less than 1. We need the option which is greater than 3/4 when squared. So the answer would be the option which is the largest since only one correct answer is there and one correct answer is definitely there!

I hope these examples showed you how you can deal with most questions without doing any long calculations. More on this, later…

You are welcome to put up calculation-intensive questions in the comments and we at Veritas will try and provide ‘logical’ solutions. We don’t promise that there are calculation-cutting tricks for all questions (especially the harder Permutation Combination questions) since sometimes, some providers give long calculations only to make the questions harder – they totally miss the point – but in most cases, we would be able to do justice.

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 in Detroit, and regularly participates in content development projects such as this blog!

9 Responses

1. G says:

Hi Karisma , thanks for the great post .Can you also discuss Permutation – Combination and probabilities

2. Mike says:

Good post. Can you explain this again?

6/7 and 7/8 are both greater than 3/4. If 6/7 were the answer and the square of 6/7 were greater than 3/4, since 7/8 is even greater than 6/7, its square would be greater than 3/4 too. But we cannot have multiple answers. Hence, only the square of 7/8 must be greater than 3/4. Answer must be 7/8.

3. Karishma says:

I will take Permutation Combination and Probability in a few weeks. From the exam point of view, they are not very important. You will see very few questions from these topics on your test so until and unless you are aiming for Q51 and want to leave no stone unturned, don’t worry too much about them. Just know your basics very well. From an intellectual point of view, they are extremely interesting and very wicked.

4. Karishma says:

Hey Mike,

Here is the logic used:

We are dealing with positive numbers between 0 and 1. Say I have two such numbers: a and b and I know that a < b.
Can I square both sides? Sure because both the numbers are positive so squaring both sides will not change the inequality sign. (If one of the numbers or both are negative, we cannot simply square both sides if there is an inequality sign. Take some examples to figure it out.)
Therefore, we get that a^2 < b^2
On the same lines, 6/7 < 7/8
Therefore, (6/7)^2 < (7/8)^2

Now, think. If (6/7)^2 were greater than 3/4, then (7/8)^2 would also be greater than 3/4. This means we would get two answers – D and E. But all GMAT questions have only one answer. Hence, only (7/8)^2 must be greater than 3/4.

The greatest number must be the answer since the square of the greatest number will be the greatest among all the squares. We can only have one answer so only that number's square must be greater than 3/4.

5. macjas says:

Hi Karishma,

This is how I did the first problem. Is my logic justified?

A certain portfolio consisted of 5 stocks, priced at \$20, \$35, \$40, \$45 and \$70, respectively. On a given day, the price of one stock increased by 15%, while the price of another decreased by 35% and the prices of the remaining three remained constant. If the average price of a stock in the portfolio rose by approximately 2%, which of the following could be the prices of the shares that remained constant?

I used the weighted averages method.

Let B be the stock that increased by 15% and A the stock that decreased by 35%

Then: A (-35%)———-2%————-B(15%)
Then the ratio of A:B is 13:39
or roughly 1:3
The only two numbers that comes close to that ratio is 20 and 70.

6. macjas says:

Sorry – edit typo: The ratio is 13:37 (or roughly 1:3)

7. Mike says:

Hi Karishma,

Can you please elaborate: “15% of A will be equal to 30% of B if A is twice of B”? How did you know that? Can you post a general rule? Thank you.

• Karishma says:

15% of A = 30% of B
(15/100)*A = (30/100)*B
A/B = 2/1

15% of a number can be equal to 30% of another number only when the first number is twice of the other number.

• Karishma says:

Also, there are no such ‘rules’ that you can memorize. These things come to you intuitively when you practice regularly.

x% of number will be equal to 2x% of another number if the first number is twice of the second number.
x% of a number will be equal to 3x% of another number if the first number is thrice the second number.

and so on…