### 8 - 760+: Answer to the Prior Question - Long Math Approaches vs. Faster Tactical Approaches

By richc On Oct 25, 2021 In  General GMAT

In the prior post, I listed the following Quant PS question:

Andrew has a certain number of coins in his pocket. He has three times as many dimes as quarters and six times as many nickels as dimes. A nick is worth \$0.05, a dime is worth \$0.10 and a quarter is worth \$0.25. If he has a total of \$10.15, then which of the following represents the number of dimes in Andrew’s pocket?

9
10
18
20
21

Many GMATers will approach these types of questions algebraically, even though that type of approach often takes longer than other, faster, more strategic options.

To start, here is how you can solve this algebraically. Let’s use the following variables:

N = number of nickels
D = number of dimes
Q = number of quarters

The second sentence allows us to create two equations using the above variables:

D = 3Q
N = 6D

The third and fourth sentences allow us to create a third equation:

(.05)N + (.10)D + (.25)Q = 10.15

Now we have a ‘system’ of equations – three variables and three unique equations, so we can solve for each of the individual variables. The prompt asks us to determine the number of DIMES, so I’ll focus the work on solving for D.

D = 3Q …. Q = D/3
N = 6D

Substituting in for N and Q, we end up with…

(.05)(6D) + (.10)(D) + (.25)(D/3) = 10.15

.3D + .1D + (.25D/3) = 10.15

We can now multiply everything by 3 to get rid of the fraction…

.9D + .3D + .25D = 30.45

1.45D = 30.45

D = 30.45/1.45

D = 21

Now, consider ALL of the work that I just did. Even if you took a slightly different approach to the algebra, how long would all of this work take…? Two minutes? Three minutes? Longer?

Instead, let’s use the ‘design’ of the GMAT to our advantage. Here, the answer choices ARE numbers, and we’re asked to solve for just one variable (the number of dimes), so let’s TEST THE ANSWERS.

To start, we’re told that the number of dimes is 3 TIMES the number of quarters, so the number of dimes MUST be a MULTIPLE OF 3. That helps us to immediately eliminate answers B and D (since 10 and 20 are NOT multiples of 3).

Let’s TEST Answer C: 18. If it’s the correct answer, then we’ll be done. If it’s “too high” or “too low”, then we’ll know exactly which of the remaining two answers is the correct one.

IF… there are 18 dimes
Then there are 6 quarters (since there are three times as many dimes as quarters) and there are 108 nickels (since there are 6 times as many nickels as dimes).

Given the respective values of the three coins, we would have a total of….

(108)(.05) = \$5.40
(18)(.10) = \$1.80
(6)(.25) = \$1.50
Total = \$8.70

However, we were told that the actual total value of the coins is \$10.15. This total (\$8.70) is TOO LOW, which means that we need there to be more nickels, dimes and quarters. There’s only one answer left that ‘fits’ what we’re looking for, so that MUST be the correct answer!