Question: Hockey versus Football

Comment on Hockey versus Football

Alternately we can solve it as follows-
Since we are dealing with ratios we can eliminate choices B and D as they are not divided by both 5 (2+3) and 8 (5+3).
Working with 40 we can see that it can be divided in ratio of 2:3 (25,15) but taking 18 from 25 to 15 doesn't give us a ratio of 3:5. Similarly 120 can be eliminated. In fact we would have got the answer 80 if we started working with option C as we usually do.
gmat-admin's picture

Great approach - I love it!
The great/interesting thing about GMAT math questions is that they can typically be solved using more than one approach.

Hi Sandy
Can I please request you to explain how you arrived at 5(2+3) and 8(5+3)


Dear JSN
On Friday the ratio is 2:3, therefore, the total number of fans has to be divisible by 5 (2+3). That means our answer cannot be 72 or 108 because these numbers cannot be split into ratio of 2:3 without using fractions. Next day the ratio becomes 5:3 when 18 fans switch over side (from football to hockey). It means that if we split our number into ratio of 2:3 (hockey:football) and then move 18 from football to hockey, the ratio of hockey to football must become 5:3. for example, if number of fans is 40 then on Friday the ratio is 16:24 (2:3) but if we move 18 from football to hockey it will become 34:6 which is not equal to (5:3). Similarly we can eliminate 120. In case of 80 the ratio on Friday will be 32:48 and on Saturday 50(32+18):30(48-18) which is equal to 5:3, hence the answer.

Thank you Sandy :)

Alternately it could be :

Since, 3H = 2F. We could just substitute it in the second eq. to get the value of F directly and hence find H.

I solved this in another way. I gave the ratios variables.
So Fri - 2x:3x and Sat - 5y:3y.
So the two equations are - 3x-18=3y and 2x+18=5y.

Now in this way, I need solve only for one variable. I solve for y, get it as 10 and add 5(10)+3(10)=80.

Is this a valid approach?
gmat-admin's picture

Perfectly valid!

and alternatively - just plug in the answer choices. Since those are in ascending order.take Answer Choice "c". 80. so 80=sum of the ratios 5:3 ---> 8. So 5 units = 50 and 3 units = 30. not we reverse the action stated (As we go back in time) and subtract 18 from 50 "giving" it back to 30. so the new ration is 32 Hockey fans to 48 Football fans. simplifying the ratio to its smallest form give you 2:3 as stated and we are done!
gmat-admin's picture


Can this be solved with 1 variable ?
gmat-admin's picture

You bet.

GIVEN: On Friday, the ratio of hockey fans to football fans was 2/3

Let 2x = number of hockey fans
So, 3x = number of football fans
(notice that these values ensure that the ratio = 2/3)

GIVEN: On Saturday, 18 football fans became hockey fans
So, 2x + 18 = number of hockey fans on SATURDAY
And 3x - 18 = number of football fans on SATURDAY

GIVEN: the ratio of hockey fans to football fans became 5/3
So: (2x + 18)/(3x - 18) = 5/3
Cross multiply: 3(2x + 18) = 5(3x - 18)
Expand: 6x + 54 = 15x - 90
Solve: x = 16

So, the number of hockey fans on Friday = 2(16) = 32
So, the number of football fans on Friday = 3(16) = 48
TOTAL number of fans = 32 + 48 = 80


You guys are awesome. Thanks a lot :)

Like more people suggested in the comments, I started by checking answer choice C:

80/ 5 (because the ratio is 2:3) = 16
Hockey players: 2 * 16 = 32 | Football players 3 * 16 = 48 -> (32 + 48 = 80)
Hockey players: 32 + 18 = 50 | Football players 48 - 18 = 30 -> 50 + 30 = 80 as well

I do understand that the algebraic equations need to be practiced for questions where the tactic of checking answer choices is less convenient ;).

gmat-admin's picture

That's a perfectly valid approach, especially when you start with the correct answer :-)

It's hard to say whether testing the answer choices would be the fastest approach if one had to test 3 or 4 answer choices.


Hi, what is the level of this question?
gmat-admin's picture

I'd say this is a medium-difficulty question (between 500 and 600).

can I write 2F - 3H = 0 ?
gmat-admin's picture

Since the ratio of hockey fans to football fans is 2 to 3, we can write: H/F = 2/3
Cross multiply to get: 2F = 3H
Subtract 3H from both sides to get: 2F - 3H

Hi Brent,

I tried using this method but could not get the answer:

Since 18 members increased the ratio from (1) 2:3 to (2) 5:3. I thought that would make sense that ratio 5-2 = 3 mean 3 ratio = 18 members. Thus, 1 ratio = 6 members. If there is a total if 8 ratios (5+3), then total number of fans is 8*6 = 48.

May I know what is wrong with this?
gmat-admin's picture

That method will work, but for a slightly different question type.
Your solution would work if the question told us that 18 NEW hockey fans were added to the group, and the ratio changed to 5 to 3.

However, in the given question, 18 people stopped being football fans and switched to being hockey fans.

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