Question: Hockey versus Football

Comment on Hockey versus Football

I think this problem is rather difficult to understand. Why did you subtract the terms after stacking them? why didn't you add them up instead? Thanks
greenlight-admin's picture

In both equations (3H - 2F and 3H - 5F = -144) , we had 3H, so by SUBTRACTING the bottom equation from the top, the H's canceled out, leaving us with 3F = 144. Perfect! We can now solve this equation for F.

Conversely, if we take the equations 3H - 2F and 3H - 5F = -144 and ADD them, we get 6H - 7F = -144
Now what do we do? We can't do much with 6H - 7F = -144. We certainly can't solve this equation for H or F.

Does that help?

I took the people in the dormitory as "X" and use only the ratios to get the answer.
Friday H : F is 2:3
Hockey fans at Friday = (2/5)X
Saturday H:F is 5: 3
H fans Saturday = (5/8)X

(2/5)X + 18 = (5/8)X
16X + 720 = 25 X
720 = 9X
80= X

greenlight-admin's picture

Awesome approach!!!

Hmm I can't quite see what he did here. Would you mind elaborating a bit? Looks like an elegant approach
greenlight-admin's picture

I'll add some information to the solution.

Let x = number of people in the dormitory
Let H = number of hockey fans
Let F = number of football fans

For Friday, we have: H:F is 2:3
In other words, for every 5 dorm people, 2 are hockey fans
Or we can say that 2/5 of the people are hockey fans.
So, the number of hockey fans on Friday = (2/5)x

For Saturday, we have: H:F is 5:3
In other words, for every 8 dorm people, 5 are hockey fans
So, the number of hockey fans on Saturday = (5/8)x

On Saturday, the number of hockey fans increased by 18
We can write: (number of hockey fans on Friday) + 18 = number of hockey fans on Saturday
In other words: (2/5)x + 18 = (5/8)x
From here, we need only solve for x.

Multiply both sides by 40 to get: 16x + 720 = 25x
Subtract 16x from both sides: 720 = 9x
Solve: 80 = x

This how I approached this question:

On Saturday,


x=144/9 = 16(Using this value in the equation: 2x+18+3x-18 we get 80.
greenlight-admin's picture


I LOVE testing the answers! A true fool-proof method for practically all word problems that are multiple choice. Testing answer C) 80...

Originally you have 2 to 3 ratio of hockey to football. So 2x + 3x = 80. 5x = 80. x = 16. Therefore, you have 32 hockey and 48 football fans.

If 18 football fans become hockey fans, this now means you have 30 football fans and 50 hockey fans. What's 50 to 30 ratio? 5 to 3!! BOOM. There it is.

I think once you have the Test The Answers strategy down, you can easily bump your quantitative score 5 to 10 points.

greenlight-admin's picture

I agree with much of what you've said!
In my opinion, students don't consider testing the answer choices as often as they should.
In fact, I always tell students that, if you encounter a word problem with 5 answer choices, Plan A should automatically be to test the answer choices. Then give yourself 15 to 20 seconds to find an alternative approach that MAY be faster. If you don't come up with anything during that time, start testing the answer choices.

Having said all of that, I wouldn't go as far as saying the strategy works for "practically all word problems that are multiple choice." I'd say MAYBE half of the questions can be solved using that approach.

Here are a couple of examples where it's either impossible or very difficult to test the answer choices:

It's also worth noting that one of the drawbacks of testing the answer choices is that the strategy can often be a lot more time-consuming than other approaches.

tldr: testing the answer choices is a must-have strategy that all GRE students should have in their mathematical tool boxes.


So the way I did it was saying 2x+18/(3x-18) = 5/3 then i solved for x, but instead of plugging into back into x I plugged into 5/3 so I did 16*8 and got 128, why is that the wrong approach?
greenlight-admin's picture

The key here is to remember what x stands for.

In your solution,
2x = the ORIGINAL number of hockey fans
3x = the ORIGINAL number of football fans
NOTE: This ensures that the ratio of hockey fans to football fans is 2/3

So, when we solve your equation for x, we get x = 16
Since 2x = the total number of hockey fans, we can see that there are 32 hockey fans.
Since 3x = the total number of football fans, we can see that there are 48 football fans.
So the total number of people = 32 + 48 = 80

If we want to use the information from the ratio 5/3 (as you did), we need to recognize what happens AFTER 18 football fans switch to become hockey fans.
In this case (as you have noted in your solution), we know that:
2x + 18 = the NEW number of hockey fans
3x - 18 = the NEW number of football fans
So, when we solve our equation to get x = 16, we need to plug x = 16 into the values above.

In other words, the NEW number of hockey fans = 2(16) + 18 = 50
And the NEW number of football fans = 3(16) - 18 = 30
So the total number of people = 50 + 30 = 80

Total number of people in dormitory should be divisible by 2,3, and 5. When we check the answers, there are only 2 numbers in the answer choice divisible by all of the 3 integers. Next you can try both of them.

H=2x F=3x H+F=5x

Rest you can try yourself.
greenlight-admin's picture

I partially agree with you.
Since the original ratio is 2:3, the total number of people must be divisible by 5 (since 2+3 = 5)
And since the resulting ratio is 5:3, the total number of people must be divisible by 8 (since 5 + 3 = 8)

Hey Brent, On the second equation at 01.49. I ended up with 5F-3H=+144. can you please tell me where did I go wrong with the approach here?
greenlight-admin's picture

You didn't do anything wrong.
The equations 3H - 5F = -144 and 5F - 3H = 144 are equivalent (just like the equations 2x = 10 and 4x = 20 are equivalent).

If you take the equation: 3H - 5F = -144
And multiply both sides by -1, you get the equivalent equation -3H + 5F = 144
Rearrange to get: 5F - 3H = 144

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