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## Comment on

Hockey versus Football## I think this problem is

## In both equations (3H - 2F

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

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

## Awesome approach!!!

Awesome approach!!!

## Hmm I can't quite see what he

## I'll add some information to

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

H/F=2X/3X

On Saturday,

2x+18/3x-18=5/3,

6x+54=15x-90

-9x=-90-54

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

## Perfect!

Perfect!

## I LOVE testing the answers!

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.

## I agree with much of what you

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:

- https://gre.myprepclub.com/forum/if-4x-is-9-greater-than-the-sum-of-x-an...

- https://gre.myprepclub.com/forum/the-sum-of-2-numbers-x-and-y-equals-twi...

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.

Cheers,

Brent

## So the way I did it was

## The key here is to remember

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

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

80/5=16

x=16

Rest you can try yourself.

## I partially agree with you.

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

## You didn't do anything wrong.

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