Question: Equation with Many Fractions

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Comment on Equation with Many Fractions

The last example is wrong, as far as I have solved it. The result should be -8/5 and not -5/8...
greenlight-admin's picture

Can you show your calculations?
Once we get to -5 = 8x, we divide both sides by 8 to get: -5/8 = 8x/8
Simplify to get: -5/8 = x

Why can you multiply both sides by 20x if one of the denominators (5 in the second fraction on the left) does not include x?
greenlight-admin's picture

Yes, it's okay to do that.
When we do this, the only thing we need to be sure of is that x does not equal zero.
We can be certain that x does not equal zero, since the original equation wouldn't make any sense if x equaled zero.

I don't understand how you got -5/8. I looked back at how you carried the x and I think it was supposed to be 10x+12=20+15x instead of 10+12x=20x+15. Is this correct? If so, wouldn't the answer change to -8/5 which would mean that B would be the correct answer? Sorry if this is a dumb question.
greenlight-admin's picture

Let's take it step-by-step. When it comes time to multiple both sides by 20x (aka 20x/1), we'll show an extra step.

Given: 1/2x + 3/5 = 1 + 3/4x
Multiple both sides by 20x/1 to get: 20x/2x + 60x/5 = 20x + 60x/4x
Simplify: 10 + 12x = 20x + 15
Subtract 10 from both sides: 12x = 20x + 5
Subtract 20X from both sides: -8x = 5
Divide both sides by -8: x = -5/8

Does that help?

Yes, thanks!

my confusion is dropping the x from 20x/2x. is it not 10x?
greenlight-admin's picture

Let's use a nice fraction property that says ab/cd = (a/c)(b/d)
So, we can write: 20x/2x = (20/2)(x/x)
= (10)(1)
= 10

Aside: We also used the fact that x/x = 1

and the 60x/4x. why not 15x. Thanks
greenlight-admin's picture

We'll use the same fraction property: ab/cd = (a/c)(b/d)
So, we can write: 60x/4x = (60/4)(x/x)
= (15)(1)
= 15

Another way to verify whether 60x/4x = 15 is to TEST some values of x.

Try x = 2
60x/4x = 60(2)/4(2) = 120/8 = 15 PERFECT!

Try x = 3
60x/4x = 60(3)/4(3) = 180/12 = 15 PERFECT!

Try x = 10
60x/4x = 60(10)/4(10) = 600/40 = 15 PERFECT!

Since you are suggesting that 60x/4x = 15x, let's test this out by testing some more values

Try x = 2
Does 60x/4x = 15x?
Plug in x = 2 to get: 60(2)/4(2) = 15(2)
Evaluate both sides to get: 15 = 30
Doesn't work. So, it is not the case that 60x/4x = 15x

Does that help?

That helps. Thanks

When I added 1/2x + 3/5 I got 6x+5/10x. And 1+ (3/4x) is (4x+3)/4x. Setting these equal to each other and cross multiplying, I got 4x*(6x+5) = 10x*(4x+3). After simplifying I get the quadratic equation 16(x^2) +10x+0=0. This has two solutions, 0 and -5/8. Why do I get two solutions instead of only one like you did with your method? Thanks.
greenlight-admin's picture

Great question! First off, your solution strategy is perfectly valid (although perhaps longer).

We get x = 0 as a solution due to the fact that, if x = 0, the fractions on each side are UNDEFINED (since we're dividing by 0). The equation you created "behaves" as though x = 0 is a solution, but we must recognize that it's actually an invalid solution.

I went at this in a completely different way using equivalent fractions. 1/2x became 2/4x and 1 became 5/5. I was then able to just add and subtract to get -1/4x = 2/5. I then cross multiplied to get -5=8x then divided by 8 to get x = -5/8.
greenlight-admin's picture

Perfect!!

Hello, you said in a previous video that for matching operations we cannot multiply or divide by variables unless we are 100% sure that the variable is positive. How can we then now multiply by 20x? Thanks :D
greenlight-admin's picture

Hi Julian,

That's not quite what I said.

In Quantitative Comparison questions, you cannot multiply/divide the TWO QUANTITIES (Quantity A and Quantity B) by variables unless we are 100% sure that the variable is positive..

In this question, we are multiplying the given EQUATION by 20x. This is okay.

Here's the video that says you cannot multiply/divide the TWO QUANTITIES by variables unless we're sure the variable is positive: https://www.greenlighttestprep.com/module/gre-quantitative-comparison/vi...

Here's the video on solving EQUATIONS through matching operations: https://www.greenlighttestprep.com/module/gre-algebra-and-equation-solvi...

Does that help?

Cheers,
Brent

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