Efficient Algebra Question Solutions: Ratios and Equations

[Miike012]

Homework Statement

I know the answers to the question however I was wonder if there was a way of figuring out the answer rather than just pluggin in answers and guessing.

first question: Find two numbers in the ratio of 7:12 so that the greater exceeds the less by 275.

second questionif 15(2x^2-y^2)=7xy, find the ratio x:y

The Attempt at a Solution

I aloready know the answers but it was only because I plugged in numbers one after the other... is there a quicker way?

 

[Doc Al]

first question: Find two numbers in the ratio of 7:12 so that the greater exceeds the less by 275.

How about using algebra? Let the numbers be X and Y. How would you express the two relationships given using algebraic equations?

 

[Miike012]

Sorry, I am drawling a blank. I don't have much alg under my belt.
The only thing that I can think of is obviously the denominator is larger than the numerator... therefore the den. must be larger then the num. by 275. And the final answer must be a ratio of 7:12. But I don't know how to represent the information I just gave you into two numbers X and Y.

 

[Doc Al]

How would you express the ratio of X and Y?

 

[Miike012]

This was what I was thinking exactly... I thought of x/y and because we are relating x/y and the ratio 7/12.. i set x/y = 7/12, then I know that y needs to be bigger than x by 275
so I set y = x + 275... but that really wouldn't make sence... and that is where I am stuck... hmmm.

 

[Doc Al]

This was what I was thinking exactly... I thought of x/y and because we are relating x/y and the ratio 7/12.. i set x/y = 7/12, then I know that y needs to be bigger than x by 275
so I set y = x + 275... but that really wouldn't make sence... and that is where I am stuck... hmmm.

Why wouldn't that make sense? It's perfect, so far. Now all you have to do is solve those two equations together.

Hint: Rewrite x/y = 7/12. (Cross multiply)

 

[Miike012]

then i would get 12x = 7y

 

[Doc Al]

then i would get 12x = 7y

Good! Now combine that with your second equation.

 

[Miike012]

Alright I am back... ok combine 12x = 7y If y = x + 275 then

12x = 7x + 1925
5x = 1925
x = 385
y = 385 + 275 = 660 !
Woohoo! thanks lol.

 

[Doc Al]

Good. Now try the other one. How can you rewrite that equation totally in terms of x/y?

 

[Miike012]

Would you first set

15(2x^2-y^2) - 7xy = x/y ?

 

[Doc Al]

Would you first set

15(2x^2-y^2) - 7xy = x/y ?

No. How did you get that?

Hint: Start with your original equation and divide. Twice.

 

[Miike012]

The only thing that I can think of is trying to get the variables on one side so I would divide by 15 and xy

then I would get 2x^2/xy - y^2/xy = 7/15

But to be honest I don't know what I am looking for.

 

[Doc Al]

Hint: What do you have to divide x^2 by to get (x/y)^2?

 

[Miike012]

divide x^2 by y^2

 

[Doc Al]

divide x^2 by y^2

Good. So divide both sides of your original equation by y^2 and see what happens.

 

[Miike012]

30x^2/y^2 -1 = 7x/y

 

[Miike012]

almost looks kinda like a quadratic but I've never seen a quadratic with x over y...

 

[Miike012]

unless I substitue x/y for u then I get 30u^2 - 7u -1 = 0
then I get u = 1/3 ; -1/10

then x/y = 1/3 ---- x = 1/3y ; x/y = -1/10 ---- x = -1/10 y

then I plug in x = 1/3y ; x/y = -1/10 ---- x = -1/10 y
to the original equation both seperatly
That wouldn't get my anyhting would it?

 

[Doc Al]

30x^2/y^2 -1 = 7x/y

Double check the coefficient of that second term.

almost looks kinda like a quadratic but I've never seen a quadratic with x over y...

Well, now you have! Since x/y is what you are asked to find, solve that equation and you're good. (But correct it first.)

 

[Miike012]

30x^2/y^2 - 15 = 7x/y

 

[Doc Al]

30x^2/y^2 - 15 = 7x/y

Good. Now solve that quadratic. (It might be easier to let z = x/y and rewrite it in terms of z.)

 

[Miike012]

your right... thank you. I hope to be good like that one day lol. I just need a tons more practice.
But thakn you.