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Incomes of $A$ and $B$ are in the ratio $4:5$ and expenditures are also in the ratio $4:5$. Who saves more?

Options:

I) A

II) B

III) both save equally

IV) cannot be determined on the basis of the information provided

We've tried solving this by taking Income of A = 4x and expA = 4y, Inc B as 5x and exp B = 5y. But these are all ratios so there's no way of actually determining the value. We tried hypothetically taking A:B actual values as 40:50, and savings as 4:5, and here were getting B saves more. But we have no way of knowing if that would always apply.

There are also different variations of the question where the expenditure is in different ratios, 5:6 as an example. The popular opinion of my group seems to be the answer would be CBD regardless.

Michael Rozenberg
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Pranjali Ingle
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    What have you tried? Where did you get stuck? Please don't just ask us to do your homework for you. – Noah Schweber Mar 26 '17 at 00:59
  • I don't know where to begin, and I came here looking for help. Isn't that the whole point of this site? This isn't "homework", I'm studying for a competitive exam and some of us believe the answer cannot be determined and want to leave it at that. I want to dig deeper. So either you can help, or you cant. Either way, please don't assume I've done nothing. – Pranjali Ingle Mar 26 '17 at 01:03
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    Welcome to Math.SE! Please do not take this personally but I downvoted this question due to the lack of context. Please provide your effort showing your thoughts and what you've tried to solve the problem (And add it to your post) so that we know exactly where you are stuck with the problem and don't repeat information you already know. Then I will not only remove my downvote: I'll replace it with an upvote. – projectilemotion Mar 26 '17 at 01:04
  • We're happy to help, but as I said you need to work with us. Have you tried to set up an equation? Do you know what incomes and expenditures being in certain ratios means? – Noah Schweber Mar 26 '17 at 01:04
  • Couldn't you elaborate more ? The information you have specified is pretty sparse. You should at least edit the question to give more details about the problem. – John Kontol Mar 26 '17 at 01:07
  • This is literally the question that was presented, guys. There's no more information. We've tried creating hypothetical equations, but it's spiralling into a mess without values. I'll edit in the options, and I'll put up the equations and perhaps you can tell us where we're going wrong. – Pranjali Ingle Mar 26 '17 at 01:10
  • "This is literally the question that was presented, guys. There's no more information." Yeah, we're not asking for more detail on the question, but rather on what you've tried. "I'll put up the equations and perhaps you can tell us where we're going wrong." Yes, that is exactly what we've asked you to do. – Noah Schweber Mar 26 '17 at 01:34
  • @Noah Shweber: As the title says, we're not sure of the concept, so the whole thing may be off target. I typed in the whole thing just to convince you helpful folk that we've racked our brains. And we have an answer that seems too easy. And we don't get that a lot. Someone did ask for more detail on the question, hence the clarification. – Pranjali Ingle Mar 26 '17 at 01:40
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    As promised, I removed my downvote and replaced it with an upvote. – projectilemotion Mar 26 '17 at 01:41
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    @projectilemotion I appreciate that. Just learnt I could reply to specific comments as well! – Pranjali Ingle Mar 26 '17 at 01:43
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    @projectilemotion As have I. To the OP: the point was never to prove you've worked on it. The answer alone won't help you very much - but if you show us what you've tried, we can see what exactly (if anything) you don't understand, where (if anywhere) you've gone wrong or what (if anything) you haven't thought of, and so actually help you. – Noah Schweber Mar 26 '17 at 01:43
  • @NoahSchweber True. I need someone to help with the concept itself, so we can apply it regardless of the ratio of expenditure (which is where the variance in questions seems to be). Can this be determined without values? I have the vague notion we're missing something, but can't quite put my finger on it. – Pranjali Ingle Mar 26 '17 at 01:47
  • @Pranjali Ingle Thanks for doing the edit. With this, I remove my downvote and replaced by an upvote. – Juniven Acapulco Mar 26 '17 at 01:51

1 Answers1

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You've started in the right direction, but you've kept your equations separate; the key part is figuring out how to combine the equations you've gotten to represent the information you've been given.

In this case, you have $$Inc(A)=4x,\quad Inc(B)=5x,\quad Exp(A)=4y, \quad Exp(B)=5y.$$

Alright, but the problem is asking about the savings $A$ and $B$ make - how does savings relate to income and expenditure?

Well, this is just: $$Sav=Inc-Exp.$$ So we have $$Sav(A)=Inc(A)-Exp(A)=4x-4y,\quad Sav(B)=Inc(B)=Exp(B)=5x-5y.$$

That's step one. Now, we want to compare these two quantities. That is, we're asking:

Which is larger, $4x-4y$ or $5x-5y$?

So let's subtract the first from the second; if the difference is positive, the second is bigger, and if it's negative the first is bigger, and if it's zero they're equal.

This difference is $$(5x-5y)-(4x-4y)=x-y.$$

So now the entire problem boils down to:

Is $x-y$ positive or negative?

Do you think this is a question that you have enough information to answer, or does it depend on what exactly $x$ and $y$ are? What does this tell you about the answer to the whole problem?

Noah Schweber
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  • So I would actually depend on the values of x and y, which we don't have. The answer would have to be CBD in my opinion. – Pranjali Ingle Mar 26 '17 at 01:55
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    @PranjaliIngle Exactly. Now the last part: let's prove you're right. If the answer can't be determined, then you have to be able to find one set of values for $x$ and $y$ making the difference negative, and another set of values making the difference positive. Can you do that? – Noah Schweber Mar 26 '17 at 01:59
  • I don't get it.. without actual values how could we possibly get the two values of differences? They could be anything. I suppose we could take hypothetical values, x=90 , y=10; so Savings A:Savings B =36:45.. but if I reverse that, I.e. x=10, y=90 I get negative values, which can't be right. How can savings be negative ? Ugh. – Pranjali Ingle Mar 26 '17 at 02:09
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    @PranjaliIngle No, you've done it right! Savings being negative makes perfect sense. HINT: "Wow, this sucks - I can't find a job, and rent costs $10000000 a month!" What's my savings in this scenario? (Also, why on earth am I still in this apartment?) – Noah Schweber Mar 26 '17 at 02:12
  • Haha, I'm already homeless, in that case. So we're saying X HAS to be more than Y. Or well, could be equal. In either case, B has to save more, because B makes more. Even if his expenses are more than As. – Pranjali Ingle Mar 26 '17 at 02:16
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    @PranjaliIngle No, we're not. It's quite possible for $x$ to be less than $y$; that just means that I'm losing money. That stinks, but it's possible (seriously, have you seen the economy lately?). So it is entirely possible for B to be saving less than A - that is, losing more. – Noah Schweber Mar 26 '17 at 02:19
  • Negative savings are not, well, savings :s But mathematically speaking, if either are possible, I'm stumped again. X can be >=< to y. Hence CBD, then? – Pranjali Ingle Mar 26 '17 at 02:23
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    @PranjaliIngle Yup. And that's a fair confusion, but I'd say losses are negative savings. – Noah Schweber Mar 26 '17 at 02:39
  • That makes sense. I'm going to try and apply that to different expenditure ratio values. Thank you so much! – Pranjali Ingle Mar 26 '17 at 02:42