Proportions & Units

[KWF]

Hello:

I want to solve the following by using a proportion, but I'm not sure what units to use on the left side of the equal sign.

Two partners, A and B, share $1000.00 in a ratio of 2:3. How much does each get?


? units/? units = $?/$1000.00 The dollar units are on this side.

Answers: A $400.00; B: $600.00

 

[mmm4444bot]

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I'd forget the units.

2 is to 3 the same as A's dollars are to B's dollars

2/3 = A/B

Express B in terms of A, then solve for A

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[KWF]

Hello mmm4444bot

How do I "Express B in terms of A, then solve for A."

I want to know the following from my first question: "I'm not sure what units to use on the left side of the equal sign."

 

[Quaid]

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How do I "Express B in terms of A, then solve for A."

You have a relationship between the numbers A and B (their sum is 1000).

In other words, A+B=1000.

This means that B is the number 1000-A. (The expression 1000-A expresses the number B, in terms of A.)

But, I just noticed that you posted on the arithmetic board, so maybe you have not studied any algebra, yet.

Let's try a different approach -- one that does not involve solving an equation.

I want to know the following from my first question: "I'm not sure what units to use on the left side of the equal sign."

Sometimes, numbers are just numbers (not measurements). In such cases, there might be no units associated with particular numbers.

2/3 = A/B

In this exercise, however, we can think of the units on the left side of the proportion as "parts".

That is, 1000 dollars is divided into five equal parts; person A gets two parts, and person B gets three parts.

So, we can state the proportion as "2 parts is to 3 parts the same as A's dollars are to B's dollars".

You know how to divide 1000 into five equal parts, yes? Give two of them to A; give the rest to B.

Cheers

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[KWF]

Thanks Quaid for the reply and your solution!

I think the 2:3 ratio indicates dollar amounts that partner A and B will share. I set the proportion with the units on the left-hand side as follows and just solved for ?. The tricky part was knowing to add the part-to-part ratio of 2:3 to get the part-to-whole ratio of 2:5

$2/$5 = ?/$1000

Now cross multiply to get the following:

$5 X ? = $2 X $1000

Now isolate the ? on the left side of the equal sign by dividing both sides of the equal sign by $5.

? = ($2 X $1000)/$5

Now cancel the dollar sign units to get 2 X $200 = $400 = A's share.

I can see that having the dollar sign unit on the ? would not cancel. The calculation would be $? = 2 X $200 = $400, which could be also expressed at $? = $400. This does not look correct.

This proportion, if I understand correctly, indicates that for every $5 partner A gets $2. For every $1000, he gets $400. $5 is to $2 as $1000 is to $400 and vice versa.

 

[Quaid]

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I think the 2:3 ratio indicates dollar amounts that partner A and B will share.

I don't see 2/3rds as a ratio of dollars; I see it as a comparison of two numbers. And, if I see dollar signs in a ratio, I cancel them.

$A/$B = A/B

I view the proportion 2/3=A/B as telling us that A's money compares to B's money the same as the number 2 compares to the number 3. No units needed, in these comparisons.

In other words, the fraction 400/600 is equivalent to the fraction 2/3.

I suggested thinking about "parts" because you seem to really want to understand the 2 and the 3 in terms of something other than numbers (even though any units would cancel, as each would be the same unit). I also suggested thinking in terms of "parts" because that approach does not require solving an equation (each part is 200).

By the way, if we multiply parts times dollars, is the resulting unit part-dollars?

If we multiply dollar units times dollar units, is the resulting unit dollars-squared?

I would forget units and simply work with the numbers involved, in both the proportion and calculations. I would report the final answer with dollar signs.

PS: Have you studied any algebra, yet? Are you familiar with dimensional analysis (a process in which units cancel)?

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