# Advanced Math/Proportion

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Question
QUESTION: Hello:

I have a question that I hope you can answer.  I want to solve the following by using a proportion, but I'm not sure what units to use on one side of the equal sign.

Two partners, partner Mr. Jones and Mr. Smith, share \$1000.00 in a ratio of 2:3. How much does each get?

I know that one side of the proportion will have dollars as the units, but I do not know what units will be on the other side of the proportion.  Can you explain?

?/? = \$/?1000.00

Answers: Mr. Jones \$400.00; Mr. Smith \$600.00.

I thank you for your reply.

ANSWER: Hi Kenneth,
The simple way to look at it is that the units on any side (on both numerator and denominator) should be the same since they would essentially cancel out. So, if Mr. Jones' share is \$M i.e M dollars
2/(2+3) = \$M / \$1000
which basically becomes
2/5 = M/1000
and
M = 400
Mr. Jones' share is therefore \$400
You can then work out Mr. Smith's share to be \$600 either by using the same method or just subtracting from \$1000.

Regards

---------- FOLLOW-UP ----------

QUESTION: Hello;

I want to thank you for the reply.

1. Would the proportion look like the following without canceled units?

\$2/\$5 = ?/\$1000 for Mr. Jones share

2. I'm not sure whether or not the units should appear on the unknown numerator.  Should it be \$?/\$1000 or ?/\$1000?

Please be sure to answer both questions!

I thank you for your reply.

ANSWER: Hi Kenneth,
1) As you can see from the earlier solution, I did not cancel any units on the left hand side as there are none. You should understand that a ratio of 2:3 simply means that the higher value should be 3/2 times the other value. That is all it says and nothing else, it just describes the relation between the magnitudes of the real quantities (the one with the units) on the right hand side. For instance we can say that \$10 is 'two' times as much as \$5. The statement has nothing to do with two dollars.

2) For this case there isn't exactly a wrong way to write it but you have to understand what the unknown (question mark) refers to. In my solution, M is 400 and not \$400. So it would be incorrect to say that Mr. Jones' share is M, as it is \$M.
If we had instead written the equality as
2/5 = M/\$1000
then M would be \$400 and not just 400.
Mr. Jones' share would now be M, and not \$M.
The most important thing is the numerical magnitude of M, the rest only matters in a manner of speaking.

Regards

---------- FOLLOW-UP ----------

QUESTION: Hello:

I want to thank you for your replies to my questions!

I understand some of what you have explained. However, I see nothing incorrect with leaving the dollar units on \$2/(\$2+\$3) = ?/\$1000.

These units cancel from determining the value of ?.

Example:  \$2/\$5 = ?/\$1000

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 = Mr. Jones' 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 Mr Jones gets \$2.  For every \$1000, he gets \$400.  \$5 is to \$2 as \$400 is to \$1000.

I thank you for your reply and comments!

Answer
Hi Kenneth,
It wouldn't exactly be incorrect to add the dollar unit on the left hand side, but why bother? If someone added oranges to the equation the answer would still come out correct since
2 oranges/5 oranges = ?/\$1000
becomes
2/5 = ?/\$1000
and
? = \$400
Now tell me, what exactly does oranges have to do with the situation? See?

Again, the difference in adding the dollar unit to the ? is that in the expression
\$5 X ? = \$2 X \$1000
? = \$400
while in the expression
\$5 X \$? = \$2 X \$1000
? = 400
It is that simple.
If you had \$10 in your pockets and someone asked you how much you have, you'd say ten dollars. But if you were instead asked how many dollars you have, you'd just say ten.
So, basically, stating the unknown in dollar units before the solution simply means that you KNOW it's in dollars. Leaving it without units mean that the units will emerge from the solution. Also, it's better to denote the unknown by an alphabet rather than the question mark sign.

Regards
Questioner's Rating
 Rating(1-10) Knowledgeability = 10 Clarity of Response = 10 Politeness = 10 Comment Thanks for your reply and assistance regarding my questions!

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