Expert: Josh - 11/19/2009

Question
QUESTION: Hello:

Can $120/5% and $120/2% be tweaked or manipulated in some manner so that the amounts of $2000.00, and $1000.00 can be determined?

$120/5% = $2400.00 and $120/2% = $6000.00

I want $100/5% = $2000 or $100/$2000 = 5% from $120/5%, and I want $20/2% = $1000 or $20/$1000 = 2% from $120/2%.

$2400.00 is $400.00 too much and $6000.00 is $5000.00 too much.
$100.00/5% = $2000.00 and $20.00/2% = $1000.00.

NOTE: we do NOT know that $120.00 is divided as $20.00 and $100.00.

Use a basic calculation if possible.

I thank you for your reply.


ANSWER: Hi Kenneth,

Once again, I will make the point that it is best to leave out the "%" symbol and "$" sign in any formal mathematics expression.

In general, "division of B by 100*A percentages" is best written as B/A, with B and A each in decimal form. This makes it easy to treat everything as just numbers.

e.g., we can write (divide $120 by 5%) as 120/0.05. If you so wish to convert this to 2000 (dollars), then we ask "what conversion factor do we multiply 120/0.05 by, to get 2000"?

We set this up as (120/0.05)*(X/Y) = 2000.
Setting Y=120 will immediately simplify the LHS of the equation. Due to cancellation of the term "120" in the numerator and denominator, we are left with X/0.05 = 2000. Finally, multiply both sides by 0.05, we get X=2000*0.05=100.

This means, if we multiply 120/0.05 by 100/120, we will get 2000 as desired.

You can use the same approach to convert 120/0.02 to 1000.
| see (120/0.02)*(X/Y) = 1000
| Let Y=120, then X/0.02 = 1000
| X=1000*0.02
| Required multiplier is (X/Y) = 20/120 = 1/6




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

QUESTION: Hello Josh:

I want to thank you for your reply. I have a follow-up question:

Using the amounts of $120 and $3000, if you do NOT know that $120.00 is divided as $20.00 and $100.00, and you do NOT know that the amount of $3000 is divided as $2000 and 1000, how would the amounts of $1000 and $2000 be determined from $120/5% = $2400.00 and $120/2% = $6000.00?

From $120/5%, I want $100/5% = $2000 or $100/$2000 = 5%, and from $120/2% I want $20/2% = $1000 or $20/$1000 = 2%.
Or use $120/$3000 to with the other information to determine the amounts of $1000 and/or $2000.

In your first reply, you used $1000 and $2000 in your calculations. These amounts are unknown, so they cannot be used in the calculation.

Use a basic calculation if possible.

I thank you for your reply.

ANSWER: Hi Kenneth,

I don't quite understand what you are asking. It confuses me when you say "the amounts" of 1000 (or 2000) "are unknown" and "cannot be used in the calculation". Why would a figure like "1000" be unknown if I intend to turn another number into "1000" via a scaling operation.

In my first reply, I thought that you wanted to manipulate a number like 120/0.05 (which has a value of 2400) into an arbitrary number. Let's call this arbitrary number A. If A=1000, for instance, then we can multiply 120/0.05 by (X/Y) to get A=1000.

The equation of interest is (120/0.05) * (X/Y) = A.

We are free to choose any number we like for Y. For convenience, we can pick, say, Y=120. Then, we are left with X/0.05 = A which implies X=0.05*A. It doesn't matter What A represents, if we let Y=120 and X=0.05*A, when we multiply 120/0.05 by (X/Y), we always get back "A".

COMMENTS
========
Mixing "$" signs with "%" is an impediment to progress. If I were you, I would translate any thought of $D/(100P)% into a number like D/P as a starting point. When we deal with pure numbers, we can multiply (D/P) by a scaling factor, e.g., X/Y, to get any number (or amount) A that you want; without worrying about percentages.

EXAMPLE
=======
e.g., Suppose you are thinking of dividing $120 by 2%. On paper, we start with 120/0.02 [This corresponds to the (D/P) part]. To turn this into another number you have in mind, say, A=4000 (this can be anything you like), suppose we multiply 120/0.02 by (X/Y) to get A.

i.e., we focus on the equation (120/0.02)*(X/Y)=A .......[#]
To make things easy, let Y=120 for the purpose of cancellation. This leaves us with X/0.02=A, which means X=0.02*A. Finally, the scale factor (X/Y)=0.02*A/120. So, if your objective is to get A=5000, for example, the scale factor would be (X/Y)=0.8333333...

According to [#], (120/0.02) * 0.8333333... equals 5000.

In fact, the scale factor (X/Y) is always "A" (the target number) multiplied by the inverse of (D/P).

Example 2: To turn 300/0.04 (=7500) into 1500, the required scale factor is 1500/(300/0.04), i.e., (X/Y) = 1500*0.04/300 = 0.2. Just to verify, ($3000/0.04) * 0.2 = $1500.



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

QUESTION: Hello:

Hello:

Can your solution be applied to determine the answers to the following?

An investor invests $3000 into two accounts earning 2% and 5% simple interest for one year.
How much will he need to invest in the two accounts to earn 4% total return or $120/$3000?

Answers: $1000 and $2000

Once again, can $120/5% and $120/2% or ($120/$3000) be tweaked or manipulated in some manner so that the amounts of $2000.00, and $1000.00 can be determined or separated from the amount of $3000?

$120/5% = $2400.00 and $120/2% = $6000.00

From $120/5%, I want $100/5% = $2000 or $100/$2000 = 5%, and from $120/2% I want $20/2% = $1000 or $20/$1000 = 2%.

$2400.00 is $400.00 too much and $6000.00 is $5000.00 too much.

$100.00/5% = $2000.00 and $20.00/2% = $1000.00.

We do NOT know that $120.00 is divided as $20.00 and $100.00. And we do NOT know that $3000 is divided as $2000 and 1000.

NOTE: Use a basic calculation if possible. I already have an algebraic calculation. I am looking for an alternative method.

I thank you for your reply.  

Answer
I think what we looked at last time is not unrelated to the question that you now pose.

Recapping on what we have done -- previously, we considered the problem of finding a multiplicative factor (X/Y), such that when given a fraction (A/B) and a target number C, (A/B)*(X/Y)=C. We concluded that this scaling factor (X/Y) always equal C*(B/A).

This has no direct connection with
| "An investor invests $3000 into two accounts earning 2% and 5% simple interest
| for one year. How much will he need to invest in the two accounts to earn 4%
| total return or $120/$3000?"

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Re: "Once again, can $120/5% and $120/2% or ($120/$3000) be tweaked or manipulated in some manner so that the amounts of $2000.00, and $1000.00 can be determined or separated from the amount of $3000?"

Ans: I fail to see how using 120/0.05 and 120/0.02 would make it any easier to find the answer. It seems to invite more trouble than what it is worth. Why not use conventional algebra techniques (which we have discussed at length in previous correspondence). I do not believe these so called "alternative" methods are anything new. They probably originate from intermediate steps that emerge when we solve simultaneous equations in the usual way, or some observations that we make in retrospect. I do not see the point of reinventing the wheel when we have a trusted, reliable method based on standard algebra.
|
| a*x + b*(P-x) = c*P, where a=0.05, b=0.02, c=0.04, P=3000.
| (a-b)*x = (c-b)P
| x = (c-b)/(a-b)*P

There is nothing that we can add that is not already contained in these steps.

The quantities you referred to, viz., 120/0.05 and 120/0.02 are simply identified as c*P/a and c*P/b, respectively. They do not naturally arise in obtaining the solution. They are not relevant to the problem. Any attempt to introduce them just complicates things needlessly.

Ans: We have taken two lines to essentially obtain the answer by standard algebra. My question is how would the things you introduced, e.g., "From $120/5%, I want $100/5% = $2000..." get us closer to finding the solution without creating extra burden. If your steps are not tied in with the underlying constraints of the problem, you are going no where. And for a problem like this, there is no simpler (and more general) way of solving it than using standard algebra.