Expert: Josh - 1/19/2011

Question
QUESTION: Hello:

I want to change this direct proportion into an indirect proportion and solve for the answer:

40 feet of wire rope weigh 26 pounds.  What is the length 65 pounds of the same kind of rope weigh?

Direct Proportion Solution:

Feet 40/? = Pounds 26/65
Answer ? = 100 feet

I think that the above would be converted as follows as an indirect proportion:

Indirect Proportion Solution:

40/? = (1/65)/(1/26)
100 = [40 X (1/26)]/(1/65)

So, the proportion is as follows:

40/100 = (1/65)/(1/26)

In the direct proportion example, 40 feet and 26 pounds indicate that 40 feet of wire rope weigh 26 pounds. Likewise, 100 feet of the same kind of wire weigh 65 pounds.

However, in the indirect proportion example, what do 40 feet and 100 feet indicate in relation to 1/65 and 1/26 pounds?

I thank you for your reply.

ANSWER: Kenneth,

Try rewriting 40/100 = 26/65 as (1/100)/(1/40) = (1/65)/(1/26).
Then, in the numerator, (1/100) identifies with (1/65). In the denominator, (1/40) identifies with (1/26).


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

QUESTION: Hello Josh:

The proportion that you indicated uses the inverse of 100, that is, 1/100, but the solution is for 100 feet.

(1/100)/(1/40) = (1/65)/(1/26) is proportional, but 40/? = (1/65)/(1/26)
100 = [40 X (1/26)]/(1/65) is the answer of 100 feet.

I want to convert a direct proportion to a indirect proportion and solve for the same answer if that is possible or makes sense.

I thank you for your follow-up reply.

Answer
I do not see any real difference between what you called "direct proportion" and "indirect proportion". If we pose the problem as 40/X=26/65, the task is to find X.

(a) Using cross multiplication, X=(40*65)/26.

(b) You could, by making things more complicated (this in my view is completely unnecessary) write the same equation [40/X=26/65] as 40/X=(1/65)/(1/26). Then, you could say, X=40*(1/26)/(1/65). It is just dancing around, without achieving anything really.

(c) You can also, if you insist, write the equation [40/X=26/65] as 1/[X*(1/40)]=(1/65)/(1/26).
Just to make this clear, this is same as (1/X)*(1/(1/40)) = (1/65)/(1/26).

So, (1/X)=(1/40)*(1/65)/(1/26).
Finally, taking the reciprocal on both sides, we get X=(1/26)/[(1/65)*(1/40)] which is same as 40*(1/26)/(1/65), which is in turn identical to 40*65/26 = 100.

This just about covers all possibilities. But why make things more complicated by doing (b) or (c) when the answer can be found simply using (a).