Expert: Josh - 2/17/2010

Question
Hello:

1. If 63 books cost $126, what will 125 books cost?

Calculation: ($126 X 125)/63

Explanation: If 63 books cost $126, each book will cost one sixty-third of $126, which is expressed by 63 below the divisor line. 125 books will cost 125 times this number of dollars, which is expressed by writing 125 above the line as a factor of the dividend. The result is $250.

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If 15 men can do a piece of work in 7 days, in how many days can 21 men do the same work?

2. I think the calculation is (15 X 7)/21

Can you use a similar explanation from the first example to explain how the answer is determined for the second example?

Which would be used (1/21 men X 15 men or 1/21 men X 7 days) or neither?

Josh, I do not want an algebra calculation. I can solve the problem. I just want to know the logic behind calculation 2. as was explained with calculation 1.

I thank you for your reply.

Answer
Personally, I would write 7 x (15/21). But it makes no difference to the calculation how you group the terms, because a*(b/c) = (a*b)/c. This is justified by the associative law of multiplication.

Another point to make is that a*b/c is the same as a*b*(1/c). You can rearrange the order of multiplication in any way you like. This is not going to affect the answer. According to the commutative law of multiplication, you can write "a*b/c" as "a/c*b". The latter simply comes from swapping the "b" with "1/c". i.e., a*b*(1/c)=a*(1/c)*b.

So, from an arithmetic point of view, it makes absolutely no difference which one you choose. One expression is NO more and NO less "correct" than another.

From the point of view of understanding what you are doing, I would simply put it in this form:

7 x (15/21)

We identify "7" as the time duration.
We interpret the fraction (15/21) as a scaling factor (or ratio).
There is nothing more to it.

To make sense of the situation, we know intuitively that 21 people are going to be more productive than 15 people (working on the same job, all things being equal). So, it is going to take a lesser amount of time to complete the work compared to the situation before. Thus, the scaling factor must be "15/21" (less than 1) rather than "21/15". If the multiplier is greater than 1, it implies that the work would take longer to complete. This reasoning can be rephrase in a dozen different ways.

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Kenneth,

I can see exactly why you are puzzle. Even though the wording sounds similar, and both questions involve some unit price (or unit work rate) calculation, the first problem is actually quite different to the second problem.

It is almost impossible to explain this to you, or help you recognize this difference, if you insist on NOT using algebra. Using only numbers to formulate a problem can be very confusing, because we very quickly get loss on what each number actually represents and where they come from.

If you really want to understand the logic behind the calculation, please read the following carefully.

In the 2nd example: The time taken to complete the job = Amount of work / Effective work rate.
Mathematically, we can write this as T=W/R ......[#1]
| T=7 is the time taken to complete the job initially.
| T' is the time taken to complete the job with more people.
| W is the amount of work to be done.
| R is the effective work rate.
| r is the individual work rate.
| N=15 is the number of people working initially.
| N'=21 is the number of people working later.

Initially, the combined work rate R=r*N ......[#2]
Substituting [#2] into [#1], we determine the work rate per person to be r=W/(N*T).

Later, the situation is governed by T'=W/R' ......[#3]
The amount of work to be done remains unchanged.
With N'=21 people working together, the effective work rate becomes R'=N'*r. i.e., R'=N'*W/(N*T)....[#4]
So, substituting [#4] into [#3], we find
T'=W/R'
 =W/[N'*W/(N*T)]
 =T*N/N'       .....[#5]

Plugging in the numbers, T'=7x(15/21).


In the 1st example: Total cost = Number of books x Unit Price.
Here, we can write this as T=R or T=N*r ......[#6]
where
| T=126  represents the total cost initially.
| T' represents the total cost later when more books are purchased.
| r  is the price of each book.
| N=63 is the number of books purchased initially.
| N'=125 is the number of books purchased later.

IMPORTANT POINT:
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Fundamentally, the 1st example is described by an equation of the form T=R  or T=N*r. Here, the unit cost "r" appears in the numerator to begin with, whereas the 2nd example has the form T=W/R or T=W/(N*r), so "r" actually appears in the denominator.

Anyway, from [#6], we deduce that each book costs r=T/N ....[#7].
Later, the situation is described by T'=N'*r. The unit cost of each book remains the same (fixed at r), but the number of books is varied. So, we have T'=T*(N'/N).....[#8]
Plugging in numbers, T'=126x(125/63).

This is enough reason why we cannot always use an adhoc strategy and apply it to another problem blindly just because they "sound" similar. Using algebra, we don't get confused by the numbers. In fact, we have shown that the two examples are in fact quite different. This would explain why "divide by N, then multiply by N' " is the right thing to do in the 1st example. By contrast, we "multiply by N, then divide by N' " in the 2nd example.  Just compare [#5] with [#8].

Don't be intimidated by algebra. It is a friend, it gives true insight, it doesn't cloud our judgment  and it makes everything crystal clear. If we skip the explanation, the numerous paragraphs above can be condensed into 4 lines.

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2nd Example:
T=W/R, R=N*r => r=W/(N*T).
T'=W/R'=T*(N/N').

1st Example:
T=N*r => r=T/N.
T'=N'*r=T*(N'/N).
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