Converting Units

[KWF]

If some unit of measure is to be converted to another unit of measure, does the process involve division or multiplication?

Example: Convert 300 seconds to minutes.

Which is more accurate?

1. 300 seconds divide by 60 seconds/one minute

2. 300 seconds multiplied by one minute /60 seconds?

I know that 60 seconds/minute is a unit rate; so I think 1. is more accurate because the process involves determining how many groups of 60 seconds are in 300 seconds. And this process requires division.

 

[JeffM]

If some unit of measure is to be converted to another unit of measure, does the process involve division or multiplication?

Example: Convert 300 seconds to minutes.

Which is more accurate?

1. 300 seconds divide by 60 seconds/one minute

2. 300 seconds multiplied by one minute /60 seconds?

I know that 60 seconds/minute is a unit rate; so I think 1. is more accurate because the process involves determining how many groups of 60 seconds are in 300 seconds. And this process requires division.

One method is correct; the other is not. In this specific case, division is correct. But conversion of units may involve multiplication in other cases. Example: convert 12 minutes to seconds. If you divide by 60, you will conclude that 0.5 seconds has the same duration as 12 minutes, which violates common sense. If instead you multiply, you get 720 seconds, which is consistent with common sense and correct.

In other words, sometimes division is right when converting units, and sometimes multiplication is right when converting units. There is a technique called dimensional analysis that will help you figure out when to multiply and when to divide. Have you heard of it?

 

[mmm4444bot]

… Which is more accurate?

1. 300 seconds divide by 60 seconds/one minute

2. 300 seconds multiplied by one minute /60 seconds …

These are two different ways of looking at the same thing; each of them simplifies to 5 min.

Division is the same as multiplication by the reciprocal.

A/B = A/(B/1) = A∙(1/B) = A/B

To convert 300 seconds to minutes, I realize that I need to cancel seconds and introduce minutes, so I would choose to multiply by the ratio that has seconds in the denominator and minutes in the numerator.

 

[MarkFL]

This is how my dad taught me to convert between units. Suppose we want to convert from mph to ft/s. So, we write:

\(\displaystyle x\dfrac{\text{mi}}{\text{hr}}\)

First, we state the relationship between miles and feet:

\(\displaystyle 1\text{ mi}=5280\text{ ft}\)

We want feet in the numerator, so we divide through by \(\displaystyle 1\text{ mi}\) to get:

\(\displaystyle 1=\dfrac{5280\text{ ft}}{1\text{ mi}}\)

Next, we state the relationship between hours and seconds:

\(\displaystyle 1\text{ hr}=3600\text{ s}\)

We want seconds in the denominator, so we divide through by \(\displaystyle 3600\text{ s}\) to get:

\(\displaystyle 1=\dfrac{1\text{ hr}}{3600\text{ s}}\)

Multiplying our two values of 1, we get:

\(\displaystyle 1=\dfrac{5280\text{ ft}}{1\text{ mi}}\cdot\dfrac{1\text{ hr}}{3600\text{ s}}\)

Since this is 1, we may multiply our original quantity by this value without changing its magnitude:

\(\displaystyle x\dfrac{\text{mi}}{\text{hr}} =x\dfrac{\text{mi}}{\text{hr}} \cdot\dfrac{5280\text{ ft}}{1\text{ mi}} \cdot\dfrac{1\text{ hr}}{3600\text{ s}}\)

Next divide out the unwanted units:

\(\displaystyle x\dfrac{\text{mi}}{\text{hr}} =x \cdot\dfrac{5280\text{ ft}}{3600\text{ s}}\)

Reduce the remaining numbers:

\(\displaystyle x\dfrac{\text{mi}}{\text{hr}} =x \cdot\dfrac{22\text{ ft}}{15\text{ s}}\)

And so we have:

\(\displaystyle x\dfrac{\text{mi}}{\text{hr}} =\dfrac{22}{15}x\, \dfrac{\text{ ft}}{\text{ s}}\)

And this tells us to convert from mph to ft/s, we multiply the speed in mph by 22/15 to convert to the same speed in ft/s. After you use this method a few times, you can streamline it so that it can all be done in one or two steps.

 

[Dr.Peterson]

If some unit of measure is to be converted to another unit of measure, does the process involve division or multiplication?

Example: Convert 300 seconds to minutes.

Which is more accurate?

1. 300 seconds divide by 60 seconds/one minute

2. 300 seconds multiplied by one minute /60 seconds?

I know that 60 seconds/minute is a unit rate; so I think 1. is more accurate because the process involves determining how many groups of 60 seconds are in 300 seconds. And this process requires division.

As has been said, mathematically these are identical, as dividing by 60 is the same thing as multiplying by 1/60. So they are also equally "accurate", in the sense of giving the right answer. But I think you knew that, and are asking which is a better way to think about it -- that is, a "more accurate" way to express what you are doing.

I would say that either can be explained as a natural way to think about the problem, so both are equally "accurate" explanations.

The first way, as you said, you can think of grouping the 300 seconds into groups of 60 seconds, each of which is 1 minute. This means dividing 300 by 60. It's a very good way to think.

The second way, you can think of a second as 1/60 of a minute; 300 seconds is therefore 300 of these, that is, 300 times 1/60. This is also a very good way to think of it.

Others have stated additional ways to think, which are perhaps more mechanical and involve less "understanding"; but they are also good.

Yet another way, incidentally, is just to consider that since a minute is bigger than a second, there must be fewer minutes than seconds; so you do the operation that produces a smaller number, which is to divide by 60 (or, equivalently, multiply by 1/60)!

 

[mmm4444bot]

… Yet another way, incidentally, is just to consider that since a minute is bigger than a second, there must be fewer minutes than seconds; so you do the operation that produces a smaller number, which is to divide by 60 (or, equivalently, multiply by 1/60)!

Yes, I've done this while working on something abstract and reasoning symbolically.

If somebody walked up to me on the street and asked, "How many minutes is 300 seconds?", I'd probably think of breaking 300 into 60 equal pieces, mentally drop zeros, and that's 30/6.

If we're working with paper and pencil, and I'm showing someone dimensional analysis, then I treat the ratios as conversion factors, and I think in terms of what needs to cancel and what needs to be introduced.

It's good to have so many viewpoints posted; students can stuff their toolkits like a turkey!