Question about this unit conversion principle (multiplying by "1")

In summary, the equation allows for multiplying 100,000 t by 1000kg/t to get 10E8 kg. This is incorrect because 0 kg = 0 t.
  • #1
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Homework Statement
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Relevant Equations
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Suppose I want to convert 100,000 metric Tonnes to kilograms, then I would perform, a unit cancellation:

Given that 1 t = 1000 kg

##100,000 t \times \frac{1000~kg}{1 t} = 1 \times 10^8 kg##, however, why are we allowed to multiply the 100,000 by that?

My reasoning is,
##1 t = 1000 kg##
##1 = \frac{1000 kg}{1 t}## therefore multiplying by ##100,000 t## by ##\frac{1000 kg}{1 t}## is the same as multiplying by 1

Does someone please know whether my reasoning is correct?

Many thanks!
 
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  • #2
Yes
 
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  • #3
Frabjous said:
Yes
Thank you for your reply @Frabjous!
 
  • #4
ChiralSuperfields said:
Suppose I want to convert 100,000 metric Tonnes to kilograms, then I would perform, a unit cancellation:

Given that 1 t = 1000 kg

##100,000 t \times \frac{1000~kg}{1 t} = 1 \times 10^8 kg##, however, why are we allowed to multiply the 100,000 by that?

My reasoning is,
##1 t = 1000 kg##
##1 = \frac{1000 kg}{1 t}## therefore multiplying by ##100,000 t## by ##\frac{1000 kg}{1 t}## is the same as multiplying by 1

Does someone please know whether my reasoning is correct?
No, it is not. Treat the dimensions in the equation as you would unknown variables.
 
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  • #5
hmmm27 said:
No, it is not. Treat the dimensions in the equation as you would unknown variables.
You need to expand on that.
 
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  • #6
##100,000t \times 1 = 100,000t## which is incorrect not the answer to the question.

##\frac t t=1## as a next step yields the correct answer.
 
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  • #7
hmmm27 said:
No, it is not. Treat the dimensions in the equation as you would unknown variables.
How is that different?

I guess you mean something like this?
##ax = by \Rightarrow x = \frac{b}{a} y##
then substitute to get ## cx = c (\frac{b}{a} y) = (c \frac{b}{a}) y ##
where ## a,b,c ## are constants and ## x, y ## are units (variables?)

This is instead of
##ax = by \Rightarrow 1 = \frac{by}{ax} ##
## cx = cx (1) = cx (\frac{by}{ax})= (c \frac{b}{a}) y ## as others suggested.

I feel like I'm missing your point here.
 
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  • #8
You can multiply 100,000t by 1 and you do indeed get 100,000t. So what? That not what is sought. What is sought is to get a final answer in kg, not in t. To do that you need to multiply 100,000t by 1000Kg/t to get 10E8Kg. Units matter.
 
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  • #9
My point is the OP used "reasoning" which, while useful in creating or validating a conversion factor, doesn't solve the equation by itself whereas, after including it in the calculation, simple cancellation does :
##100,000 \cancel t \times \frac{1,000kg}{\cancel t} = 100,000,000kg##
 
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  • #10
ChiralSuperfields said:
##1 = \frac{1000 kg}{1 t}##
I believe that such equation is not correct.
##1000~kg/t## is simply a rate, a proportion.
It is used only because it is useful in mathematical operations, where it can be cancelled to obtain the desired units.

IMHO, it is not different from ##3600~s/h## or ##1000~km/m##
 
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  • #11
Lnewqban said:
I believe that such equation is not correct.
##1000~kg/t## is simply a rate, a proportion.
It is used only because it is useful in mathematical operations, where it can be cancelled to obtain the desired units.

IMHO, it is not different from ##3600~s/h## or ##1000~km/m##
Is there a situation where assuming the equation is correct will get you in trouble?
I agee that it is not different from 3600s/h or 1000m/km.
 
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  • #12
Frabjous said:
Is there a situation where assuming the equation is correct will get you in trouble?
I agee that it is not different from 3600s/h or 1000m/km.
It is not incorrect if used as a conversion factor ; but the OP specifically asked "Are we allowed to multiply like that"... then proceeded to not bother, or at least not show the bother.
 
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  • #13
hmmm27 said:
It is not incorrect if used as a conversion factor ; but the OP specifically asked "Are we allowed to multiply like that"... then proceeded to not bother, or at least not show the bother.
I am looking for the substantive reason that you believe it is sometimes wrong.
 
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  • #14
Frabjous said:
I am looking for the substantive reason that you believe it is wrong.
That I believe what is wrong ? That ##100,000t = 100,000t## is not a useful answer for "convert from tonnes to kg's" ?
 
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  • #15
Frabjous said:
Is there a situation where assuming the equation is correct will get you in trouble?
I agree that it is not different from 3600s/h or 1000m/km.
No, unless it is taken out of context.
It may be mathematically correct, but I don't know enough to see any value in something like
##1=1~kilo-banana/1000~bananas##.

sddefault.jpg
 
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  • #16
Lnewqban said:
No, unless it is taken out of context.
It may be mathematically correct, but I don't know enough to see any value in something like
##1=1~kilo-banana/1000~bananas##.
Obviously you have never studied economics on the Planet of the Apes. :wink: Low utility situations do not invalidate a concept.

There are known pitfalls when dimensionality comes into play. For example, the addition of non-dimensional numbers or the dimensional equivalence of torque and energy. We struggle through.
 
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  • #17
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FAQ: Question about this unit conversion principle (multiplying by "1")

What is the principle of multiplying by "1" in unit conversions?

The principle of multiplying by "1" in unit conversions involves using conversion factors that are equivalent to 1 to change units without altering the value of the quantity. For example, since 1 inch is equal to 2.54 centimeters, the conversion factor 1 inch / 2.54 cm or 2.54 cm / 1 inch is effectively "1". This allows you to multiply by these factors to convert units while keeping the measurement's value consistent.

How do you choose the correct conversion factor when multiplying by "1"?

To choose the correct conversion factor, you need to identify the units you are converting from and to. The conversion factor should cancel out the original units and introduce the desired units. For example, to convert inches to centimeters, you would use the factor 2.54 cm / 1 inch, because multiplying by this factor will cancel out the inches and leave you with centimeters.

Can you provide an example of a unit conversion using the multiplying by "1" principle?

Sure! Let's convert 10 inches to centimeters. The conversion factor is 2.54 cm / 1 inch. By multiplying 10 inches by this factor: 10 inches * (2.54 cm / 1 inch) = 25.4 cm. Here, the inches unit cancels out, leaving you with the measurement in centimeters.

Why is it important to use the multiplying by "1" principle in unit conversions?

Using the multiplying by "1" principle ensures accuracy and consistency in unit conversions. It helps avoid errors that can arise from incorrect arithmetic or misunderstanding of conversion relationships. This principle maintains the integrity of the measurement by using equivalent values that do not alter the original quantity.

Are there any common pitfalls to avoid when using the multiplying by "1" principle?

Common pitfalls include using incorrect conversion factors, not properly canceling out units, and misinterpreting the direction of the conversion. Always double-check the conversion factor to ensure it is correct and that units are appropriately canceled. For example, using 1 inch / 2.54 cm instead of 2.54 cm / 1 inch would lead to an incorrect conversion from inches to centimeters.

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