Basic Algebra & Varible Cancellation: is 5/a * 2/(a/b) equal to 10b

KWF

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Have I solved the following correctly?

1. 5/a * 2/(a/b) = 10b

2. 5/a * 2/(a * b) = 10/(a^2 b)

3. 5/a * (2a)/b = 10b

Are equations 1. and 3. the same?
 
Have I solved the following correctly?

1. 5/a * 2/(a/b) = 10b

2. 5/a * 2/(a * b) = 10/(a^2 b)

3. 5/a * (2a)/b = 10b

Are equations 1. and 3. the same?

Your answers to problems 2 is correct, but problem 1 and problem 3 are incorrect. Many times, expressions can end up being equivalent even though they look wildly different. However, that is not the case here. Let's begin be rewriting the given expression from problem 1 to help better see what's going on:

\(\displaystyle \dfrac{5}{a} \cdot \dfrac{2}{\left( \dfrac{a}{b} \right)}\)

There's a fraction in the denominator of the second fraction, so we can use the fact that multiplication and division are inverses to help us out. When we multiply by something, we can divide by that same something to "undo" it and vice versa. But we also can see that multiplying by 1/x is the same as dividing by x, and dividing by 1/x is the same as multiplying by x. With any fraction \(\displaystyle \dfrac{p}{q}\), we know that \(\displaystyle \dfrac{1}{\left( \dfrac{p}{q} \right)} = \dfrac{q}{p}\). In other words, "one over" a fraction is the same as taking its reciprocal. Let's do that now and change it to multiplication:

\(\displaystyle \dfrac{5}{a} \cdot 2 \cdot \dfrac{b}{a}\)

Can you finish up from here? What do you get? Do you see why it's not the same as the given expression from problem 3?

You're also very close on problem 3, but I suspect you made an error when multiplying. The given expression looks like:

\(\displaystyle \dfrac{5}{a} \cdot \dfrac{2a}{b}\)

Multiplying straight across, as per the rules of multiplication, gives:

\(\displaystyle \dfrac{5 \cdot 2a}{a \cdot b}\)

What would your final step be? How does this new answer differ from your original answer? Do you see why this is correct and your original answer was not?
 
Your answers to problems 2 is correct, but problem 1 and problem 3 are incorrect. Many times, expressions can end up being equivalent even though they look wildly different. However, that is not the case here. Let's begin be rewriting the given expression from problem 1 to help better see what's going on:

\(\displaystyle \dfrac{5}{a} \cdot \dfrac{2}{\left( \dfrac{a}{b} \right)}\)

There's a fraction in the denominator of the second fraction, so we can use the fact that multiplication and division are inverses to help us out. When we multiply by something, we can divide by that same something to "undo" it and vice versa. But we also can see that multiplying by 1/x is the same as dividing by x, and dividing by 1/x is the same as multiplying by x. With any fraction \(\displaystyle \dfrac{p}{q}\), we know that \(\displaystyle \dfrac{1}{\left( \dfrac{p}{q} \right)} = \dfrac{q}{p}\). In other words, "one over" a fraction is the same as taking its reciprocal. Let's do that now and change it to multiplication:

\(\displaystyle \dfrac{5}{a} \cdot 2 \cdot \dfrac{b}{a}\)

Can you finish up from here? What do you get? Do you see why it's not the same as the given expression from problem 3?

You're also very close on problem 3, but I suspect you made an error when multiplying. The given expression looks like:

\(\displaystyle \dfrac{5}{a} \cdot \dfrac{2a}{b}\)

Multiplying straight across, as per the rules of multiplication, gives:

\(\displaystyle \dfrac{5 \cdot 2a}{a \cdot b}\)

What would your final step be? How does this new answer differ from your original answer? Do you see why this is correct and your original answer was not?



I want to thank you for the reply!

The solution to 1. would be 10b/a^2

3. I still get 10/b because the "a" variables cancel out from 5/a * (2a)(/b)

If I understand correctly, variables and numbers function the same way. For example, 5/2 * 2/(3/4) = 5/2 * 2* 4/3 = 40/6.

Thanks again for your assistance!
 
I want to thank you for the reply!

The solution to 1. would be 10b/a^2

3. I still get 10/b because the "a" variables cancel out from 5/a * (2a)(/b)

If I understand correctly, variables and numbers function the same way. For example, 5/2 * 2/(3/4) = 5/2 * 2* 4/3 = 40/6.

Thanks again for your assistance!

Yes, 10/b is the correct answer to problem 3. In your original post, however, you wrote 10b, without the division sign. From your writing here, I assume this was a typo?
 
Yes, 10/b is the correct answer to problem 3. In your original post, however, you wrote 10b, without the division sign. From your writing here, I assume this was a typo?

Yes, I believe it is a typographical error from my first posting.

Thanks again for your assistance!
 
Correct. Same as 20/3.

If variables and numbers function the same way as fractions in the denominator, will units of measure function the same way?

Examples: If I'm not mistaken, the following should be as follows: 2/(a/b) = (2 * b)/a just as 2/(3/4) = (2 * 4) /3

But what about $0.75/(share/quarter)? Is it incorrect to express this as ($0.75 * quarter)/share?
 
If variables and numbers function the same way as fractions in the denominator, will units of measure function the same way?

Examples: If I'm not mistaken, the following should be as follows: 2/(a/b) = (2 * b)/a just as 2/(3/4) = (2 * 4) /3

But what about $0.75/(share/quarter)? Is it incorrect to express this as ($0.75 * quarter)/share?
The unit conversions are correct - however "kind of" meaningless.

Instead of "share/quarter" possibly you meant to write "quarter/share"?
 
The unit conversions are correct - however "kind of" meaningless.

Instead of "share/quarter" possibly you meant to write "quarter/share"?

Sorry, I'm confused. Do units of measure function as numbers and variables when they are denominators in fractions.
 
If variables and numbers function the same way as fractions in the denominator, will units of measure function the same way?

Examples: If I'm not mistaken, the following should be as follows: 2/(a/b) = (2 * b)/a just as 2/(3/4) = (2 * 4) /3

But what about $0.75/(share/quarter)? Is it incorrect to express this as ($0.75 * quarter)/share?
A variable is a symbol that represents a single number just as a numeral does. The difference is that a numeral stands for a known number whereas a variable stands for an unknown number. It is a conceptual mistake to think that a variable is somehow different from any other number.

Units of measure are not numbers themselves, but they are measured in numbers. So measures are numbers. To keep track of what the measuring numbers mean, we keep the units of measurement attached to the numbers. However, when we are dealing with one unit of measure, we frequently ignore the number: the 1 is implied.

\(\displaystyle 75 \text { cents for each share in each quarter IS REPRESENTED mathematically }\)

\(\displaystyle \dfrac{\dfrac{ 75 \text { cents }}{1 \text { share}}}{1 \text { quarter}} =\)

\(\displaystyle \dfrac{\dfrac{ 75 \text { cents }}{1 \text { share}}}{\dfrac{1 \text { quarter}}{1}} =\)


\(\displaystyle \dfrac{ 75 \text { cents }}{1 \text { share}} * \dfrac{1}{1 \text { quarter}} =\)

\(\displaystyle \dfrac {75 \text { cents}}{\text { share quarter}}\)
 
And yes, in division, we can cancel units of measurement like numbers.
 
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