About reciprocal in algebra

[Unit_02]

Here is something i was able to understand in theory but not in its practical use.

I understand the example below only if compared to, for example, the example of "1/(1/2)". In this case, without using the reciprocal technique, i understand that the result its "2".



Looking at the "solution" for this exercise it says that the answer is "w" because the reciprocal of 1/w is w/1 (???). I get it, again, i understand that specific example, but can i use it like that, without any restriction when i encounter a bigger equation? (I know that i cant, but just wanna know why he did it and how can i do it properly)

My question is, for more complexes exercises, how can i use the reciprocal technique in a more consistent way? Is it a valid technique or something that has no real value when solving exercises?

( I know that the question is not that clear but any information about how to use it will be helpful)

 

[Dr.Peterson]

Why do you think you can't always do it (when applicable)? In effect, in this problem, you have the reciprocal of a reciprocal, which is the original number.

But if you are unsure when you can interpret 1/something as the reciprocal of that something, you can use other techniques to simplify. For example, you can treat division as multiplication by the reciprocal:

[MATH]\frac{1}{\frac{1}{w}} = 1\div\frac{1}{w} = 1\cdot\frac{w}{1} = w[/MATH]​

Or, you can multiply the numerator and denominator by w:

[MATH]\frac{1}{\frac{1}{w}} = \frac{1\cdot w}{\frac{1}{w}\cdot w} = \frac{w}{1} = w[/MATH]​

Can you give an example where you would want to use "the reciprocal technique" but would not be sure?

 

[Steven G]

Note that 10/2 = 5 and that 2*5 = 10. Same with 12/4 = 3 and 3*4 =12. This pattern always work. That is if you did not know what 10/2 equals you can always ask yourself 2* what = 10 and then know what 10/2 equals.

In 1/(1/w) just ask yourself (1/w)*what = 1? That answer is w.


Alternatively you can multiply 1/(1/w) by w/w (which is 1 and multiplying by 1 does not change the original value) and get w/1 which is w.

 

[Unit_02]

Why do you think you can't always do it (when applicable)? In effect, in this problem, you have the reciprocal of a reciprocal, which is the original number.

But if you are unsure when you can interpret 1/something as the reciprocal of that something, you can use other techniques to simplify. For example, you can treat division as multiplication by the reciprocal:

[MATH]\frac{1}{\frac{1}{w}} = 1\div\frac{1}{w} = 1\cdot\frac{w}{1} = w[/MATH]​

Or, you can multiply the numerator and denominator by w:

[MATH]\frac{1}{\frac{1}{w}} = \frac{1\cdot w}{\frac{1}{w}\cdot w} = \frac{w}{1} = w[/MATH]​

Can you give an example where you would want to use "the reciprocal technique" but would not be sure?

I either use the least common multiple technique, or i multiply everything by something that i think will give me a good shoot to get rid of the denominators, but both this process are sometimes time consuming.

For example, if a want to "solve" 4a/5 =100 -12/b (i mean isolate "ab = x") but using the reciprocal technique. When there is only one fraction i understand (i think), my problem is how to use it consistently when there are different fractions around.

 

[Dr.Peterson]

I'm not sure what "the reciprocal technique" means, exactly, applied to an equation like [MATH]\frac{4x}{5} = 100 - \frac{12}{b}[/MATH], but I suspect it is not applicable. The reciprocal itself is not a "technique", but a concept that can be used when it's useful. It would take extra work to use reciprocals when you are solving an equation with multiple fractions. Your other example was entirely different -- not an equation to solve, but an expression to simplify.

Here, I would clear fractions by multiplying everything by the LCD, 5b. (But there is no "ab = x" to solve for!)

 

[Unit_02]

I'm not sure what "the reciprocal technique" means, exactly, applied to an equation like [MATH]\frac{4x}{5} = 100 - \frac{12}{b}[/MATH], but I suspect it is not applicable. The reciprocal itself is not a "technique", but a concept that can be used when it's useful. It would take extra work to use reciprocals when you are solving an equation with multiple fractions. Your other example was entirely different -- not an equation to solve, but an expression to simplify.

Here, I would clear fractions by multiplying everything by the LCD, 5b. (But there is no "ab = x" to solve for!)

"The reciprocal itself is not a "technique", but a concept that can be used when it's useful." That was gold! So its not a thing you use it whenever one wants, that's what was missing for the puzzle. I looked online and could not find a statement like that. That cleared my mind a lot. So its useful to simplify expressions and not to solve equations.

Thank you very much, that was puzzling me since this morning. I love this forum!