Is (ab = ac) = (b = c) for all values a?

  • Thread starter CuriousBanker
  • Start date
In summary, when simplifying a fraction like a/(b/c), you can multiply both the numerator and denominator by the denominator's reciprocal (in this case c) to get ac/b. However, this is only valid if both b and c are not equal to 0, as division by 0 is undefined. It's important to be careful when using this method, as it may lead to incorrect results if used with 0 as a value for any variable.
  • #1
CuriousBanker
190
24
I know that by definition, a/(b/c)= a x 1(b/c)...but from there I am lost. Help please!
 
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  • #2
Well, I'm terrible at formal proofs, but consider:

a/(b/c)

multiply numerator and denominator by c, so that we get b by itself as the denominator (nobody likes pesky fractions in their denominator):

(c/c)*[a*(b/c)]

You can do this because c/c = 1.

now you have:

(c*a) / [c*(b/c)]

which equals:

(c*a) / b
 
  • #3
Nevermind I got it!
 
  • #4
Ah, king you posted that just as I got it. Thanks though!
 
  • #5
a/(b/c) = ac/b

Just a note that you have to be careful. The above is only true when both b and c are not 0. The reason is that division by 0 is undefined. In the above proof (c/c) is used which is 0/0 when c=0. 0/0 is undefined.

I just got shut out of physics thread by a Mentor poster who believed that:

ab = ac implies b = c for all values a. That is not correct when a is 0.
 

FAQ: Is (ab = ac) = (b = c) for all values a?

What is the equation for proving that a/(b/c) = ac/b?

The equation for proving that a/(b/c) = ac/b is (a/b) * (c/b) = ac/b.

What is the meaning of the variables in this equation?

The variable "a" represents the numerator of the first fraction, "b" represents the denominator of the first fraction, and "c" represents the numerator of the second fraction.

Why is it important to prove this equation?

Proving this equation is important because it shows the relationship between fractions and their reciprocals, and helps us understand how to simplify complex fractions.

What are the steps to prove this equation?

The steps to prove this equation are:
1. Write out the equation: (a/b) * (c/b) = ac/b
2. Multiply the fractions on the left side: (a * c)/(b * b) = ac/b
3. Simplify the denominator on the left side: (a * c)/b² = ac/b
4. Multiply both sides by b: (a * c)/b = ac
5. Simplify the fraction on the left side: ac/b = ac/b

Can this equation be applied to any values of a, b, and c?

Yes, this equation can be applied to any values of a, b, and c as long as b and c are not equal to 0. If either b or c is equal to 0, the equation becomes undefined.

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