When is it NOT okay to simplify?

[RebeccaD696]

Question: 20 is 16% of what number?

When I simplify, I get the wrong answer (see work below). My question is, when is it okay vs. not okay to simplify in a problem? Can you never cross-simplify across an equals sign?

How I solved:
20/x = 16/100 --> 1/x = 16/5 --> 16x = 5 --> x = 5/16

Correct way to solve:
20/x = 16/100 --> 20/x = 4/25 --> 4x = 500 --> x = 125

Thank you!

[esledge]

Hi Rebecca, when you did this first step...

20/x = 16/100 --> 1/x = 16/5

...you divided by 20 on both sides of the equation. However:

On the left side you divided the top number by 20 (making the whole left side smaller by a factor of 20)
On the right side you divided the bottom number by 20 (making the whole right side larger by a factor of 20).

Thus, your answer was off by a factor of 20*20 = 400. (Your answer times 400 is the correct answer: 400 * 5/16 = 25 * 5 = 125.)

So, you can only simplify if you are following the golden rule of algebra/arithmetic:
(1) You only do steps that keep the value equivalent to the original (e.g. 16/100 = 4/25)
OR
(2) You can change the values, but only if you are doing the same change to both sides of an equation, so that the sides of the equation are not changing relative to each other.