Cross multiplying proportions. In 40/x = 4 why can't I do x = 4 * 1/40?

[JUWON]

This doesn't necessarily relate directly to cross multiply but I have a question.

In 40/x = 4 why can't I do x = 4 * 1/40? 4x = 40 makes sense and all but, why can't I multiply 1/40 on the other side to isolate the x? By doing this however, the answer becomes x = 1/10

 

[ksdhart2]

So, if I understand you correctly, you're trying to use cross multiplication to solve 40/x = 4. This would work just fine. Setting up the problem gives us:

\(\displaystyle \dfrac{40}{x}=\dfrac{4}{1}\)

Cross multiplying shows us that:

\(\displaystyle 40 \cdot 1 = 4 \cdot x \implies 4x=40 \implies x=10\)

Your other proposed solution sounds sensible at first glance, and it too will work, but I suspect you messed up a bit on the algebra to end up with x = 1/10. As you (should) know, when you multiply one side of an equation by something, you must also multiply the other side of the equation by the same something. Let's try multiplying both sides by 1/40 and see what happens:

\(\displaystyle \dfrac{40}{x} \cdot \dfrac{1}{40} = 4 \cdot \dfrac{1}{40} \implies \dfrac{40}{40x} = \dfrac{1}{10}\)

If we do a bit of simplifying, we're left with:

\(\displaystyle \dfrac{1}{x} = \dfrac{1}{10}\)

Then taking the reciprocal of both sides tells us that x = 10. This checks with the cross multiplication method, as well as simply plugging the solution back in. 40/10 = 4, as stipulated.

 

[mathmania]

Cross Multiplying

The way you tried to do it, you cross multiplied the 40 perfectly, but that leaves the x still in the denominator (bottom) on the left side.
So instead, try cross-multiplying both the 4 and the x, so that the x ends up on the right, on top, by itself, and the left side is 40 / 4.

 

[mmm4444bot]

The way you tried to do it, you cross multiplied the 40 [times 1] perfectly, but that leaves the x still in the denominator

If we cross-multiply "perfectly", we eliminate the fractions, so x is not in the denominator.

JUWON indicated in their post that they got 40 = 4x, after cross-multiplying. So, they did that correctly.

JUWON's question has to do with a different method, and the issue is that they did not multiply each side by 1/40.

So instead, try cross-multiplying both the 4 and the x, so that the x ends up on the right, on top, by itself, and the left side is 40 / 4.

Again, if we cross-multiply, we get 40 = 4x. That does not leave x by itself.

Your thoughts may be correct, mathmania, but you are not stating them correctly.