Originally Posted by

**HallsofIvy** To point out a rather obvious mistake, you have

$\displaystyle \frac{660}{x+ 10}$ as the "rate" of machine A and

$\displaystyle \frac{660}{x}$ as the rate of machine B

but then say "Let x = the amount of sprockets per hour produced by Machine A."

The numerator, 660, is a number of sprockets so "sprockets" divided by "sprockets per hour" has units of hours, which is NOT a "rate"!

In order that $\displaystyle \frac{660}{x}$ be the rate of machine B, in sprockets per hour, x must be the time in hours that it takes machine B to make 660 sprockets. Then, since "It takes machine A ten hours longer to produce 660 sprockets than machine B" it is true that A's rate, again in "sprockets per hour" is $\displaystyle \frac{660}{x+ 10}$. So in one hour, A will make $\displaystyle \frac{660}{x+ 10}$ and B will make $\displaystyle \frac{660}{x}$ sprockets. We are also told that the second of those is 10% more than the first. Saying "P is 10% more than Q" means P= Q+ 0.10Q= 1.1Q.