Why is a machine producing 7 nails in 9 seconds faster than one producing 11 nails in 14 seconds?

Question

Machine A Produces 11 nails in 14 seconds.

Machine B produces 7 nails in 9 seconds.

Which is faster?

I thought that $\frac{11}{14}$ is larger than $\frac{7}{9}$ and thus A is the answer. However, the answer says that $\frac{9}{7}$ is bigger than $\frac{14}{11}$ so B is the answer. How does this work?

Answer 1

You are right indeed

$$\frac 7 9 < \frac {11}{14}\iff7\cdot 14 <11\cdot9 \iff98<99$$

Answer 2

Your answer is correct. In 126 seconds (= 14 $\times$ 9), machine A produces $11 \times 9 = 99$ nails while machine B produces $7 \times 14 = 98$ nails. Therefore machine A is faster.

Answer 3

If you make their denominators equal, you obtain $11/14={99\over14×9}$ and $7/9={7×14\over14×9}$. Since $99>7×14=98$, we have that $11/14>7/9$.

Answer 4

What is the meaning of "Which is faster"? The answer depends on specific value. A is faster than B, If you need enough many nails.

However I understood the question. Your text's answer is correct, too. This answer compares each speed in one second.