Question

hair grows at a rate of around $5 \times 10^{-2}$ inches per month, whereas nails grow around $1 \times 10^{-2}$ inches per month. use these numbers to complete the sentence below.
nails grow about \square times as fast as hair does.

Explanation:

Step1: Identify the growth rates

Hair growth rate: $5 \times 10^{-2}$ inches per month.
Nail growth rate: $1 \times 10^{-2}$ inches per month.

Step2: Calculate the ratio of nail growth rate to hair growth rate

To find how many times faster nails grow than hair, we divide the nail growth rate by the hair growth rate:
$$\frac{1 \times 10^{-2}}{5 \times 10^{-2}}$$

Step3: Simplify the ratio

The $10^{-2}$ terms cancel out, leaving $\frac{1}{5} = 0.2$. Wait, no—wait, actually, we want nails' rate compared to hair? Wait, no, wait: Wait, hair is $5\times10^{-2}$, nails are $1\times10^{-2}$. Wait, that would mean hair grows faster. But the question says "Nails grow about [ ] times as fast as hair does." Wait, maybe I mixed up. Wait, no—wait, maybe the numbers are reversed? Wait, no, the problem says hair is $5\times10^{-2}$, nails $1\times10^{-2}$. Wait, that would mean hair grows 5 times faster than nails. But the question is asking how many times nails grow as fast as hair. So that would be nails' rate divided by hair's rate: $(1\times10^{-2}) / (5\times10^{-2}) = 1/5 = 0.2$. But that would mean nails grow 0.2 times as fast as hair (i.e., hair is 5 times faster). But maybe there's a typo, but according to the numbers given:

Wait, let's do the calculation again. Nail rate: $1e-2$, hair rate: $5e-2$. So ratio is $(1e-2)/(5e-2) = 1/5 = 0.2$. So nails grow 0.2 times as fast as hair (or hair grows 5 times as fast as nails). But the question is phrased as "Nails grow about [ ] times as fast as hair does." So according to the numbers, that's 0.2. But maybe I misread. Wait, maybe the hair is $1\times10^{-2}$ and nails $5\times10^{-2}$? But the problem says hair is $5\times10^{-2}$, nails $1\times10^{-2}$. So proceeding with the given numbers:

So the calculation is $(1\times10^{-2}) / (5\times10^{-2}) = 1/5 = 0.2$. So nails grow 0.2 times as fast as hair. But that seems odd. Wait, maybe the problem has a typo, but based on the given numbers, that's the calculation.

Wait, no—wait, maybe I made a mistake. Wait, the question is "Nails grow about [ ] times as fast as hair does." So we take nails' rate divided by hair's rate. So:

Nail rate: $1e-2$, hair rate: $5e-2$. So $(1e-2)/(5e-2) = 1/5 = 0.2$. So the answer is 0.2. But that would mean nails grow 1/5 as fast as hair. But maybe the problem intended hair to be $1e-2$ and nails $5e-2$? But as per the problem statement, we have to use the given numbers. So:

Answer:

0.2