We can use fractions like $\frac{1}{10}$ and $\frac{1}{100}$ to reason about the location of the decimal point in a product of two decimals.  

Let’s take $24 \boldcdot (0.1)$ as an example. There are several ways to find the product:

• We can interpret it as 24 groups of 1 tenth (or 24 tenths), which is 2.4.
• We can think of it as $24 \boldcdot \frac{1}{10}$, which is equal to $\frac {24}{10}$ (and also equal to 2.4).
• Because multiplying by $\frac {1}{10}$ has the same result as dividing by 10, we can also think of it as $24 \div 10$, which is equal to 2.4.

Here is another example: To find $(1.5) \boldcdot (0.43)$, we can think of 1.5 as 15 tenths and 0.43 as 43 hundredths. We can write the tenths and hundredths as fractions and rearrange the factors.

$\displaystyle \left(15 \boldcdot \frac{1}{10}\right) \boldcdot \left(43 \boldcdot \frac{1}{100}\right) = 15 \boldcdot 43 \boldcdot \frac{1}{1,000}$

Multiplying 15 and 43 gives us 645, and multiplying $\frac{1}{10}$ and $\frac{1}{100}$ gives us $\frac{1}{1,000}$. So $(1.5) \boldcdot (0.43)$ is $645 \boldcdot \frac{1}{1,000}$, which is 0.645.