Question

Riley is buying pencils and erasers from the store. Pencils come in packages of 8, but erasers are sold in packages of 12. If Riley wishes to purchase the same number of pencils as erasers, what is the smallest number of erasers that she can buy?

Answer 1

To find the smallest number of erasers Riley can buy such that the number of pencils is equal to the number of erasers, we start by defining the number of packages of pencils and erasers that Riley buys.

Let \( p \) be the number of packages of pencils and \( e \) be the number of packages of erasers. Each package contains 8 pencils and 12 erasers, so the total number of pencils and erasers can be expressed as follows:

- Total pencils = \( 8p \)
- Total erasers = \( 12e \)

We want the total number of pencils to equal the total number of erasers:

\[
8p = 12e
\]

To simplify this equation, we can divide both sides by 4:

\[
2p = 3e
\]

This equation shows the relationship between \( p \) and \( e \). We can rearrange it to express \( p \) in terms of \( e \):

\[
p = \frac{3e}{2}
\]

From this equation, for \( p \) to be an integer, \( e \) must be an even number because \( 3e \) must be divisible by 2. The smallest even integer is 2. By substituting \( e = 2 \) into the equation, we can find \( p \):

\[
p = \frac{3 \times 2}{2} = 3
\]

Now we can calculate the total number of pencils and erasers when \( e = 2 \):

Total erasers:

\[
12e = 12 \times 2 = 24
\]

Total pencils:

\[
8p = 8 \times 3 = 24
\]

Since both totals are equal, it confirms that Riley can buy 24 erasers and 24 pencils. To ensure this is the smallest number, we conclude that:

\[
\text{Smallest number of erasers Riley can buy} = \boxed{24}
\]