Question

Cara is buying pencils and erasers from the store. Pencils come in packages of 12, but erasers are sold in packages of 4. If Cara 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 that Cara can buy while ensuring she purchases the same number of pencils, we need to relate the packages of pencils and erasers:

- Pencils come in packages of 12.
- Erasers come in packages of 4.

Let \( p \) be the number of packages of pencils that Cara buys, and \( e \) be the number of packages of erasers. The total number of pencils \( P \) and the total number of erasers \( E \) can be expressed as:

\[
P = 12p
\]
\[
E = 4e
\]

We want the total number of pencils to equal the total number of erasers, so we set \( P = E \):

\[
12p = 4e
\]

To solve for \( e \) in terms of \( p \), we can rearrange this equation:

\[
e = \frac{12p}{4}
\]
\[
e = 3p
\]

This means that for every \( p \) packages of pencils, Cara needs to buy \( 3p \) packages of erasers. Consequently, the total number of erasers \( E \) can be calculated as:

\[
E = 4e = 4(3p) = 12p
\]

The smallest integer value for \( p \) is \( 1 \) (buying 1 package), which gives us:

\[
E = 12p = 12(1) = 12
\]

Thus, when Cara buys \( 1 \) package of pencils, she needs to buy \( 3 \) packages of erasers, resulting in a total of:

\[
E = 4 \times 3 = 12
\]

Therefore, the smallest number of erasers that Cara can buy is \(\boxed{12}\).