To find the smallest number of bars of soap and bottles of soap sold by Norma's Bath Shop, we need to consider the packaging sizes:
- Bars of soap are sold in boxes of 6.
- Bottles of soap are sold in boxes of 12.
Let:
- \( x \) be the number of boxes of bars sold.
- \( y \) be the number of boxes of bottles sold.
The total number of bars of soap sold is \( 6x \) and the total number of bottles sold is \( 12y \). According to the problem, these totals are equal:
\[ 6x = 12y \]
To simplify, we can divide both sides by 6:
\[ x = 2y \]
This equation tells us that the number of boxes of bars sold \( x \) is twice the number of boxes of bottles sold \( y \).
To find the smallest number of bars and bottles that satisfy this equation, we can use the smallest positive integer for \( y \). If we set \( y = 1 \):
\[ x = 2 \times 1 = 2 \]
Now we can calculate the total number of bars and bottles sold:
-
Total bars sold: \[ 6x = 6 \times 2 = 12 \]
-
Total bottles sold: \[ 12y = 12 \times 1 = 12 \]
Thus, the shop sold 12 bars and 12 bottles of soap. Therefore, the smallest number of each type of soap that the shop could have sold is:
\[ \boxed{12} \]