A party planner is preparing gift bags. She has a box containing 12 toy cars and another b… — ACT

A party planner is preparing gift bags. She has a box containing 12 toy cars and another box containing 18 stickers. She wants to purchase the minimum number of boxes of each toy so that she has the exact same number of toy cars and stickers, with no items left over. If she puts all of these purchased items into the gift bags, what is the minimum total number of items (cars and stickers combined) she will have?

A
6
B
12
C
30
D
36
72Answer

Answer

To find the minimum total number of items, we must first find the least common multiple (LCM) of the box sizes, 12 and 18. The prime factorizations are

12 = 2^2 \times 3

and

18 = 2 \times 3^2

. The LCM is

2^2 \times 3^2 = 4 \times 9 = 36

. This means the planner must buy 36 toy cars (3 boxes of 12) and 36 stickers (2 boxes of 18) to have an equal amount of each. The minimum total number of items combined is

36 + 36 = 72

Step-by-Step Solution

Identify that the number of toy cars and stickers must be equal and must be a multiple of the box sizes (12 and 18). Find the least common multiple (LCM) of 12 and 18 to determine the minimum number of each item needed.

LCM(12, 18) = 36

Since the planner wants the minimum number of boxes and equal amounts of both items, the number of each item must be the smallest common multiple of the two box sizes.

Calculate the total number of items by adding the number of toy cars and stickers together.

36 + 36 = 72

The question asks for the total number of items combined, so we must add the 36 toy cars to the 36 stickers.

Key Concept

Least Common Multiple (LCM)

Estimated Time:1m 30s

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