Question

At a party store, packages of 12 noisemakers are $2 each. Packages of 9 party hatsare $15 each. Mia wants to buy equal numbers of noisemakers and hats.What is the least amount of money Mia can spend?1. To have an equal number of each, the smallest number of noisemakers and hats Mia can buy is the least common multiple (LCM) of 12 and 9.What is the LCM of 12 and 9? Explain how you found the LCM. 2. To get that many noisemakers and hats:How many packages of noisemakers does Mia need to buy?How many packages of hats?Explain. 3. If Mia buys that many packages of each item, what is her total cost? Show your work

Answer 1

We have to find the minimum mount of noisemakers and hats she has to buy and the total cost of that purchase.

As the number of noisemakers (N) and hats (H) has to be the same, we have to find the least common multiple (LCM) between the units that came in the packages of each product: 9 and 12.

We can find the least common multiple of 9 and 12 by decomposing each of the numbers in factors. Then, the least common denominator (LCM) will be the product of the two numbers divided by the greatest common factor between the two numbers.

We can start by decomposing the numbers:

`\begin{gathered} 9=3\cdot3 \\ 12=3\cdot2\cdot2 \end{gathered}`

The greatest common factor between the two is 3, so we can calculate the LCM as:

`\text{LCM}(9;12)=(9\cdot12)/(3)=3\cdot12=36`

Then, she has to buy 36 noisemakers and 36 hats.

To know how many packages she need of each product, we divide the number of hats or noisemakers she has to buy by the number of units per package of each product.

This means that she has to buy 36/12 = 3 packages of noisemakers and 36/9 = 4 packages of hats.

Then, we can calculate the total cost by multiplying the number of packages of each product by its price: 3 packages of noisemakers at $2 each and 4 packages of hats at $15 each.

`3\cdot2+4\cdot15=6+60=66`

Her total cost is $66.

Answer:

1) The LCM of 9 and 12 is 36.

2) She has to buy 3 packages of noisemakers and 4 packages of hats.

3) The total cost is $66