HCF and LCM Word Problems Worksheet No 2 (with solutions ... - Free Printable

Problem 4:

Anna and Mary both had a dance lesson today. Anna has a dance lesson every 8 days, and Mary has a dance lesson every 12 days. How many days will it be until Anna and Mary both have a dance lesson on the same day?

#### Solution:
To determine how many days it will be until Anna and Mary both have a dance lesson on the same day, we need to find the Least Common Multiple (LCM) of their respective lesson intervals (8 days and 12 days). The LCM is the smallest number that is a multiple of both 8 and 12.

##### Step 1: Prime Factorization
- Prime factorization of 8:
$$
8 = 2 \times 2 \times 2 = 2^3
$$
- Prime factorization of 12:
$$
12 = 2 \times 2 \times 3 = 2^2 \times 3
$$

##### Step 2: Determine the LCM
The LCM is found by taking the highest power of each prime factor that appears in the factorizations:
- For the prime factor 2: The highest power is $2^3$.
- For the prime factor 3: The highest power is $3^1$.

Thus, the LCM is:
$$
\text{LCM} = 2^3 \times 3^1 = 8 \times 3 = 24
$$

##### Step 3: Interpret the Result
The LCM of 8 and 12 is 24. This means that Anna and Mary will both have a dance lesson on the same day after 24 days.

#### Final Answer for Problem 4:
$$
\boxed{24}
$$

---

Problem 7:

A baker made 42 butter cookies and 180 chocolate chip cookies. He wants to create identical packages of cookies to sell, and he must use all of the cookies.
- a) What is the greatest number of packages the baker can make?
- b) How many cookies of each type will there be in each package?

#### Solution:
To solve this problem, we need to determine the greatest number of identical packages that can be created using all the cookies. This involves finding the Greatest Common Divisor (GCD) of the number of butter cookies (42) and chocolate chip cookies (180). The GCD will tell us the maximum number of packages that can be made, and dividing the total number of each type of cookie by the GCD will give the number of cookies per package.

##### Step 1: Find the GCD of 42 and 180
We use the Euclidean algorithm to find the GCD.

1. Divide 180 by 42 and find the remainder:
$$
180 \div 42 = 4 \quad \text{remainder} \quad 180 - (42 \times 4) = 180 - 168 = 12
$$
So, $180 = 42 \times 4 + 12$.

2. Now, divide 42 by 12 and find the remainder:
$$
42 \div 12 = 3 \quad \text{remainder} \quad 42 - (12 \times 3) = 42 - 36 = 6
$$
So, $42 = 12 \times 3 + 6$.

3. Next, divide 12 by 6 and find the remainder:
$$
12 \div 6 = 2 \quad \text{remainder} \quad 12 - (6 \times 2) = 12 - 12 = 0
$$
So, $12 = 6 \times 2 + 0$.

Since the remainder is now 0, the GCD is the last non-zero remainder, which is 6.

##### Step 2: Determine the Number of Packages
The greatest number of identical packages the baker can make is the GCD of 42 and 180, which is 6.

##### Step 3: Determine the Number of Cookies per Package
- Number of butter cookies per package:
$$
\frac{42}{6} = 7
$$
- Number of chocolate chip cookies per package:
$$
\frac{180}{6} = 30
$$

#### Final Answers for Problem 7:
- a) The greatest number of packages the baker can make:
$$
\boxed{6}
$$
- b) Number of cookies of each type in each package:
$$
\boxed{7 \text{ butter cookies and } 30 \text{ chocolate chip cookies}}
$$

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Summary of Solutions:

1. Problem 4:
$$
\boxed{24}
$$

2. Problem 7:
- a) Greatest number of packages:
$$
\boxed{6}
$$
- b) Cookies per package:
$$
\boxed{7 \text{ butter cookies and } 30 \text{ chocolate chip cookies}}
$$