Question
Ms. Winter makes homemade bath soaps and bottles of lotion. In her inventory, she has 48 bath soaps and 64 bottles of lotion. She wants to use all of her inventory to make as many gift baskets as she can using the same number of bath soaps and the same number of lotion bottles in each gift basket.
She will be able to make at most gift baskets. Each basket will contain bath soaps and bottles of lotion.
She will be able to make at most gift baskets. Each basket will contain bath soaps and bottles of lotion.
Answers
GPT-4o mini
To find out how many gift baskets Ms. Winter can make using her inventory of 48 bath soaps and 64 bottles of lotion, we want to determine the greatest common divisor (GCD) of the two quantities. The GCD will give us the maximum number of gift baskets that can be made with the same number of bath soaps and lotion bottles in each basket.
1. **Find the GCD of 48 and 64:**
To find the GCD, we can list the factors of each number or use the prime factorization method.
- The prime factorization of 48:
\[
48 = 2^4 \times 3^1
\]
- The prime factorization of 64:
\[
64 = 2^6
\]
The GCD is found by taking the lowest power of each common prime factor.
The common prime factor is \(2\):
- The lowest power in the factorizations is \(2^4\).
Therefore, the GCD of 48 and 64 is:
\[
GCD(48, 64) = 2^4 = 16
\]
2. **Determine the number of gift baskets:**
The total number of gift baskets is equal to the GCD, which is 16.
3. **Determine how many bath soaps and bottles of lotion go into each basket:**
- Each basket will contain:
- Bath soaps in each basket: \(\frac{48}{16} = 3\)
- Bottles of lotion in each basket: \(\frac{64}{16} = 4\)
In summary:
- Ms. Winter will be able to make **16 gift baskets**.
- Each basket will contain **3 bath soaps** and **4 bottles of lotion**.
1. **Find the GCD of 48 and 64:**
To find the GCD, we can list the factors of each number or use the prime factorization method.
- The prime factorization of 48:
\[
48 = 2^4 \times 3^1
\]
- The prime factorization of 64:
\[
64 = 2^6
\]
The GCD is found by taking the lowest power of each common prime factor.
The common prime factor is \(2\):
- The lowest power in the factorizations is \(2^4\).
Therefore, the GCD of 48 and 64 is:
\[
GCD(48, 64) = 2^4 = 16
\]
2. **Determine the number of gift baskets:**
The total number of gift baskets is equal to the GCD, which is 16.
3. **Determine how many bath soaps and bottles of lotion go into each basket:**
- Each basket will contain:
- Bath soaps in each basket: \(\frac{48}{16} = 3\)
- Bottles of lotion in each basket: \(\frac{64}{16} = 4\)
In summary:
- Ms. Winter will be able to make **16 gift baskets**.
- Each basket will contain **3 bath soaps** and **4 bottles of lotion**.