To find the total number of CDs Emma, Sophia, and Olivia have and to demonstrate the use of the Greatest Common Factor (GCF) and the Distributive Property, we need to follow these steps:
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Identify the number of CDs each person has:
- Emma: 45 CDs
- Sophia: 25 CDs
- Olivia: 65 CDs
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Calculate the total number of CDs: \( \text{Total CDs} = 45 + 25 + 65 \)
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Perform the addition step-by-step: First, add Emma's and Sophia's CDs: \[ 45 + 25 = 70 \] Now, add Olivia's CDs: \[ 70 + 65 = 135 \]
Therefore, the total number of CDs is \( 135 \).
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Find the GCF of the three numbers:
- Factors of 45: \( 1, 3, 5, 9, 15, 45 \)
- Factors of 25: \( 1, 5, 25 \)
- Factors of 65: \( 1, 5, 13, 65 \)
The common factors are \( 1 \) and \( 5 \), and the GCF is \( 5 \).
- Use the GCF and the Distributive Property:
We can express each number in terms of the GCF:
- \( 45 = 5 \times 9 \)
- \( 25 = 5 \times 5 \)
- \( 65 = 5 \times 13 \)
Now, we can factor out the GCF from the total sum: \[ \text{Total CDs} = 45 + 25 + 65 = 5 \times 9 + 5 \times 5 + 5 \times 13 \] Using the Distributive Property: \[ \text{Total CDs} = 5 \times (9 + 5 + 13) \] Now calculate the sum inside the parentheses: \[ 9 + 5 + 13 = 27 \] Now apply the factor: \[ \text{Total CDs} = 5 \times 27 = 135 \]
Thus, the total number of CDs Emma, Sophia, and Olivia have is \[ \boxed{135} \]