How to Calculate Two Numbers from Their Product and Ratio
Given:
- \( P \): the product of two numbers
- \( m:n \): the ratio of the two numbers
Steps:
- Express the numbers using the ratio: Assume \( x = m \cdot k \) and \( y = n \cdot k \), where \( k \) is a constant.
- Relate the product and ratio: Substitute into the product equation \( x \cdot y = P \):
\( (m \cdot k) \cdot (n \cdot k) = P \)
\( m \cdot n \cdot k^2 = P \)
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Solve for \( k \):
\( k = \sqrt{\frac{P}{m \cdot n}} \)
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Find \( x \) and \( y \):
\( x = m \cdot k\)
\(y = n \cdot k \)
Examples
Example 1: The product of two numbers is 72, and their ratio is 3:4. Find the numbers.
Solution:
1. \( m \cdot n = 3 \cdot 4 = 12 \)
2. \( k = \sqrt{\frac{72}{12}} = \sqrt{6} \)
3. \( x = 3 \cdot \sqrt{6} \approx 7.35 \)
4. \( y = 4 \cdot \sqrt{6} \approx 9.8 \)
Result: The two numbers are approximately 7.35 and 9.8.
Example 2: The product is 1008, and the ratio is 4:7. Find the numbers.
Solution:
1. \( m \cdot n = 4 \cdot 7 = 28 \)
2. \( k = \sqrt{\frac{1008}{28}} = \sqrt{36} = 6 \)
3. \( x = 4 \cdot 6 = 24 \)
4. \( y = 7 \cdot 6 = 42 \)
Result: The two numbers are 24 and 42.
Example 3: The product is 1575, and the ratio is 7:9. Find the numbers.
Solution:
1. \( m \cdot n = 7 \cdot 9 = 63 \)
2. \( k = \sqrt{\frac{1575}{63}} = \sqrt{25} = 5 \)
3. \( x = 7 \cdot 5 = 35 \)
4. \( y = 9 \cdot 5 = 45 \)
Result: The two numbers are 35 and 45.