The sum of two numbers is 72. If the larger number is increased by 8 and the smaller number is decreased by 6, then the resulting numbers are in the ratio \(3 : 2\), respectively. Find the smaller of the original two numbers.
35.6
Let the smaller number be \(s\). Then the larger number is \(72 - s\).
After the changes, the larger becomes \((72 - s) + 8 = 80 - s\) and the smaller becomes \(s - 6\).
These are in the ratio \(3 : 2\): \(\frac{80 - s}{s - 6} = \frac{3}{2}\).
Cross-multiplying: \(2(80 - s) = 3(s - 6)\), so \(160 - 2s = 3s - 18\).
Then \(160 + 18 = 5s\), giving \(5s = 178\) and \(s = 35.6\).
Hence, the smaller of the original two numbers is 35.6.
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