Divide 595 into two parts so that 15 times the first and 5 times the second…

Concept: When a total quantity is split into two parts with a known sum and a separate weighted-sum condition on those parts, the two unknowns can be found by solving the sum equation and the weighted equation together as a pair of simultaneous linear equations.

Application:

1. Let the first part be x and the second part be y, so x + y = 595.

2. The weighted condition states that 15 times the first plus 5 times the second equals 3975: 15x + 5y = 3975.

3. Substitute y = 595 - x into the weighted equation: 15x + 5(595 - x) = 3975.

4. Simplify: 15x + 2975 - 5x = 3975, which gives 10x = 1000, so x = 100.

5. Then y = 595 - 100 = 495.

6. The ratio of the first part to the second part is x : y = 100 : 495, which reduces to 20 : 99 after dividing both terms by 5.

Cross-check:

• Sum check: 100 + 495 = 595, which matches the total given in the question.

• Weighted check: 15(100) + 5(495) = 1500 + 2475 = 3975, which matches the given condition.

Result: The ratio of the first part to the second part is 20 : 99.