34) There are 12 pieces of five, ten and twenty rupee currencies whose total value is Rs. 105. When first 2 sorts are interchanged in their numbers its value will be increased by Rs. 20. Find the number of currencies in each sort.
Answer: Let the number of five, ten and twenty rupee currencies be x, y and z respectively.
Given:
Total number of currencies: \(x + y + z = 12\) ... (1)
Total value: \(5x + 10y + 20z = 105\) ... (2)
Value after interchanging x and y: \(10x + 5y + 20z = 105 + 20 = 125\) ... (3)
Solving the system of linear equations:
From the calculations:
Subtracting (3) from (2): \(-5x + 5y = -20\) ... (4)
Solving (1) and (2) by eliminating z, we get: \(15x + 10y = 135\) ... (6)
Solving equations (4) and (6), we get \(y = 3\).
Substituting \(y=3\) in (4): \(-5x + 15 = -20 \implies -5x = -35 \implies x = 7\).
Substituting \(x=7, y=3\) in (1): \(7 + 3 + z = 12 \implies z = 2\).
- The number of five rupee currencies is 7.
- The number of ten rupee currencies is 3.
- The number of twenty rupee currencies is 2.