Question

There are 12 pieces of five, ten and twenty rupee currencies whose total value is ₹ 105. When first 2 sorts are interchanged in their numbers its value will be increased by ₹ 20. Find the number of currencies in each sort

Answer 1

Let the number of ₹ 5 currencies be “x”

Let the number of ₹ 10 currencies be “y”

and the number of ₹ 20 currencies be “z”

By the given first condition

x + y + z = 12  ...(1)

By the given second condition

5x + 10y + 20z = 105

x + 2y + 4z = 21 (÷5)  ...(2)

By the given third condition

10x + 5y + 20z = 105 + 20

10x + 5y + 20z = 125

2x + y + 4z = 25  ...(3)

(1) × 4 ⇒  4x + 4y + 4z = 48   ....(1)
(2) × 1 ⇒    x + 2y + 4z = 21   ....(2)
                  (–)  (–)    (–)      (–) 
(1) – 2 ⇒    3x + 2y + 0 = 27
                  3x + 2y = 27       ....(4)
Subrtacting (2) and (3)
(2) ⇒   x + 2y + 4z = 21
(3) ⇒   2x + y + 4z = 25
          (–)  (–)  (–)  (–)             
                 – x + y + 0 = – 4
                      x – y =  4          ....(5)
(4) × 1   ⇒ 3x + 2y = 27        ....(4)
(5) × 2   ⇒ 2x – 2y = 8          ....(5)
(4) + (5) ⇒      5x + 0 = 35
                               x = `35/5` = 7

Substituting the value of x = 7 in (5)

7 – y = 4 ⇒ – y = 4 – 7

– y = – 3 ⇒ y = 3

Substituting the value of x = 7, y = 3 in  ...(1)

7 + 3 + z = 12

z = 12 – 10 = 2

x = 7, y = 3, z = 2

Number of currencies in ₹ 5 = 7

Number of currencies in ₹ 10 = 3

Number of currencies in ₹ 20 = 2