there are 12 pieces of five, ten, twenty rupee currencies whose total value is rupees 105 . when first 2 sorts are interchanged in their number it's value will be increased by rupees 20 . find the number of currencies in each sort.
Answers
Given :
- There are 12 pieces of five, ten, twenty rupee currencies.
- Total value = rupees 105
To Find :
- Number of currencies in each sort
Solution :
x + y + z = 12 -----------> (1)
5x + 10y + 20z = 105 -----------> (2)
As per given in the question,
We've to interchange the first two sorts, and then we've to add 20 in the given number 105.
10x + 5y + 20z = 125 -----------> (3)
Now multiply 1st equation with 5
(1) × 5 → 5x + 5y + 5z = 60
(-) (-) (-) (-)
(2) → 5x + 10y + 20z = 105
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-5y -15z = -45 ------------> (4)
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Now multiply 2nd equation with 2
(2) × 2 → 10x + 20y + 40z = 210
(-) (-) (-) (-)
(3) → 10x + 5y + 20z = 125
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15y + 20z = 85 ------------> (5)
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Now multiply 4th equation with 3
(4) × 3 → -15y - 45z = -135
(5) → 15y + 20z = 85
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-25z = -50
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⇒ -25z = -50
⇒ z = -50/25
⇒ z = 2
So the value of z = 2
Substitute, z = 2 in (5) we get
⇒ 15y + 20z = 85
⇒ 15y + 20 × 2 = 85
⇒ 15y + 40 = 85
⇒ 15y = 85 - 40
⇒ 15y = 45
⇒ y = 45/15
⇒ y = 3
Now, Substitute y = 3 and z = 2 in (1)
⇒ x + y + z = 12
⇒ x + 3 + 2 = 12
⇒ x + 5 = 12
⇒ x = 12 - 5
⇒ x = 7
•°• The solutions are,
- The number of rupees 5 are 7
- The number of rupees 10 are 3
- The number of rupees 20 are 2