A vending machine's coin box contains nickels, dimes, and quarters. The total number of coins in the box is 308. The number of dimes is three times the number of nickels and quarters together. If the box contains 30 dollars and 75 cents, find the number of nickels, dimes and quarters that it contains.

number of nickels = 58

dimes = 231

quarters = 19

Step-by-step explanation:

Let x, y, z represent the number of nickels, dimes and quarters that it contains.

The total number of coins in the box is 308

x+y+z = 308 ... 1

The number of dimes is three times the number of nickels and quarters together;

y = 3 (x+z) ... 2

Nickel = $0.05

Dime = $0.10

Quarters = $0.25

the box contains 30 dollars and 75 cents

0.05x + 0.10y + 0.25z = 30.75 ... 3

Let y = 3 (x+z)

0.05x + 0.30 (x+z) + 0.25z = 30.75

0.35x + 0.55z = 30.75

x + (0.55/0.35) z = 30.75/0.35 ... 4

From equation 1

x+y+z = 308

Substituting equation 2

x + 3 (x+z) + z = 308

4x + 4z = 308

x+z = 77 ... 5

Subtract equation 5 from 4

(0.55/0.35 - 1) z = 30.75/0.35 - 77

z = 19

from equation 5

x = 77-z = 77-19 = 58

x = 58

From equation 2

y = 3 (x+z) = 3 (77) = 231

y = 231

number of nickels x = 58

dimes y = 231

quarters z = 19