72 coins nickels and dimes equal 4.95? - Answers

This is a simultaneous-equation problem. Because there are two variables - the number of nickels and the number of dimes - we have to set up two relationships between them in order to solve the system.Let n be the number of nickels and d be the number of dimes.Because there's a total of 72 coins, the first equation is n + d = 72.The second thing we know is that the total value is $4.95. It's easiest to express each amount in cents, n nickels are worth 5n cents, d dimes are worth 10d cents, and $4.95 is 495 cents so the second equation is 5n + 10d = 495.From here there are two ways to obtain the answer.Method 1 - SubstitutionBecause the first equation is n + d = 72, we can express n in terms of n as n = 72 - d.Then replace the "n" term of the second equation with the new expression in terms of d: 5(72 - d) + 10d = 495Expanding the left side gives 5*72 - 5d + 10d = 495.Next collect like terms: 360 + 5d = 495Finally solve for d: 5d = 495 - 360, or 5d = 135so d = 135/5, or d = 27; i.e. there are 27 dimes.Because we know there are 72 coins in total (d + n = 72) there must be 45 nickels.Method 2 - BalancingWrite the equations on two successive lines: n + d = 725n + 10d = 495The coefficient of n in the first equation is 1 (implied) and the coefficient in the second equation is 5 (explicit). To balance the two, multiply every term in the first equation by 5 so the "n" term has the same coefficient in both: 5n + 5d = 3605n + 10d = 495Both equations now share a "5n" term. The "d" coefficient is larger in the second equation so subtract the first equation from the second: 5n + 10d = 495-[5n + 5d = 360]------------------------or (5n - 5n) + (10d - 5d) = 495 - 360The 5n and -5n terms cancel each other out, leaving 5d = 135, or d = 27 and n = 45, the same answer as in Method 1 (which it should be!!) Check: 27 dimes are worth 27*10 = $2.70. 45 nickels are worth 45*5 = $2.25. The total value is $4.95.