SOLUTION: A man has a collection of 19 coins worth $2.55. There are twice as many nickels as dimes. If the rest are quarters, how many quarters are there?

Question 1188526: A man has a collection of 19 coins worth $2.55. There are twice as many nickels as dimes. If the rest are quarters, how many quarters are there?
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52788) (Show Source): You can put this solution on YOUR website!
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A man has a collection of 19 coins worth $2.55. There are twice as many nickels as dimes.
If the rest are quarters, how many quarters are there?
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x dimes, 2x nickels and 19-x-2x = 19-3x quarters. Total money equation 5*(2x) + 10*x + 25*(19-3x) = 255 cents. Simplify and find x 10x + 10x + 25*19 - 75x = 255 (10 + 10 - 75)x = 255 - 25*19 -55x = -220 x = = 4. 4 dimes, 8 nickels and 19-4-8 = 7 quarters. ANSWER. 7 quarters.

Solved.

Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

The other tutor showed a formal algebraic solution, in which she unfortunately started with the wrong equation and so ended up with the wrong answer. Undoubtedly she will see my alternate solution and fix hers.

Since she provided an example of a formal algebraic solution, I will show a less formal solution using logical reasoning and simple mental arithmetic.

The number of dimes is a whole number, so the total value of the dimes is a multiple of 10 cents.

The number of nickels is an even number (because it is twice the number of dimes), so the total value of the nickels is a multiple of 10 cents.

The total value of any even number of quarters is a multiple of 10 cents; the total value of an odd number of quarters is not.

The total value of all the coins is NOT a multiple of 10 cents; therefore, the number of quarters is odd.

With a total value of $2.55, or 255 cents, for the 19 coins, the total value in cents of the quarters must be either 225, 175, 125, 75, or 25. Do some quick mental arithmetic to see which of those is likely (or possible).

$2.25 in quarters means 9 quarters, leaving 10 nickels and dimes to make the remaining 30 cents, which is clearly not possible.

$1.75 in quarters means 7 quarters, leaving 12 nickels and dimes to make the remaining 80 cents. 12 nickels makes 60 cents, and 12 dimes makes 120 cents; since 80 cents is between 60 and 120 cents, there should be a solution here.

So now our problem is to find x for which x dimes at 10 cents each and 2x nickels at 5 cents each makes those other 80 cents:

10(x)+5(2x)=80
20x=80
x=4
2x=8

ANSWER: 7 quarters, 4 dimes, and 8 nickels

CHECK:
7+4+8=19
7(25)+4(10)+8(5)=175+40+40=255

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