Question 952187: My piggy bank contains 29 coins, each of which is either a penny, nickel, dime, or quarter. The total value of all the coins is $2.86, and the number of quarters is equal to the sum of the nickels and dimes, and is less than the number of pennies. How many nickels are there?
Answer by ankor@dixie-net.com(22740)   (Show Source):
You can put this solution on YOUR website! p = no.of pennies
n = no. of nickles
d = no. of dimes
q = no. of quarters
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write an equation for each statement
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My piggy bank contains 29 coins, each of which is either a penny, nickel, dime, or quarter.
p + n + d + q = 29
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The total value of all the coins is $2.86,
.01p + .05n + .10d + .25q = 2.86
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and the number of quarters is equal to the sum of the nickels and dimes,
q = n + d
d = q - n
n = q - d
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and is less than the number of pennies.
q < p
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In the 1st equation replace n + d with q and you have
p + 2q = 29
p = -2q + 29
From the above equation and the total$ equation we know the number of pennies has to be 21 or 11 pennies which would make it 4 or 9 quarters
.01p + .05n + .10d + .25q = 2.86
lets say 9 quarters and 11 pennies
.11 + .05n + .10d + .25(9) = 2.86
.11 + .05n + .10d + 2.25 = 2.86
.05n + .10d + 2.36 = 2.86
.05n + .10d = 2.86 - 2.36
.05n + .10d = .50
simplify, divide by .05
n + 2d = 10
we know the d = q - n, d = 4 - n
n + 2(9-n) = 10
n + 18 - 2n = 10
-n = 10 - 18
-n = -8
n = 8 nickels
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Check this find the dimes
8 + 2d = 10
2d = 2
d = 1 dime
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11 + 8 + 1 + 9 = 29
and
.01(11) + .05(8) + .10(1) + .25(9) = 2.86
.11 + .40 + .10 + 2.25 = 2.86
--------------------Response to Comment-----------------------
Tried sending this to the given email address, came back as "Milbox not available"
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Yes, well to start with, we have 4 unknowns and only 3 meaningful equations, which means we will have to apply some logical assumptions
Assuming you followed me Ok up to the equation:
p = –2q + 29
Since we have a total of 2.86, and the total of the rest of the coins have to be a multiple of 5, the number of pennies have to end in a 1 or a 6
and looking at the above equation, subtract 2q from 29 will give you an odd number of pennies less than 29 of course, leaves us with a choice of 11 or 21 pennies
This narrows it down to a manageable, try it and see” situation
We can find the number of quarters with either p=11 or p=21
-2q + 29 = 11
-2q = 11 – 29
-2q = –18
q = –18/-2
q = 9 quarters
The other alternative, when p = 21
-2q + 29 = 21
-2q = 21 – 29
q = –8/-2
q = 4 quarters
Hence the statement,
”From the above equation and the total$ equation we know the number of pennies has to be 21 or 11 pennies which would make it 4 or 9 quarters”
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Hopefully it made made sense to you after that. If not, email me your concerns about this, email me at ankor@att.net
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