Question 1003406: A restaurant owner estimates that she needs in small change the same number of dimes as pennies and nickels together and the same number of pennies as nickels.If she gets $26 worth of pennies, nickels, and dimes, how should they be distributed?
a) Write the equations using p(pennies), n(nickels) and d(dimes) only
b) Solve the equations for the variables p, n and d. Show your work
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! d = number of dimes
10d = number of cents that d number of dimes are worth.
p = number of pennies
1p = number of cents that p number of pennies are worth.
n = number of nickels
5n = number of cents that n number of nickels are worth.
you are given that:
d = p + n
p = n
this means:
the number of dimes is equal to the number of pennies and nickels combined.
the number of pennies is equal to the number of nickels.
the total money is equal to 2600 cents.
you needed to total value in cents because the value of the coins is expressed in cents.
they could have expressed in dollars which i'll show you down further.
for now, though, everything is expressed in cents.
start with:
1p + 5n + 10d = 2600
this equation tells you that the total value of each of the coins is equal to the total value of 2600 cents.
your goal is to reduce the number of variables to 1 so you can solve the equation for that variable.
once you have the value of that variable, you can then use that to solve for the value of the rest of the variables in turn.
from the givens, you have that:
d = p + n
p = n
since p = n, you can replace p in d = p + n with n to get:
d = n + n = 2n
p = n
now start with:
1p + 5n + 10d = 2600
replace p with n and replace d with 2n to get:
1n + 5n + 10(2n) = 2600
simplify to get:
1n + 5n + 20n = 2600
combine like terms to get:
26n = 2600
divide both sides of this equation by 26 to get:
n = 100
you now know that you need 100 nickels.
since p = n, you also know that you need 100 pennies.
the value of 100 nickels and 100 pennies is 500 cents + 100 cents = 600 cents.
subtract that from 2600 cents and you have 2000 cents left for dimes.
divide 2000 cents by 10 to get 200 dimes.
you now know that you need 100 pennies and 100 nickels and 200 dimes.
1 * 100 = 100 cents in pennies.
5 * 100 = 500 cents in nickels.
10 * 200 = 2000 cents in dimes.
100 + 500 + 2000 = 2600 so you're solution is good.
divide each one of them by 100 and you get.
1 dollar in pennies, 5 dollars in nickels, and 20 dollars in dimes.
your answer is now expressed in dollars rather than cents.
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you could have solved this using dollars instead of cents right from the beginning.
the total value is 26 dollars.
one nickel is equal to .05 dollars.
1 penny = .01 dollars
1 dime = .10 dollars
your overall equation becomes:
.01p + .05n + .10d = 26
do the same substitutions as above to get:
d = n + n = 2n
p = n
now start with:
.01p + .05n + .10d = 26
replace p with n and replace d with 2n to get:
.01n + .05n + .10(2n) = 26
simplify to get:
.01n + .05n + .20n = 26
combine like terms to get:
.26n = 26
divide both sides by .26 to get n = 100
since p = n, p = 100
go back to the original equation of .01p + .05n + .10d = 26 and replace p and n with 100 to get .01(100) + .05(100) + .10(d) = 26
you get 1 + 5 + .10(d) = 26
combine like terms to get:
6 + .10(d) = 26
subtract 6 from both sides to get .10(d) = 20
divide both sides by .10 to get d = 20/.10 = 200
you get:
p = 100
n = 100
d = 200
all the requirements of the problem have been satisfied.
.01(100) + .05(100) + .10(200) = 26
p = n
d = p + n
your solution is good and it's the same as before, as it should be.
either method works.
you can equate the value to cents or you can equate the value to dollars.
the key is that you have to be consistent and convert everything to the same denomination.
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