Question 670863: I am not sure how to figure out a coin problem without how much it's worth for example: A bank contains 44 coins (nickles, dimes, quarters). There are twice as many dimes as nickles and 8 fewer nickles than quarters. How much money is in the bank?
Found 5 solutions by Theo, ikleyn, timofer, greenestamps, n2:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! 44 coins in total.
n = number of nickels.
d = number of dimes
q = number of quarters.
there are twice as many dimes as nickels.
d = 2n
8 fewer nickels than quarters.
n = q - 8
how much money is in the bank.
the amount of money in the bank is determined as follows:
.05n = amount of money in nickels.
.1d = amount of money in dimes.
.25q = amount of money in quarters.
you have 2 equations.
n + d + q = 44
.05n + .1d + .25q = amount of money in the bank.
you need to solve for n, d, and q and then you can determine how much money is in the bank.
start with the equation of n + d + q = 44
you are given that d = 2n so you can substitute 2n for d which makes the equation becomes:
n + 2n + q = 44
you are given that n = q - 8 so you can substitute q - 8 for n which makes the equation become:
q - 8 + 2(q - 8) + q = 44
since you now have 1 unknown in 1 equation, you can solve for q.
simplify the equation to get:
q - 8 + 2q - 16 + q = 44
combine like terms to get:
4q - 24 = 44
add 24 to both sides of the equation to get:
4q = 44 + 24 = 68
divide both sides of the equation by 4 to get:
q = 68/4 = 17
since you know that n = q - 8, this means that n = 9
you have:
q = 17
n = 9
since you know that d = 2n, this means that d = 18
now you have:
q = 17
n = 9
d = 18
the first equation you had to solve is n + d + q = 44
replace q with 17 and n with 9 and d with 18 and you get:
9 + 18 + 17 = 44 which becomes 44 = 44 which is true so the numbers for n and d and q look good.
now to the money.
the equation is:
.05n + .1d + .25q = the money
replace n with 9 and d with 18 and q with 17 and you get:
.05(9) + .1(18) + .25(17) = .45 + 1.8 + 4.25 which is equal to 6.5 dollars that you have in the bank.
Answer by ikleyn(53250)  (Show Source):
You can put this solution on YOUR website! .
I am not sure how to figure out a coin problem without how much it's worth for example:
A bank contains 44 coins (nickles, dimes, quarters). There are twice as many dimes as nickles
and 8 fewer nickles than quarters. How much money is in the bank?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
I will give a simple, straightforward, short and instructive solution,
using only one single equation in one single unknown.
Let x be the number of nickels.
Then the number of dimes is 2x, according to the problem,
and the number of quarters is (x+8).
Having this, write equation for the total number of coins
x + 2x + (x+8) = 44. (here 44 is the given total number of coins).
Simplify this equation and find x
4x + 8 = 44,
4x = 44 - 8 = 36,
x =36/4 = 9.
So, there are 9 nickels in the bank, 2*9 = 18 dimes and 9+8 = 17 quarters.
Now you can easily find the value of the bank
5*9 + 10*18 + 17*25 = 650 cents, or 6.50 dollars.
ANSWER. The value of the bank is $6.50.
Solved and explained in a simple and instructive way, using only one unknown,
and without introducing system of equation, which is not necessary here.
/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/
I wrote this my solution as an alternative to the long, wordy solution by @Theo.
We live now at the time when every our solution, produced at this forum,
goes directly to the database for Artificial Intelligence.
So not tens, but tens millions students/readers will read them.
While 10 years ago so wordy solutions as in the posts by @Theo
did not cause protest, now they become INAPPROPRIATE
for teaching millions of readers of AI.
It is why I work here now checking and re-developing many @Theo' solutions.
Because they SHOULD BE re-developed.
Answer by timofer(124)  (Show Source):
Answer by greenestamps(13240)  (Show Source):
You can put this solution on YOUR website!
A comment about the two responses you have received to this point, one using three variables and one using a single variable.
The easiest way to set the problem up is directly from the given information, which means using three variables and three equations; but then solving the problem means solving a system of three equations in three variables.
It takes a bit of effort (not much, really) to set the problem up using a single variable and a single equation; but then the effort needed to solve the problem is much less.
So the most efficient way to solve the problem using formal algebra is using a single variable and a single equation.
The given information compares the number of dimes to the number of nickels, and it compares the number of nickels to the number of quarters. The common item there is the number of nickels, so the clear logical choice for the single variable is the number of nickels.
So proceed as shown in the response from the tutor who uses a single variable:
let x = # of nickels
then 2x = # of dimes (there are twice as many dimes as nickels)
and x+8 = # of quarters (... and 8 fewer nickels than quarters -- i.e., 8 more quarters than nickels)
The equation then says the total number of coins is 44:
(x) + (2x) + (x+8) = 44
The solution from there is easy....
Answer by n2(19) (Show Source):
You can put this solution on YOUR website! .
A bank contains 44 coins (nickels, dimes, quarters). There are twice as many dimes as nickels
and 8 fewer nickels than quarters. How much money is in the bank?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Let x be the number of nickels.
Then the number of dimes is 2x, according to the problem,
and the number of quarters is (x+8).
Having this, write equation for the total number of coins
x + 2x + (x+8) = 44. (here 44 is the given total number of coins).
Simplify this equation and find x
4x + 8 = 44,
4x = 44 - 8 = 36,
x =36/4 = 9.
So, there are 9 nickels in the bank, 2*9 = 18 dimes and 9+8 = 17 quarters.
Now you can easily find the value of the bank
5*9 + 10*18 + 17*25 = 650 cents, or 6.50 dollars.
ANSWER. The value of the bank is $6.50.
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