Nadia has 32 coins made up of nickles, dimes, and quarters. The sum of the number of nickels and the number of quarters is three times the number of dimes. If the total value of the coins is $4.60, how many of each kind does she have?
I can't figure out how to make this word problem into 3 equations.
To solve this word problem, we can break it down into three equations based on the given information.
Let's start by assigning variables for the number of nickels, dimes, and quarters. We'll use N for nickels, D for dimes, and Q for quarters.
1. The sum of the number of nickels and quarters is three times the number of dimes:
N + Q = 3D
2. Nadia has a total of 32 coins:
N + D + Q = 32
3. The total value of the coins is $4.60:
0.05N + 0.10D + 0.25Q = 4.6
Now we have a system of three equations that we can solve to find the values of N, D, and Q.
One method to solve this system is through substitution. We can start by solving equation 1 for Q in terms of N and D:
Q = 3D - N
Next, substitute this expression for Q in equation 2:
N + D + (3D - N) = 32
Simplify the equation:
4D = 32
Divide both sides by 4:
D = 8
Now, substitute the value of D back into equation 1 to solve for N:
N + Q = 3(8)
N + Q = 24
We know that Q = 3D - N, so:
N + (3D - N) = 24
Simplify the equation:
3D = 24
Divide both sides by 3:
D = 8
Now, substitute the values of N and D into equation 2:
N + D + Q = 32
N + 8 + Q = 32
Subtracting 8 from both sides:
N + Q = 24
Since we know that N + Q = 24 from equation 1, we find that:
24 = 24
This means that N can be any value. Since we're looking for a specific solution, we need to substitute the values of N and D into equation 3:
0.05N + 0.10D + 0.25Q = 4.6
(0.05 * 8) + (0.10 * 8) + 0.25Q = 4.6
0.4 + 0.8 + 0.25Q = 4.6
Simplify the equation:
0.25Q = 4.6 - 0.4 - 0.8
0.25Q = 3.4
Divide both sides by 0.25:
Q = 13.6
Since we can't have a fraction of a coin, we need to round down to the nearest whole number:
Q = 13
Now we can substitute the values of D and Q back into equation 1 to solve for N:
N + 13 = 24
N = 11
So, Nadia has 11 nickels, 8 dimes, and 13 quarters.
x nickels
y dimes and
z quarters:
equation base on number of coins
x + y + z = 32 ------ #1
equation based on value of coins
5x + 10y + 25z = 460, which reduces to
x + 2y + 5z = 92 ----- #2
"The sum of the number of nickels and the number of quarters is three times the number of dimes"
----> x + z = 3y
x -3y + z = 0 ----- #3
There are your 3 equations,
if you do #1 - #3 , you get
4y = 32
y = 8
so #1 becomes
x + 8 + z = 32
x + z = 24
#2 becomes
x + 16 + 5z = 92
x + 5z = 76
subtract those two ...
4z = 52
z = 13
back into the 1st
x + 8 + 13 = 32
x = 11
looks like 11 nickels, 8 dimes and 13 quarters
Check:
number of coins = 11+8+13 = 32
value of coins = 5(11) + 10(8) + 25(13) = 460
is 11+13 equal to 3 times 8 ??
yes!
All is good