Use a system of equations to solve problem
Emily emptied her bank ana totaled the nickels, dimes, and quarters that she had saved. The total amount was 14.50
The number of dimes was 2 more than 3 times the number of nickels. How many of each coin were there
Can you show the steps i just cant do this can you show every step so i can understand this
Let N be number of nickels, D dimes, and q quarters.
VAlue equation:
.25q+ .10d + .05n=14.50
d=3N+2
.25q+.10(3N+2)+.05n=14.50
.25q+.30n +.20+.05n=14.50
.25q+.35n=14.30
Now q, n have to be integers...you cant have a fraction of a coin.
So lets get the nickels out of the way first by getting the quarter total value to some multiple of a dollar, or 25c, or 50 cents...
How many .35 will it take to make the total a multiple of .25? Turns out, three...If n is 3, then .35n=1.05, so
.25q=14.30-1.05=13.25 so q is you can do that.
let the number of nickels, dimes and quarters be N, D, and Q
but D= 3N+2
5N + 10(3N+2) + 25Q = 1450
25Q + 35N = 1430
5Q + 7N = 286
we want integer solutions for N and Q
by some trial and error I got
Q=25
and N = 23, so D = 71
check: 25*25 + 71*10 + 23*5 = 1450
My question to you "Is this answer unique" ?
Sure! Let's solve this problem step-by-step using a system of equations.
Step 1: Define the variables
Let's say the number of nickels is represented by 'n',
the number of dimes is represented by 'd', and
the number of quarters is represented by 'q'.
Step 2: Set up the equations
We are given two pieces of information:
1. The total amount of money Emily saved is $14.50, which can be expressed as: 0.05n + 0.10d + 0.25q = 14.50
(Each nickel is worth $0.05, each dime is worth $0.10, and each quarter is worth $0.25)
2. The number of dimes is 2 more than 3 times the number of nickels, which can be expressed as: d = 3n + 2
Step 3: Solve the system of equations
We can now substitute equation 2 into equation 1 to eliminate the 'd' variable.
Substituting d from equation 2 into equation 1:
0.05n + 0.10(3n + 2) + 0.25q = 14.50
Step 4: Simplify and solve
Multiply 0.10 by 3n and 2:
0.05n + 0.30n + 0.20 + 0.25q = 14.50
Combine like terms:
0.35n + 0.25q = 14.30
Step 5: Simplify further if needed
To make the coefficients whole numbers, multiply the entire equation by 100:
35n + 25q = 1430
Step 6: Continue solving the system of equations
We will now set up another equation using the information from the problem.
Since we know that the total amount of money Emily saved was $14.50, we can write the equation:
0.05n + 0.10d + 0.25q = 14.50
Simplify and substitute the value of 'd' from equation 2:
0.05n + 0.10(3n + 2) + 0.25q = 14.50
Again, simplify and combine like terms:
0.05n + 0.30n + 0.20 + 0.25q = 14.50
0.35n + 0.25q = 14.30
Step 7: Multiply through by 100 to eliminate decimals:
35n + 25q = 1430
Step 8: Solve the system of equations
Now we have the following two equations:
35n + 25q = 1430 (Equation 1)
0.35n + 0.25q = 14.30 (Equation 2)
We can solve this system of equations using different methods, such as substitution, elimination, or graphing.
Let's use the substitution method to solve the system:
From Equation 2, let's isolate 'n':
0.35n = 14.30 - 0.25q
Divide by 0.35:
n = (14.30 - 0.25q) / 0.35
Now we substitute this value of 'n' into Equation 1:
35[(14.30 - 0.25q) / 0.35] + 25q = 1430
Simplify:
(499 - 7q) + 25q = 1430
Combine like terms:
499 + 18q = 1430
Subtract 499 from both sides:
18q = 931
Divide by 18:
q = 51.72
Step 9: Substitute the value of 'q' back into Equation 1 to find 'n':
35n + 25(51.72) = 1430
35n + 1293.00 = 1430
35n = 1430 - 1293
35n = 137
n = 3.91
Step 10: Substitute the values of 'n' and 'q' back into Equation 2 to find 'd':
d = 3(3.91) + 2
d = 11.73 + 2
d ≈ 13.73
Step 11: Round the values
Since you cannot have a fraction of a coin, round the values to the nearest whole number:
n ≈ 4
d ≈ 14
q ≈ 52
Therefore, there were approximately 4 nickels, 14 dimes, and 52 quarters.
Sure! Let's solve this problem step by step using a system of equations.
Let's assign variables to the unknowns in the problem:
Let's say the number of nickels is 'n'.
And let's say the number of dimes is 'd'.
Step 1: Translate the problem into equations.
We are given:
1) The total amount is $14.50. We can express this as an equation in terms of the number of nickels, dimes, and quarters:
0.05n + 0.10d + 0.25q = 14.50
(Note: We don't know specifically how many quarters there are, but we can represent it as 'q'.)
2) The number of dimes is 2 more than 3 times the number of nickels:
d = 3n + 2
Step 2: Set up a system of equations.
We now have two equations:
0.05n + 0.10d + 0.25q = 14.50
d = 3n + 2
Step 3: Solve the system of equations.
We have two equations and two unknowns, so we can solve it by substitution or elimination. Let's use substitution:
Substitute the value (3n + 2) for 'd' in the first equation:
0.05n + 0.10(3n + 2) + 0.25q = 14.50
Simplify the equation:
0.05n + 0.30n + 0.20 + 0.25q = 14.50
0.35n + 0.20 + 0.25q = 14.50
0.35n + 0.25q = 14.30 (Equation 3)
Now, we have two equations:
0.35n + 0.25q = 14.30 (Equation 3)
d = 3n + 2 (Equation 2)
Step 4: Solve the equations.
We can solve this system by substituting the value of 'd' from Equation 2 into Equation 3:
0.35n + 0.25q = 14.30
Substitute d = 3n + 2:
0.35n + 0.25q = 14.30
Now we can solve this equation for one variable. Let's solve for 'q':
0.25q = 14.30 - 0.35n
q = (14.30 - 0.35n) / 0.25 (Equation 4)
Step 5: Substitute the value of 'q' from Equation 4 into Equation 2:
Substitute q = (14.30 - 0.35n) / 0.25 into Equation 2:
d = 3n + 2
Now, we have an equation in terms of 'n' and 'd'. We can solve this equation to find the values of 'n' and 'd'.
Step 6: Solve the equation.
Substitute q = (14.30 - 0.35n) / 0.25 into Equation 2:
d = 3n + 2
Simplify the equation:
(14.30 - 0.35n) / 0.25 = 3n + 2
Now, you can solve this equation for 'n'. Once you find the value of 'n', you can substitute it back into Equation 2 to find the value of 'd'. Finally, you can substitute the values of 'n', 'd', and 'q' into the original equation from Step 1 to verify that the total amount is indeed $14.50.