A box contains $7.80 in nickels, dimes, and quarters. There are 46 coins in all, and the sum of the numbers of nickels and dimes is 2 less than the number of quarters. How many coins of each kind are there?
Here is my work so far:
x + y + z = 46
x + y = z - 2
.5x + .10y + .25y = 7.80
Typo...
.5x + .10y + .25y = 7.80
is actually
.5x + .10y + .25z = 7.80
no, it is actually
.05x + .10y + .25z = 7.80
Now solve 'em!
x+y+z=46
x+y=z-2
.05x+.10y+.25z=7.80
x+y=z-2
z=x+y+2
x+y+x+y+2=46
2x+2y=44
Does this look right so far..?
a farmer has 500 feet of fence with which to fence a rectangular plot of land. The plot lies along a river so that only three sides need to be fenced. Estimate the largest area that can be fenc
To solve this problem, we can set up a system of equations based on the given information.
Let's assign variables for the number of nickels, dimes, and quarters. Let:
x = number of nickels
y = number of dimes
z = number of quarters
From the problem statement, we have the following equations:
Equation 1: x + y + z = 46 (since there are a total of 46 coins)
Equation 2: x + y = z - 2 (since the sum of the numbers of nickels and dimes is 2 less than the number of quarters)
To introduce decimals, let's convert the values to cents:
0.05x + 0.10y + 0.25z = 780 cents (since the total value is $7.80)
Now we have a system of three equations:
Equation 1: x + y + z = 46
Equation 2: x + y = z - 2
Equation 3: 0.05x + 0.10y + 0.25z = 780
To solve this system of equations, we can use different methods like substitution, elimination, or matrices. In this case, let's solve it using the substitution method.
Let's solve Equation 2 for x:
x = z - 2 - y
Now substitute this expression for x in Equations 1 and 3:
Equation 1 becomes:
(z - 2 - y) + y + z = 46
Simplifying:
2z - 2 = 46
2z = 48
z = 24
Equation 3 becomes:
0.05(z - 2 - y) + 0.10y + 0.25z = 780
Simplifying:
0.05z - 0.10 - 0.05y + 0.10y + 0.25z = 780
Combine like terms:
0.30z - 0.05y - 0.10 = 780
0.30z - 0.05y = 780 + 0.10
0.30z - 0.05y = 780.10
Since we know the value of z is 24, we can now solve for y:
0.30(24) - 0.05y = 780.10
7.20 - 0.05y = 780.10
-0.05y = 780.10 - 7.20
-0.05y = 772.90
y = 772.90 / -0.05
y ≈ -15458
This result doesn't make sense because we cannot have a negative number of dimes. It seems there might be an error in the problem statement or calculations. Please double-check your problem statement and calculations to find any mistakes.