1. Ms.lynch has 21 coins in nickels and dimes. Their total value is $1.65. How many of each coin does she have?
2. A vending machine that takes only dimes and quarters contains 30 coins, with a total value of $4.20. How many of each coin are there?
this is similar to the question I answered on the question after this:
pretend nickels=x
dimes=y
x+y=21 *total number of dimes and nickels is 21
.05x+.1y=$1.65 *the value of x is .5 and the value of y is .1 so when you multiply it by the number of x and y you get $1.65
now use elimination:
first solve for one variable
x+y=21
y=-x+21
now plug "x+21" as "y" in the other equation
.05x+.1(-x+21)=$1.65
.05x-.1x+2.1=1.65 *distribute
-.05x=-.45
x=9
now plug 9 as x into the first equation
x+y=21
9+y=21
y=12
she has 9 nickels and 12 dimes
check by plugging in answers into both equations:
9+12=21 *correct
and
.05(9)+.1(12)=1.65 *correct
now apply the same concept to number 2
22 Dimes
8 quarters
1. Let's assign variables to represent the number of nickels and dimes.
Let x be the number of nickels.
Let y be the number of dimes.
Based on the given information, we can create two equations:
Equation 1: x + y = 21 (since Ms. Lynch has a total of 21 coins)
Equation 2: 0.05x + 0.10y = 1.65 (since the total value of the coins is $1.65)
To solve this system of equations, we can use the method of substitution or elimination.
Let's use the method of substitution to solve this system of equations:
From Equation 1, we can express x in terms of y:
x = 21 - y
Substituting x in Equation 2 with 21 - y:
0.05(21 - y) + 0.10y = 1.65
Now, let's simplify and solve for y:
1.05 - 0.05y + 0.10y = 1.65
0.05y = 1.65 - 1.05
0.05y = 0.60
y = 0.60 / 0.05
y = 12
Substitute the value of y back into Equation 1 to find the value of x:
x + 12 = 21
x = 21 - 12
x = 9
Therefore, Ms. Lynch has 9 nickels and 12 dimes.
2. Let's assign variables to represent the number of dimes and quarters.
Let x be the number of dimes.
Let y be the number of quarters.
Based on the given information, we can create two equations:
Equation 1: x + y = 30 (since there are a total of 30 coins)
Equation 2: 0.10x + 0.25y = 4.20 (since the total value of the coins is $4.20)
To solve this system of equations, we can use the method of substitution or elimination.
Let's use the method of substitution to solve this system of equations:
From Equation 1, we can express x in terms of y:
x = 30 - y
Substituting x in Equation 2 with 30 - y:
0.10(30 - y) + 0.25y = 4.20
Now, let's simplify and solve for y:
3 - 0.10y + 0.25y = 4.20
0.15y = 4.20 - 3
0.15y = 1.20
y = 1.20 / 0.15
y = 8
Substitute the value of y back into Equation 1 to find the value of x:
x + 8 = 30
x = 30 - 8
x = 22
Therefore, in the vending machine, there are 22 dimes and 8 quarters.
To solve both of these problems, we can use a system of equations. Let's assign variables to represent the number of nickels and dimes in the first problem, and the number of dimes and quarters in the second problem.
1. Let's assign variables to represent the number of nickels and dimes:
- Let's say x is the number of nickels.
- Let's say y is the number of dimes.
We can write two equations based on the given information:
- The total number of coins is 21: x + y = 21. (Equation 1)
- The total value is $1.65: 0.05x + 0.10y = 1.65. (Equation 2)
To solve this system of equations, we can use the method of substitution or elimination. Let's use substitution:
From Equation 1, we can rewrite it as: x = 21 - y.
Substituting x in Equation 2 with (21 - y), we get:
0.05(21 - y) + 0.10y = 1.65.
Simplifying this equation, we have:
1.05 - 0.05y + 0.10y = 1.65.
-0.05y + 0.10y = 1.65 - 1.05.
0.05y = 0.60.
y = 0.60 / 0.05.
y = 12.
Now that we know the number of dimes (y), we can substitute it back into Equation 1 to find the number of nickels:
x + 12 = 21.
x = 21 - 12.
x = 9.
Therefore, Ms. Lynch has 9 nickels and 12 dimes.
2. Let's assign variables to represent the number of dimes and quarters:
- Let's say x is the number of dimes.
- Let's say y is the number of quarters.
We can write two equations based on the given information:
- The total number of coins is 30: x + y = 30. (Equation 1)
- The total value is $4.20: 0.10x + 0.25y = 4.20. (Equation 2)
Again, we can solve this system of equations using either substitution or elimination. Let's use substitution:
From Equation 1, we can rewrite it as: x = 30 - y.
Substituting x in Equation 2 with (30 - y), we get:
0.10(30 - y) + 0.25y = 4.20.
Simplifying this equation, we have:
3 - 0.10y + 0.25y = 4.20.
0.15y = 1.20.
y = 1.20 / 0.15.
y = 8.
Now that we know the number of quarters (y), we can substitute it back into Equation 1 to find the number of dimes:
x + 8 = 30.
x = 30 - 8.
x = 22.
Therefore, there are 22 dimes and 8 quarters in the vending machine.