Q) Julie has $3.10 in change in her pocket. If she has only 50 cent and 20 cent pieces and the total number of coins is 11, how many coins of each type does she have?
50x + 20y = 310
x + y = 11
multiply bottom equation by 20
50x + 20y = 310
20x + 20y = 220
substitute into 1st equation via elimination
30x = 90
x = 90 ÷ 30
x = 3
solve for y
3 + y = 11
y = 11 - 3
y = 8
Answer: 3 50c pieces and 8 20c pieces
Let's solve this problem step-by-step:
Let's assume that Julie has x 50 cent pieces and y 20 cent pieces.
1. The value of x 50 cent pieces is 50x cents.
2. The value of y 20 cent pieces is 20y cents.
3. According to the problem, the total value of the coins is $3.10, which is equal to 310 cents.
So, we can write the equation:
50x + 20y = 310
4. We also know that the total number of coins is 11, so we can write a second equation:
x + y = 11
Now we have a system of two equations. We can solve this system to find the values of x and y.
Let's solve the second equation for x:
x = 11 - y
Substitute this value of x into the first equation:
50(11 - y) + 20y = 310
Now, we can solve this equation for y.
550 - 50y + 20y = 310
-30y = 310 - 550
-30y = -240
y = -240 / -30
y = 8
Substitute the value of y into the equation x = 11 - y:
x = 11 - 8
x = 3
So, Julie has 3 fifty cent pieces and 8 twenty cent pieces.
To solve this problem, we can set up a system of equations based on the given information.
Let's say Julie has "x" 50 cent pieces and "y" 20 cent pieces. We know that the total number of coins is 11, so we can write the equation:
x + y = 11 -- Equation 1
We also know that the value of her change is $3.10. Each 50 cent piece is worth 50 cents or $0.50, and each 20 cent piece is worth 20 cents or $0.20. Therefore, we can write the equation for the total value of her change:
0.50x + 0.20y = 3.10 -- Equation 2
Now we have a system of equations. We can solve this system to find the values of "x" and "y".
One way to solve this system is by substitution. We can rearrange Equation 1 to solve for "x" in terms of "y":
x = 11 - y
Now we substitute this expression for "x" in Equation 2:
0.50(11 - y) + 0.20y = 3.10
Simplify:
5.50 - 0.50y + 0.20y = 3.10
Combine like terms:
-0.30y = 3.10 - 5.50
-0.30y = -2.40
To isolate "y", divide both sides by -0.30:
y = (-2.40) / (-0.30)
y = 8
Now, substitute this value of "y" back into Equation 1 to solve for "x":
x + 8 = 11
x = 11 - 8
x = 3
So, Julie has 3 50 cent pieces and 8 20 cent pieces.
50 f + 20 t = 310
f + t = 11 ... 20 f + 20 t = 220
subtracting equations (to eliminate t)
30 f = 90 ... f = 3
substitute back to find t