Mr. Clarkson and Mrs. Rose had an average of $180. After Mr. Clarkson spent $73 and Mrs. Rose received $38 from her father, Mr. Clarkson had 1/4 as much money as Mrs. Rose. How much money did each have at first?
Clarkson --- x
Rose ------ y
but (x+y)/2 = 180
x+y = 360
y = 360-x
clarkson spent 73:
amount left = x-73
Rose got 38
her amount now = 360-x + 38 =398-x
clarkson has 1/4 as much as Rose
x-73 = (1/4)(398-x)
4x-292 = 398 - x
5x = 690
x = 138
Clarkson had $138
Rose had $222
Let's assign variables to represent the amounts of money Mr. Clarkson and Mrs. Rose had initially.
Let x represent the amount of money Mr. Clarkson had at first.
Let y represent the amount of money Mrs. Rose had at first.
We know that the average of Mr. Clarkson and Mrs. Rose's money is $180, so the equation is:
(x + y) / 2 = 180
Mr. Clarkson spent $73, so he had x - $73 left.
Mrs. Rose received $38 from her father, so she had y + $38.
According to the given condition, Mr. Clarkson had 1/4 as much money as Mrs. Rose, so the equation is:
x - $73 = (1/4)(y + $38)
Now we can solve the two equations simultaneously to find the values of x and y.
To solve this problem, we can set up a system of equations based on the given information.
Let's assume that Mr. Clarkson initially had x dollars and Mrs. Rose initially had y dollars.
1. We are given that the average of their initial money is $180, so we can write the equation:
(x + y) / 2 = 180
2. After Mr. Clarkson spent $73, his remaining money became (x - 73).
Mrs. Rose's money remained the same, y.
3. After Mrs. Rose received $38 from her father, her total money became (y + 38).
Mr. Clarkson's remaining money remained the same, (x - 73).
4. We are also given that Mr. Clarkson had 1/4 as much money as Mrs. Rose after these transactions, so we can write another equation:
(x - 73) = 1/4 (y + 38)
Now, let's solve this system of equations to find the values of x and y.
First, we can simplify equation 4 by multiplying both sides by 4:
4(x - 73) = y + 38
Expanding the equation:
4x - 292 = y + 38
Rearranging the equation:
4x - y = 330 ...(equation 5)
Now, we can solve equations 5 and 1 simultaneously.
Substituting equation 5 into equation 1:
(x + y) / 2 = 180
x + y = 360 ...(equation 6)
Next, we can solve equations 5 and 6 using the elimination method.
Multiply equation 6 by 4 to match the x-coefficient in equation 5:
4(x + y) = 4(360)
4x + 4y = 1440 ...(equation 7)
Subtract equation 5 from equation 7:
(4x + 4y) - (4x - y) = 1440 - 330
5y = 1110
Divide both sides by 5:
y = 222
Substitute the value of y into equation 6:
x + 222 = 360
Subtract 222 from both sides:
x = 138
Therefore, Mr. Clarkson initially had $138 and Mrs. Rose initially had $222.