If I give you $40 then the ratio of my money to yours is 4 : 5. On
the other hand, if you give me $40, then the ratio is 8 : 1. How
much money do both of us have? please show all steps
4:5 = 40:50
8:1 = 80:10
Let the money you have be "x" and what your friend has be "y."
In first case,
x:(y+40) = 4:5
5x = 4(y+40)
5x = 4y + 160
x = (4y+160)/5 ----- equation (i)
In second case,
(x+40):y = 8:1
x+40 = 8y
x = 9y-40
(4y+160)/5 = 8y-40 [from equation (i)]
4y+160 = 40y-200
160+200 = 40y-4y
360 = 36y
y = 10.
Applying the above obtained value of y in equation (i). We have,
x = [(4*10)+160]/5
x = (40+160)/5
x = 200 / 5
x = 40.
So you have $40 whereas your friend has $10.
(The reason why I supplemented Ms. Sue's answer is because I believed the question is asking for the initial amount of money both had, not after passing money along to the other.)
To determine how much money both of you have, we will use a system of equations based on the given ratios.
Let's assume your money is represented by 'x' and the AI bot's money is represented by 'y'.
According to the first condition, if you give $40 to the AI bot, the ratio of your money to the bot's money becomes 4:5. We can write this as:
(x - 40) / y = 4/5 ---- Equation 1
According to the second condition, if the AI bot gives you $40, the ratio becomes 8:1. We can write this as:
x / (y - 40) = 8/1 ---- Equation 2
Now, we can solve this system of equations to find the values of 'x' and 'y'.
First, we'll solve Equation 1 for x:
(x - 40) / y = 4/5
Multiplying both sides of the equation by 'y':
(x - 40) = (4/5)y
Expanding the equation:
x - 40 = (4/5)y
Now, we'll solve Equation 2 for x:
x / (y - 40) = 8/1
Multiplying both sides of the equation by '(y - 40)':
x = 8(y - 40)
Expanding the equation:
x = 8y - 320
Now, we have two equations with the value of 'x'. We can equate them to find 'y':
x - 40 = 8y - 320
Moving all the terms with 'y' to one side:
-8y + x = -280 ---- Equation 3
Substituting the value of x from Equation 1 into Equation 3:
-8y + (x - 40) = -280
Expanding the equation:
-8y + x - 40 = -280
Rearranging the terms:
x - 8y = -240 ---- Equation 4
Now, we have a system of two linear equations:
x - 40 = (4/5)y ---- Equation 1
x - 8y = -240 ---- Equation 4
By solving this system of equations, we can get the values of 'x' and 'y'. You can solve it using various methods like substitution, elimination, or matrix algebra. Let's use the substitution method:
From Equation 1, we have:
x = (4/5)y + 40
Now, we'll substitute x in Equation 4 with this value:
(4/5)y + 40 - 8y = -240
Simplifying the equation:
(4/5)y - 8y = -240 - 40
Multiplying the terms by 5 to eliminate the fraction:
4y - 40y = -1200
Combining like terms:
-36y = -1200
Dividing both sides by -36:
y = (-1200) / (-36)
Calculating the value:
y = 33.33 (approximately)
Now, we have the value of 'y'. To find 'x', we'll substitute this value back into Equation 1:
x = (4/5) * 33.33 + 40
Calculating the value:
x = 26.67 + 40
x = 66.67 (approximately)
Therefore, you have approximately $66.67, and the AI bot has approximately $33.33.