Paul and Ian have $280 altogether. 1/6 of Paul's money is $12 less than 1/5 of lan's money.
(a) How much money does each boy have?
(b) Find the ratio of Paul's money to Ian's money.
Let P be Paul's money and I be Ian's money.
We know that P + I = 280.
We also know that 1/6P = 1/5I - 12.
Multiplying both sides of the equation by 30, we get 5P = 6I - 360.
Multiplying both sides of the equation by 6, we get 30P = 36I - 2160.
Subtracting the first equation from the second equation, we get 29P = 35I - 1880.
Dividing both sides of the equation by 29, we get P = (35I - 1880)/29.
Substituting the value of P into the first equation, we get (35I - 1880)/29 + I = 280.
Multiplying both sides of the equation by 29, we get 35I - 1880 + 29I * 29*280.
Combining like terms, we get 64I = 1880 + 8120.
Adding the two numbers, we get 64I = 10000.
Dividing both sides of the equation by 64, we get I = 156.25.
Substituting the value of I into the equation P = (35I - 1880)/29, we get P = (35(156.25) - 1880)/29 = 133.75.
Therefore, Paul has $133.75 and Ian has $156.25.
The ratio of Paul's money to Ian's money is 133.75/156.25 = 0.855/1.
Let's start by assigning variables to each boy's money. Let's say Paul has x dollars, and Ian has y dollars.
According to the given information, we have two equations:
Equation 1: x + y = 280 (since Paul and Ian have $280 altogether)
Equation 2: (1/6)x = (1/5)y - 12 (since 1/6 of Paul's money is $12 less than 1/5 of Ian's money)
To solve this system of equations, we can use the substitution method.
First, let's solve Equation 1 for x:
x = 280 - y
Substitute this value of x into Equation 2:
(1/6)(280 - y) = (1/5)y - 12
Multiply both sides of the equation by 6 and distribute:
280 - y = (6/5)y - 72
Multiply both sides of the equation by 5 to eliminate the fraction:
1400 - 5y = 6y - 360
Combine like terms:
1400 + 360 = 6y + 5y
1760 = 11y
Divide both sides by 11:
y = 1760/11 = 160
Substitute the value of y back into Equation 1 to find x:
x + 160 = 280
x = 280 - 160
x = 120
(a) Paul has $120, and Ian has $160.
(b) The ratio of Paul's money to Ian's money is x : y = 120 : 160 = 3 : 4.