Nathan and Timothy work as plumbers. In October, they made a combined income of $5,069 after working a combined total of 132 hours. Timothy’s hourly wage is $4 more than two times Nathan’s. How much did Timothy earn in October?
Let x be Nathan's hourly wage.
Therefore, Timothy's hourly wage is 2x + 4.
In October, Nathan worked N hours and Timothy worked T hours.
We know that N + T = 132 and xN + (2x + 4)T = 5069.
Substitute N = 132 - T into the second equation:
x(132 - T) + (2x + 4)T = 5069
132x - Tx + 2xT + 4T = 5069
132x + T(2x - x + 4) = 5069
132x + T( x + 4) = 5069
132x + xT + 4T = 5069
133x + 4T = 5069
133x = 5069 - 4T
x = (5069 - 4T)/133
x = (5069/133) - (4T/133)
x = 38 - (4T/133)
Since Nathan's hourly wage cannot be negative, we have:
38 - (4T/133) >= 0
38 >= 4T/133
38*133 >= 4T
[38*133]/4 >= T
353.5 >= T
If Timothy worked less than 353.5 hours in October, Nathan's hourly wage would be negative, which is not possible. Therefore, Timothy worked more than 353.5 hours in October.
We know that Timothy worked T hours and Nathan worked N hours, and they together worked 132 hours. Thus T > 132 - T and T > 66.
Timothy worked 67 hours (rounded to the nearest whole number) and Nathan worked 65 hours in October.
Plugging in T=67 hours into the equation 133x = 5069 - 4T
133x = 5069 - 4(67)
133x = 5069 - 268
133x = 4801
x = 36
Therefore, Nathan's hourly wage is $36 and Timothy's hourly wage is 2*36 + 4 = $76.
Timothy earned 76 * 67 = $5084 in October.