# Work

The first worker would need less than 4 hours to complete the task than the other worker. In fact, both workers worked for two hours together, then the first worker did the remaining work himself. In what proportion should the remuneration of the workers be based on the work was done?

Result

f =  1.781

#### Solution:

$a=b - 4 \ \\ b=a+4 \ \\ 2 \cdot \ (1/a+1/b) + 2/a=1 \ \\ 2 \cdot \ (b+a) + 2b=ab \ \\ \ \\ 2 \cdot \ (a+4+a) + 2(a+4)=a(a+4) \ \\ -a^2 +2a +16=0 \ \\ a^2 -2a -16=0 \ \\ \ \\ p=1; q=-2; r=-16 \ \\ D=q^2 - 4pr=2^2 - 4\cdot 1 \cdot (-16)=68 \ \\ D>0 \ \\ \ \\ a_{1,2}=\dfrac{ -q \pm \sqrt{ D } }{ 2p }=\dfrac{ 2 \pm \sqrt{ 68 } }{ 2 }=\dfrac{ 2 \pm 2 \sqrt{ 17 } }{ 2 } \ \\ a_{1,2}=1 \pm 4.1231056256177 \ \\ a_{1}=5.1231056256177 \ \\ a_{2}=-3.1231056256177 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ (a -5.1231056256177) (a +3.1231056256177)=0 \ \\ a=a_{1}=5.1231 \doteq 5.1231 \ \\ b=a+4=5.1231+4 \doteq 9.1231 \ \\ f=b/a=9.1231/5.1231 \doteq 1.7808 \doteq 1.781$

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