AFCAT I 22 Jan 2025 Shift 1 Paper

Work done by Ajay and Vijay per day together $=$ Ajay's rate + Vijay's rateWork done by Vijay alone in 27 days $=$ Vijay's daily rate $\times 27$ daysUse the concept of work done to set up an equation and solve for the number of days Ajay worked before leaving.Work rate of Ajay:⇒ Ajay's rate $=\frac{1}{25}$ (work/day)Work rate of Vijay:⇒ Vijay's rate $=\frac{1}{40}$ (work/day)Work rate of Ajay and Vijay together:⇒ Combined rate $=\left(\frac{1}{25}\right)+\left(\frac{1}{40}\right)$Find a common denominator and add:⇒ Combined rate $=\frac{8+5}{200}$⇒ Combined rate = $\frac{13}{200}$ (work/day)Work done by Vijay alone in 27 days:⇒ Work done by Vijay $=27 \times \left(\frac{1}{40}\right)$⇒ Work done by Vijay = 27/40Let's say Ajay worked for x days. During these x days, both Ajay and Vijay were working.Work done by Ajay and Vijay together in x days:⇒ Work done together $= x \times \left(\frac{13}{200}\right)$The total work = Work done by Ajay and Vijay together + Work done by Vijay alone:$\Rightarrow 1=\left(x \times \frac{13}{200}\right)+\frac{27}{40}$Convert 27/40 to a fraction with a denominator of 200 :$\Rightarrow \frac{27}{40}=\frac{27 \times 5}{40 \times 5}=\frac{135}{200}$Substitute and solve for x :$\Rightarrow 1=\frac{13x}{200}+\frac{135}{200}$Multiply both sides by 200 to clear the fractions:$\Rightarrow 200=13x+135$Subtract 135 from both sides:$\Rightarrow 65=13x$Divide both sides by 13 :$\Rightarrow x=\frac{65}{13}$$\Rightarrow x=5$