Ajay and Vijay can do a work in 25 days and 40 days respectively. They start the work together, but Ajay left after some days and Vijay finished the remaining work in 27 days. After how many days did Ajay leave?
Work done by Ajay and Vijay per day together = Ajay's rate + Vijay's rate Work done by Vijay alone in 27 days = Vijay's daily rate ×27 days Use 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 =1∕25 (work/day) Work rate of Vijay: ⇒ Vijay's rate =1∕40 (work/day) Work rate of Ajay and Vijay together: ⇒ Combined rate =(1∕25)+(1∕40) Find a common denominator and add: ⇒ Combined rate =(8+5)∕200 ⇒ Combined rate = 13∕200 (work/day) Work done by Vijay alone in 27 days: ⇒ Work done by Vijay =27×(1∕40) ⇒ Work done by Vijay = 27/40 Let'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×(13∕200) The total work = Work done by Ajay and Vijay together + Work done by Vijay alone: ⇒1=(x×13∕200)+27∕40 Convert 27/40 to a fraction with a denominator of 200 : ⇒27∕40=(27×5)∕(40×5)=135∕200 Substitute and solve for x : ⇒1=(13x∕200)+135∕200 Multiply both sides by 200 to clear the fractions: ⇒200=13x+135 Subtract 135 from both sides: ⇒65=13x Divide both sides by 13 : ⇒x=65∕13 ⇒x=5