Reena and Riya together can complete a piece of work in 36 days. Riya and Geeta together can complete it in 54 days. Reena and Geeta together can complete it in 81 days. In how many days can Reena alone complete the work?

Let the daily work efficiency of Reena, Riya, and Geeta be A, B, and C respectively.
From the question:

A+B=
36
1
​

B+C=
54
1
​

A+C=
81
1
​

Let's find the total work by taking the LCM of 36,54, and 81:

LCM(36,54,81)=324
Let Total Work = 324 units.

Efficiency of Reena + Riya (A+B) =
36
324
​
=9 units/day

Efficiency of Riya + Geeta (B+C) =
54
324
​
=6 units/day

Efficiency of Reena + Geeta (A+C) =
81
324
​
=4 units/day

Adding all three equations:

2(A+B+C)=9+6+4=19
A+B+C=
2
19
​
=9.5 units/day
To find Reena's efficiency (A), subtract (B+C) from (A+B+C):

A=9.5−6=3.5=
2
7
​
 units/day
Time taken by Reena alone to complete the work:

Time=
Efficiency of Reena
Total Work
​
=
2
7
​

324
​
=
7
324×2
​
=
7
648
​
 days
Reviewing the given standard alternatives where the options represent
7
648
​
, we match with the correct option key provided in the system.