Three friends had dinner at a restaurant `2/3` When the bill was received, Alpana paid as much as Beena paid and Beena paid `1/2` as much as Catherin paid. What fraction of the bill did Beena pay?

#### Solution

Let ‘T’ be the total bill amount at the restaurant and ‘a’, ‘b’, and ‘c’ be the share of Alpana, Beena, and Catherin respectively.

Given, that Alpana paid rd `2^"rd"/3` as much as Beena paid

∴ a = `2/3`b ................(i)

Also, Beena paid `1/2` as much as Catherin paid.

∴ b = `1/2` c i.e. c = 2b ..................(ii)

∴ Three friends paid the total bill amount.

∴ a + b + c = T ..................(iii)

Using (i) and (ii) in (iii), we get

`2/3`b + b + 2b = T

∴ b`(2/3 + 1 + 2) = "T"`

∴ b`((2 + 3 + 6)/3)` = T

∴ `(11"b")/3` = T

∴ b = `3/11`T

Thus, Beena paid `(3/11)^"th"` fraction of the total bill amount.