Question

A, B, C, D, and E went to a restaurant. A paid $\dfrac{1}{2}$of the bill, B paid $\dfrac{1}{5}$of the bill and rest of the bill was shared equally by C, D, and E. what fraction of the bill was paid by each?

Answer 1

Hint: First, let us assume the overall total amount that needs to pay is $1$(this is the most popular concept that used in the probability that the total fraction will not exceed $1$and everything will be calculated under the number $0 - 1$as zero is the least possible outcome and one is the highest outcome)
Given that there is a total $5$person and $2$of them have already paid their bills and we need to calculate the remaining amount to the rest of the $3$person.
The remaining three-person shares the bill equally.

Complete step by step answer:
As per the given that A, B, C, D, and E went to a restaurant together.
Let A paid the bill in the fraction of $\dfrac{1}{2}$and B paid the bill in the fraction of $\dfrac{1}{5}$
Since five of them will need to pay the bill so that the balance will need to be paid in the fraction that is remaining of the required amount.
Hence, we need to find the fraction of the bill that needs to be paid by person C, D, and E.
Also, the total fraction will not exceed $1$ the total amount assumed as $1$
Thus combine the given statement into mathematically we get, $A + B + C + D + E = 1$
From the given that we know A paid the bill in the fraction of $\dfrac{1}{2}$and B paid the bill in the fraction of $\dfrac{1}{5}$
By applying the above values in the above equation we get, $A + B + C + D + E = 1 \Rightarrow \dfrac{1}{2} + \dfrac{1}{5} + C + D + E = 1$
Further solving the equation, we get, $\dfrac{1}{2} + \dfrac{1}{5} + C + D + E = 1 \Rightarrow C + D + E = 1 - \dfrac{1}{2} - \dfrac{1}{5} \Rightarrow \dfrac{3}{{10}}$
Thus, we get; $C + D + E = \dfrac{3}{{10}}$is the possibility that C, D, and E are able to pay the bill.
Given that they said C, D, and E paid the bill equally which means three of them give the same amount for the bill.
Hence assume the three people with the same variable (because three persons paid the bill equally)
Thus, we get $C = D = E = x$ and apply this in the above equation $C + D + E = \dfrac{3}{{10}} \Rightarrow x + x + x = \dfrac{3}{{10}}$
Further solving this we get $3x = \dfrac{3}{{10}} \Rightarrow x = \dfrac{1}{{10}}$(by the division operation)
Hence, we get the fraction of C, D, E as $C = D = E = \dfrac{1}{{10}}$and $A = \dfrac{1}{2},B = \dfrac{1}{5}$is the amount paid to the bill in the fraction.

Note: As we mention in the hint, the overall total amount that needs to be paid is $1$ (assumed), now we calculate each person who paid the bill.
Let us take all person bill paid in fraction, thus we get $A = \dfrac{1}{2},B = \dfrac{1}{5},C = D = E = \dfrac{1}{{10}}$and add every fraction then we get $A + B + C + D + E = \dfrac{1}{2} + \dfrac{1}{5} + \dfrac{1}{{10}} + \dfrac{1}{{10}} + \dfrac{1}{{10}}$
With the help of LCM write every denominator in the same values, $A + B + C + D + E = \dfrac{5}{{10}} + \dfrac{2}{{10}} + \dfrac{1}{{10}} + \dfrac{1}{{10}} + \dfrac{1}{{10}}$
Finally, we get $A + B + C + D + E = \dfrac{5}{{10}} + \dfrac{2}{{10}} + \dfrac{1}{{10}} + \dfrac{1}{{10}} + \dfrac{1}{{10}} \Rightarrow \dfrac{{10}}{{10}} = 1$, hence the total amount that is assumed as one is correct.