Find the number of people in the family

Question

PRE-RMO 2014 question 14 (set-A)

One morning,each member of Manjul's family drank an 8-ounce mixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Manjul drank $\frac17 ^{th}$ of the total amount of milk and $\frac2{17} ^{th}$ of the total amount of coffee. How many people are in the family?

---

What I did: let total milk = x and coffee = y.
Therefore after removal of Manjul's share milk: 102x/119 and coffee = 105y/119.
HCF (GCD) of (102,105) is 3. Therefore number of people = 3+1. Which is probably wrong because I didn't use the total amount of mixture anywhere... And after I tried placing y in terms of x I got a messy equation, which gave different HCF for different value of x<8.

Any help would be appreciated. :)

Answer 1

So we have $x+y = 8n$ and $\dfrac x7 + \dfrac{2y}{17}=8$, where $x, y \in \mathbb R^+$ and $n \in \mathbb N$.

$$\implies 17x + 14y = 952 \implies 3x+112n=952 \quad \text{and } y = -952+120n$$

Now $x> 0 \implies n \le 8$ and $y > 0 \implies n \ge 8$, so ...

Answer 2

If 6 more member take same amount as manjul 3/17 coffee will be left

1+6=7 3 coffee which is left, 2coffee + 1 milk = 8ounce so let us assume 2 coffee is gone but no milk is left hence this 1 left coffee will be drank by the same person 7+1=8 So 8 persons