One morning, each member of manjul's family drank an 8-ounce mixture of coffee and milk. The amount of coffee and milk varied from cup to cup but never zero. Manjul drank `(1/7)^(th)` of the total amount of milk and `(2/17)^(th)` of the total amount of coffee. How many people are there in manjul's family?

Answer

To solve the problem, we need to determine how many members are in Manjul's family based on the information given about the amounts of coffee and milk they drank.

### Step-by-Step Solution:

1. **Define Variables**:
- Let \( k \) be the number of members in Manjul's family.
- Let \( x \) be the total amount of milk (in ounces).
- Let \( y \) be the total amount of coffee (in ounces).

2. **Set Up the Equations**:
- Each member drank an 8-ounce mixture of coffee and milk, so:
\[ x + y = 8k \]
- Manjul drank \( \frac{1}{7} \) of the total amount of milk and \( \frac{2}{17} \) of the total amount of coffee:
\[ \frac{x}{7} + \frac{2y}{17} = 8 \]

3. **Clear the Fractions**:
- To eliminate the fractions in the second equation, find a common denominator, which is 119 (the least common multiple of 7 and 17):
\[ 17x + 14y = 8 \times 119 \]
- Calculate \( 8 \times 119 \):
\[ 8 \times 119 = 952 \]
- Therefore, the equation becomes:
\[ 17x + 14y = 952 \]

4. **Substitute for \( y \)**:
- From the first equation \( x + y = 8k \), we can express \( y \) in terms of \( x \):
\[ y = 8k - x \]
- Substitute \( y \) in the second equation:
\[ 17x + 14(8k - x) = 952 \]
- Simplify:
\[ 17x + 112k - 14x = 952 \]
\[ 3x + 112k = 952 \]
\[ 3x = 952 - 112k \]
\[ x = \frac{952 - 112k}{3} \]

5. **Determine Constraints for \( x \)**:
- Since \( x \) must be positive, we have:
\[ 952 - 112k > 0 \implies k < \frac{952}{112} \approx 8.5 \]
- Also, since \( x \) must be less than \( 8k \):
\[ \frac{952 - 112k}{3} < 8k \]
\[ 952 - 112k < 24k \]
\[ 952 < 136k \implies k > \frac{952}{136} \approx 7 \]

6. **Combine the Inequalities**:
- From the inequalities, we have:
\[ 7 < k < 8.5 \]
- The only integer value satisfying this condition is:
\[ k = 8 \]

7. **Conclusion**:
- Therefore, the total number of members in Manjul's family is \( \boxed{8} \).