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We are given two key pieces of information about the group of students:

1. \(\frac{5}{9}\) of the students are male.
2. \(\frac{5}{6}\) of the female students are right-handed.

We are asked to find the smallest possible number of students in the group.

### Step 1: Define Variables
Let \( N \) be the total number of students in the group. Then:
- The number of male students is \( \frac{5}{9}N \).
- The number of female students is \( N - \frac{5}{9}N = \frac{4}{9}N \).

### Step 2: Female Right-Handed Students
Out of the female students, \( \frac{5}{6} \) are right-handed. So the number of right-handed female students is:
\[
\frac{5}{6} \times \frac{4}{9}N = \frac{20}{54}N = \frac{10}{27}N.
\]

### Step 3: Find the Smallest Possible \( N \)
For the number of students to be an integer, \( N \) must be a value such that:
- \( \frac{5}{9}N \) is an integer (for the number of male students),
- \( \frac{10}{27}N \) is an integer (for the number of right-handed female students).

To satisfy both conditions, \( N \) must be a multiple of the least common denominator of 9 and 27, which is 27.

### Step 4: Calculate the Group Breakdown for \( N = 27 \)
- Number of male students: \( \frac{5}{9} \times 27 = 15 \).
- Number of female students: \( 27 - 15 = 12 \).
- Number of right-handed female students: \( \frac{5}{6} \times 12 = 10 \).

Thus, the smallest possible number of students in the group is **27**.

Would you like more details, or do you have any questions?

Here are some related questions for practice:

1. What would the number of right-handed students be if the group had 54 students?
2. How would the solution change if \(\frac{2}{3}\) of the female students were right-handed?
3. If the group had 36 students, how many female left-handed students would there be?
4. Can you find the smallest possible number of students if only \(\frac{4}{5}\) of the students are male?
5. What if \(\frac{5}{9}\) of the female students were left-handed instead of right-handed?

**Tip:** When dealing with fractions and whole-number problems, always look for the least common multiple (LCM) to ensure integer solutions.

5/9 of the students in a group are male. 5/6 of the female students in the group are right-handed. Find the smallest possible number of students in the group.

Learn how to solve a fractional group distribution problem where 5/9 of the students are male and 5/6 of the female students are right-handed. This step-by-step solution explains how to find the smallest possible number of students in the group using least common multiples and fraction calculations. Suitable for Grades 6-8.

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