Let's solve the problem step by step by assigning variables and setting up equations.

Step 1: Define the variables

• Let $S$ be the total number of students.
• Let $L$ be the number of left-handed students.
• Let $R$ be the number of right-handed students.

Since every student is either left-handed or right-handed: $S = L + R$

Step 2: Express the number of blue-eyed students

• One fifth of the total students have blue eyes, so the number of students with blue eyes is: $\frac{1}{5}S$
• One tenth of the left-handed students have blue eyes, so the number of left-handed students with blue eyes is: $\frac{1}{10}L$
• One quarter of the right-handed students have blue eyes, so the number of right-handed students with blue eyes is: $\frac{1}{4}R$

Step 3: Set up the equation for the blue-eyed students

The total number of blue-eyed students is the sum of the left-handed and right-handed students with blue eyes: $\frac{1}{5}S = \frac{1}{10}L + \frac{1}{4}R$

Step 4: Substitute $R = S - L$ into the equation

Substitute $R = S - L$ into the equation to express everything in terms of $L$ and $S$: $\frac{1}{5}S = \frac{1}{10}L + \frac{1}{4}(S - L)$

Step 5: Simplify the equation

First, eliminate the fractions by multiplying the entire equation by 20 (the least common multiple of 5, 10, and 4): $4S = 2L + 5(S - L)$ Simplify the equation: $4S = 2L + 5S - 5L$ Combine like terms: $4S - 5S = -3L$ $-S = -3L$ Divide both sides by -1: $S = 3L$

Step 6: Solve for the fraction of left-handed students

Since $S = 3L$, the fraction of left-handed students is: $\frac{L}{S} = \frac{L}{3L} = \frac{1}{3}$

Final Answer

The fraction of the students who are left-handed is $\frac{1}{3}$.

Would you like more details or have any questions? Here are some additional questions related to this problem:

1. If the total number of students is 300, how many are left-handed?
2. What fraction of the right-handed students do not have blue eyes?
3. If 60 students have blue eyes, how many are left-handed and have blue eyes?
4. What fraction of the students do not have blue eyes?
5. How would the result change if one eighth of the right-handed students had blue eyes instead?

Tip: Setting up equations with variables and systematically eliminating them can simplify complex word problems.