In a school, one fifth of the students have blue eyes. One tenth of the left-handed students have blue eyes. One quarter of the right- handed students have blue eyes. What fraction of the students are left-handed?

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.

Explore how to determine the fraction of left-handed students in a school using proportions and equations. Step-by-step solution provided for a scenario involving blue-eyed students and handedness. Suitable for Grades 5-6.

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