The sides of a triangle are in the ratio of `1/3 : 1/4 : 1/5` and its perimeter is 94 cm. Find the length of the smallest side of the triangle.

Options

A. 18 cm
B. 22.5 cm
C. 24 cm
D. 27 cm

Answer

To solve the problem step by step, we will follow the given information about the sides of the triangle and its perimeter.

### Step 1: Set up the ratio of the sides
The sides of the triangle are given in the ratio \( \frac{1}{3} : \frac{1}{4} : \frac{1}{5} \). To work with these ratios more easily, we can express them in terms of a common variable \( x \).

Let:
- The first side = \( \frac{1}{3}x \)
- The second side = \( \frac{1}{4}x \)
- The third side = \( \frac{1}{5}x \)

### Step 2: Write the equation for the perimeter
The perimeter of the triangle is the sum of all its sides. According to the problem, the perimeter is 94 cm. Therefore, we can write the equation as follows:

\[ \frac{1}{3}x + \frac{1}{4}x + \frac{1}{5}x = 94 \]

### Step 3: Find a common denominator
To combine the fractions, we need to find the least common multiple (LCM) of the denominators 3, 4, and 5. The LCM of 3, 4, and 5 is 60. We will convert each term to have a denominator of 60:

\[ \frac{1}{3}x = \frac{20}{60}x, \quad \frac{1}{4}x = \frac{15}{60}x, \quad \frac{1}{5}x = \frac{12}{60}x \]

Now substituting these into the equation gives:

\[ \frac{20}{60}x + \frac{15}{60}x + \frac{12}{60}x = 94 \]

### Step 4: Combine the fractions
Now, we can combine the fractions:

\[ \frac{20 + 15 + 12}{60}x = 94 \]

Calculating the sum in the numerator:

\[ \frac{47}{60}x = 94 \]

### Step 5: Solve for \( x \)
To isolate \( x \), we multiply both sides by 60:

\[ 47x = 94 \times 60 \]

Calculating \( 94 \times 60 \):

\[ 47x = 5640 \]

Now, divide both sides by 47:

\[ x = \frac{5640}{47} = 120 \]

### Step 6: Find the lengths of the sides
Now that we have \( x \), we can find the lengths of the sides:

1. First side:
\[ \frac{1}{3}x = \frac{1}{3} \times 120 = 40 \text{ cm} \]
2. Second side:
\[ \frac{1}{4}x = \frac{1}{4} \times 120 = 30 \text{ cm} \]
3. Third side:
\[ \frac{1}{5}x = \frac{1}{5} \times 120 = 24 \text{ cm} \]

### Step 7: Identify the smallest side
Among the sides 40 cm, 30 cm, and 24 cm, the smallest side is:

\[ \text{Smallest side} = 24 \text{ cm} \]

### Conclusion
Thus, the length of the smallest side of the triangle is **24 cm**.

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