Are the following statements true?
40 persons : 200 persons = Rs. 15 : 75

Answer

To determine if the statements \( 40 \text{ persons} : 200 \text{ persons} = Rs. 15 : 75 \) are true, we will compare the two ratios step by step.

### Step 1: Express the Left-Hand Side (LHS) as a Fraction
The left-hand side is \( 40 \text{ persons} : 200 \text{ persons} \).

We can express this ratio as a fraction:
\[ \text{LHS} = \frac{40}{200} \]

### Step 2: Simplify the LHS Fraction
To simplify \( \frac{40}{200} \), we can divide both the numerator and the denominator by their greatest common divisor (GCD), which is 40:
\[ \frac{40 \div 40}{200 \div 40} = \frac{1}{5} \]

### Step 3: Express the Right-Hand Side (RHS) as a Fraction
The right-hand side is \( Rs. 15 : 75 \).

We can express this ratio as a fraction:
\[ \text{RHS} = \frac{15}{75} \]

### Step 4: Simplify the RHS Fraction
To simplify \( \frac{15}{75} \), we can divide both the numerator and the denominator by their GCD, which is 15:
\[ \frac{15 \div 15}{75 \div 15} = \frac{1}{5} \]

### Step 5: Compare the Two Fractions
Now we have:
\[ \text{LHS} = \frac{1}{5} \quad \text{and} \quad \text{RHS} = \frac{1}{5} \]

Since both fractions are equal, we conclude that:
\[ 40 \text{ persons} : 200 \text{ persons} = Rs. 15 : 75 \]

### Conclusion
Thus, the statement is **TRUE**.

---