Step 1: Understanding the Question:
This problem belongs to the topic of Ratio and Proportion. A proportion is a mathematical statement that asserts the equality of two ratios. In the expression $15 : x = 25 : 35$, we are given four terms (two antecedents and two consequents) where one of the values is an unknown variable $x$. Solving for $x$ requires us to find a value that maintains the same fractional relationship on the left side as exists on the right side.
Step 2 : Key Formulas and approach:
1. Fractional representation: A ratio $a : b$ is the same as the fraction $\frac{a}{b}$. Therefore, $a : b = c : d$ becomes $\frac{a}{b} = \frac{c}{d}$.
2. Product of Extremes and Means: In any proportion, the product of the outer terms (extremes) equals the product of the inner terms (means): $a \times d = b \times c$.
The approach involves setting up an equation using cross-multiplication to isolate the variable.
Step 3 : Detailed Explanation:
We start by rewriting the given proportion in a standard algebraic fractional format: $\frac{15}{x} = \frac{25}{35}$.
Before proceeding with large multiplications, it is often easier to simplify the known ratio. Looking at the right side, $\frac{25}{35}$ can be reduced. Both 25 and 35 are divisible by 5. Dividing both by 5, we get $\frac{5}{7}$.
Our updated equation is $\frac{15}{x} = \frac{5}{7}$. This simplification makes the subsequent steps much more manageable.
Next, we use the property of cross-multiplication. We multiply the numerator of the first fraction by the denominator of the second, and vice-versa: $15 \times 7 = 5 \times x$.
Calculating the left side, $15 \times 7 = 105$. So, our equation becomes $105 = 5x$.
To isolate $x$, we divide both sides by 5: $x = \frac{105}{5}$.
Performing the division, $105$ divided by $5$ equals $21$. This means that $15 : 21$ is the same ratio as $5 : 7$ (since $15/3=5$ and $21/3=7$), which matches our simplified original ratio.
Step 4 : Final Answer:
The value of the unknown variable $x$ is 21, which corresponds to option (C).