The third proportional to $38$ and $15$ is

### Concept & Formula When finding the third proportional to two numbers $a$ and $b$, the numbers are considered to be in continued proportion. This means the ratio of the first to the second is equal to the ratio of the second to the third (the unknown $x$). $$ a : b :: b : x \implies \frac{a}{b} = \frac{b}{x} $$ ### Step-by-Step Solution 1. Identify the given values: $a = 38$ and $b = 15$. Let the third proportional be $x$. 2. Set up the continued proportion equation: $$ \frac{38}{15} = \frac{15}{x} $$ 3. Cross-multiply to solve for $x$: $38 \times x = 15 \times 15$ 4. Isolate $x$ by dividing by 38: $$ x = \frac{15 \times 15}{38} $$ 5. Look at the provided options. The options are in uncalculated fraction form, so we do not need to compute $225 / 38$. ### Exam Strategy & Shortcut The formula for the third proportional of $a$ and $b$ is simply $x = \frac{b^2}{a}$. You can jump straight to this formula: square the second number and divide by the first number. Thus, $\frac{15^2}{38}$ or $\frac{15 \times 15}{38}$. ### Common Pitfall Confusing the third proportional with the third term in a sequence of three different given numbers. For two numbers, the middle number is always repeated to form the proportion $a:b::b:x$. ### Final Answer Therefore, the correct answer is **$\frac{15 \times 15}{38}$**.