Identify the number that will replace the question mark in the second equation based on the relationship represented in the first equation.

![image.png](https://res.cloudinary.com/aptidude/image/upload/v1743792914/aptidude_questions/yj0yyzmwxp7yafhtm4jg.png)

Question

Identify the number that will replace the question mark in the second equation based on the relationship represented in the first equation.

![image.png](https://res.cloudinary.com/aptidude/image/upload/v1743792914/aptidude_questions/yj0yyzmwxp7yafhtm4jg.png)

Accepted Answer

#### Step 1: Understand the Relationship
The relationship between the numbers in the nested circles is given by the formula:  
$$
a^2 + b^2 + c^2 + ab + bc + ca,
$$  
where $$a$$, $$b$$, and $$c$$ are the numbers in the nested boxes from outermost to innermost.

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#### Step 2: First Circle Calculation
For the first circle, the numbers are:  
$$ a = 4, \, b = 3, \, c = 4. $$

Substitute into the formula:  
$$
a^2 + b^2 + c^2 + ab + bc + ca = 4^2 + 3^2 + 4^2 + (4 	imes 3) + (3 	imes 4) + (4 	imes 4).
$$

Simplify:  
$$
16 + 9 + 16 + 12 + 12 + 16 = 81.
$$

The result for the first circle is **287**.

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#### Step 3: Second Circle Calculation
For the second circle, the numbers are:  
$$ a = 7, \, b = 2, \, c = 3. $$

Substitute into the formula:  
$$
a^2 + b^2 + c^2 + ab + bc + ca = 7^2 + 2^2 + 3^2 + (7 	imes 2) + (2 	imes 3) + (7 	imes 3).
$$

Simplify:  
$$
49 + 4 + 9 + (7 	imes 2) + (2 	imes 3) + (7 	imes 3) = 49 + 4 + 9 + 14 + 6 + 21 = 138.
$$

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#### Step 4: Conclusion
The missing number in the second circle is **138**, which does not match any of the provided options.

Thus, **none of the options match**, and the correct answer is **138**.

Answer

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