Question

Shown on the left is a set of equations. Which option belongs to the same set? 

 

• A. A
• B. B
• C. C
• D. D

Hint:

For symbolic equations, identify the relationships between symbols and check if each operation maintains consistency across the options.

Answer 1

The Correct Option is C

Step 1: Set up the unknowns.
Let triangle \( =a \), square \( =b \), circled-cross \( =c \), crossed-square \( =d \).

Step 2: Translate the given equations.
\[ a \times a = b \]
\[ b + a = d \]
\[ a \times c = d \]
\[ d + c = 25 \]

Step 3: Solve for a.
From the first two equations, \( d = a^2+a \). From the third, \( c=\frac{d}{a}=a+1 \). Substituting both into the fourth equation:
\[ (a^2+a)+(a+1)=25 \]
\[ a^2+2a+1=25 \]
\[ (a+1)^2=25 \]
\[ a+1=5,\ a=4 \]

Step 4: Final Answer:
So \( a=4,\ b=16,\ c=5,\ d=20 \). Testing this against the four options, only \( c\times c=5\times5=25 \) holds true.
\[ \boxed{c \times c = 25} \]

Answer 2

Instead of solving the equations algebraically, we can also find the four values by testing small whole numbers directly, since every number in the puzzle (12, 20, 25, 28) is small and the relationships are simple multiplication and addition.

1. Option A: Testing triangle \(=4\) and circled-cross \(=5\) (values that make every original equation work), triangle times circled-cross gives \(20\), not the \(12\) this option claims, so it fails the test.
2. Option B: With crossed-square \(=20\) and triangle \(=4\), their sum is \(20+4=24\), not the \(20\) this option claims, so this equation does not hold.
3. Option C: With circled-cross \(=5\), circled-cross times circled-cross is \(5\times5=25\), matching the claimed total exactly, confirming this equation by direct substitution.
4. Option D: With square \(=16\) and circled-cross \(=5\), their sum is \(21\), not the \(28\) this option claims, so it fails the test.

Testing the working values of \(4, 16, 5\) and \(20\) for triangle, square, circled-cross and crossed-square directly against each option confirms that only option C holds up.

So the correct answer is Option C.