Question

In some code, letters a, b, c, d and e represent numbers 2, 4, 5, 6 and 10. We just do not know which letter represents which number. Consider the following relationships:
I. \(a + c = e\),
II. \(b - d = d\) and
III. \(e + a = b\)
Which of the following statements is true?

• A. \(b = 4,\ d = 2\)
• B. \(a = 4,\ e = 6\)
• C. \(b = 6,\ e = 2\)
• D. \(a = 4,\ c = 6\)

Hint:

Relationship II gives \(b = 2d\), so only two pairs of numbers from the list can be \(b\) and \(d\). Try both and see which one lets I and III hold.

Answer 1

The Correct Option is B

Step 1: Use relationship II:
\(b - d = d\) means \(b = 2d\). From the numbers 2, 4, 5, 6, 10 the only pairs with one double the other are \((b, d) = (4, 2)\) and \((b, d) = (10, 5)\).

Step 2: Try \(b = 4,\ d = 2\):
The remaining numbers for \(a, c, e\) are 5, 6, 10. III says \(e + a = 4\), but the smallest sum of two of these is \(5 + 6 = 11\). This is impossible.

Step 3: Try \(b = 10,\ d = 5\):
The remaining numbers for \(a, c, e\) are 2, 4, 6. I says \(a + c = e\), so \(e = 6\) with \(\{a, c\} = \{2, 4\}\). III says \(e + a = 10\), so \(a = 4\) and then \(c = 2\).
So \(a = 4,\ b = 10,\ c = 2,\ d = 5,\ e = 6\). Check: \(4 + 2 = 6\), \(10 - 5 = 5\), \(6 + 4 = 10\). All three hold.

Step 4: Compare with the options:
(1) \(b = 4\) is false. (2) \(a = 4,\ e = 6\) is true. (3) \(b = 6\) is false. (4) \(c = 6\) is false.

Final Answer:
The correct option is (2): \(a = 4,\ e = 6\).