Question

Given below are two statements:
Statement I : The set of numbers (7,8,9,a,b,10,8,7) has an arithmetic mean of 9 and mode (most frequently occurring number) as 8, Then a x b = 120.
Statement II : Let a and b be two positive integers such that a + b + a x b = 84, then a + b =20.
In the light of the above statements, choose the most appropriate answer from the options given below:

• A. Both Statement I and Statement II are correct
• B. Both Statement I and Statement II are incorrect
• C. Statement I is correct but Statement II is incorrect
• D. Statement I is incorrect but Statement II is correct

Answer 1

The Correct Option is A

Let's analyze both statements to determine whether they are correct.

Statement I Analysis

The set of numbers given is \( (7, 8, 9, a, b, 10, 8, 7) \). We are told that the arithmetic mean of this set is 9 and that the mode is 8. We need to verify whether \( a \times b = 120 \).

1. First, calculate the arithmetic mean:
   • The sum of the numbers in the set is \( 7 + 8 + 9 + a + b + 10 + 8 + 7 \).
   • This simplifies to \( 49 + a + b \).
   • We are given that the mean is 9. Therefore, \(\frac{49 + a + b}{8} = 9\).
   • Solving for \( a + b \): \(49 + a + b = 72\) 
     \( a + b = 23 \).
2. Next, verify the mode:
   • The mode is the most frequently occurring number. Here, 8 appears twice, while others appear less frequently.
   • This confirms 8 is indeed the mode as required.
3. Finally, check the product \( a \times b \):
   • To satisfy \( a + b = 23 \) and calculate \( a \times b = 120 \):
   • Set up the equations for \( a \) and \( b \):
   • Consider the quadratic equation \( x^2 - 23x + 120 = 0 \).
   • Factor the equation: \( (x - 15)(x - 8) = 0 \).
   • This gives solutions: \( a = 15 \), \( b = 8 \) or vice versa.
   • This results in \( a \times b = 120 \) confirming Statement I is correct.

Statement II Analysis 

We are given: \(a + b + a \times b = 84\). We need to check if \(a + b = 20\).

1. Re-arrange the given expression:
   • Consider: \( a + b + ab = 84 \).
   • Letting \( a + b = s \) and \( ab = p \), then: \( s + p = 84 \).
   • If \( s = 20 \), then \( p = 84 - 20 = 64 \).
2. Verify by solving the quadratic:
   • With \( a \) and \( b \) as solutions to \( x^2 - sx + p = 0 \):
   • Substituting \( s = 20 \) and \( p = 64 \) yields:
   • \( x^2 - 20x + 64 = 0 \).
   • This gives roots: \( x = 16 \) and \( x = 4 \).
3. Verify:
   • This confirms numbers 16 and 4 satisfy both \( a + b = 20 \) and \( a \times b = 64 \).

Therefore, both Statement I and Statement II are correct.