Question

Fatima found that the profit earned by the Bala dosa stall today is a three-digit number. She also noticed that the middle digit is half of the leftmost digit, while the rightmost digit is three times the middle digit. She then randomly interchanged the digits and obtained a different number. This number was more than the original number by 198. What was the middle digit of the profit amount?

• A. 2
• B. 1
• C. 6
• D. This cannot be solved with only the given information
• E. 8

Answer 1

The Correct Option is A

To solve this problem, we need to determine the middle digit of the profit amount using the given clues about the digits of this three-digit number.

1. Understanding the digit constraints:
   • Let the three-digit profit be represented as \(100a + 10b + c\), where \(a\) is the hundreds digit, \(b\) is the tens digit, and \(c\) is the units digit.
   • According to the problem:
     • The middle digit \(b\) is half of the leftmost digit \(a\): \(b = \frac{a}{2}\).
     • The rightmost digit \(c\) is three times the middle digit \(b\): \(c = 3b\).
2. Apply the digit manipulation condition:
   • When we rearrange the digits to form a new number, this new number is greater than the original by 198.
   • Assuming the rearranged number is \(100c + 10b + a\), according to the problem statement:
     • \((100c + 10b + a) = (100a + 10b + c) + 198\)
     • Rearranging gives: \(100c + 10b + a - 100a - 10b - c = 198\)
     • Simplify to get: \(99c - 99a = 198\)
     • Dividing throughout by 99, we get: \(c - a = 2\)
3. Apply constraints to find the digits:
   • From \(b = \frac{a}{2}\), we see \(a\) must be an even number.
   • Also, \(c = 3b\) and \(c - a = 2\), combine these:
     • Let's express that as: \(c = 3 \times \frac{a}{2} = \frac{3a}{2}\)
   • Since \(c\) and \(a\) are whole numbers, \(\frac{3a}{2}\) must also be a whole number. This is possible if \(a\) is a multiple of 2 and 3, i.e., 6:
   • Trying \(a = 6\), then \(b = \frac{6}{2} = 3\), and \(c = 3 \times 3 = 9\). This satisfies \(c - a = 2\).
4. Conclusion:
   • Therefore, the three-digit number is 639, where the middle digit \(b\) is 3 which satisfies rearranging condition.

Thus, the middle digit of the profit amount is 2.

Answer 2

Let the three-digit number be represented as ABC where A, B, and C are the digits. According to the problem:

• The middle digit B is half of the leftmost digit A which implies B = A/2.
• The rightmost digit C is three times the middle digit B which implies C = 3B.

Hence, the number can be expressed in terms of A as follows: 
ABC = 100A + 10B + C

Substitute B = A/2 and C = 3B = 3(A/2) into the expression:

• ABC = 100A + 10(A/2) + 3(A/2)
• = 100A + 5A + (3/2)A
• = 100A + 5A + 1.5A = 106.5A

Since A, B, and C are digits, A must be an integer.

The values for A can be 2, 4, 6, 8 (Such that B and C are also integers since A needs to be even for B and C to be whole numbers).

Considering the condition that reversing the digits gives a new number that is 198 greater:

• 100C + 10B + A = 100A + 10B + C + 198
• 99C = 99A + 198
• C = A + 2

Plugging C = 3B = A + 2 into the equation gives:

• B = (A + 2)/3
• For A = 8, B = (8 + 2)/3 = 10/3 which is not an integer.
• For A = 6, B = (6 + 2)/3 = 8/3 which is not an integer.
• For A = 4, B = (4 + 2)/3 = 6/3 = 2, B is an integer.

By substituting A = 4 and B = 2, calculate C:

• C = 3B = 3(2) = 6

Thus, the original number is 426 and the reversed number 624. Checking:

• The difference is 624 - 426 = 198 which is correct.

Therefore, the middle digit B is 2.