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For a 4-digit number, the sum of its digits in the thousands, hundreds and tens places is 14, the sum of its digits in the hundreds, tens and units places is 15, and the tens place digit is 4 more than the units place digit. Then the highest possible 4-digit number satisfying the above conditions is

Updated On: Jan 15, 2026
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Correct Answer: 4195

Solution and Explanation

4-Digit Number Maximization Problem

Let the 4-digit number be represented by its digits \(a, b, c, d\), where \(a\) is the thousands digit, \(b\) is the hundreds digit, \(c\) is the tens digit, and \(d\) is the units digit. The problem specifies the following constraints:

  • Sum of thousands, hundreds, and tens digits: \( a + b + c = 14 \).
  • Sum of hundreds, tens, and units digits: \( b + c + d = 15 \).
  • Relationship between tens and units digits: \( c = d + 4 \).

The objective is to determine the largest possible 4-digit number that adheres to these conditions.

From the third condition, \( c = d + 4 \).

Substitute \(c\) into the other equations:

Equation 1: \(a + b + c = 14\)

Substituting \(c = d + 4\) yields: \( a + b + (d + 4) = 14 \), which simplifies to \( a + b + d = 10 \).

Equation 2: \(b + c + d = 15\)

Substituting \(c = d + 4\) yields: \( b + (d + 4) + d = 15 \), which simplifies to \( b + 2d = 11 \).

The aim is to find the maximal values for \(a, b, c, d\).

Step 1: Solve for \(b\) in terms of \(d\)

From \(b + 2d = 11\), we isolate \(b\): \( b = 11 - 2d \).

Since \(b\) must be a single digit (0-9), \( 11 - 2d \geq 0 \), implying \( d \leq 5 \).

Step 2: Solve for \(a\) in terms of \(d\)

Substitute \(b = 11 - 2d\) into \(a + b + d = 10\): \( a + (11 - 2d) + d = 10 \), which simplifies to \( a = d - 1 \).

As \(a\) is the leading digit, it must be between 1 and 9. Therefore, \( d - 1 \geq 1 \), implying \( d \geq 2 \).

Step 3: Combine constraints on \(d\)

The valid range for \(d\) is established as \( 2 \leq d \leq 5 \).

To maximize the 4-digit number, we select the largest possible value for \(d\), which is 5.

Step 4: Calculate digits using \(d = 5\)

With \(d = 5\):

  • \( a = d - 1 = 5 - 1 = 4 \)
  • \( b = 11 - 2d = 11 - 2 \times 5 = 11 - 10 = 1 \)
  • \( c = d + 4 = 5 + 4 = 9 \)

The resulting number is 4195.

Step 5: Verify the solution

  • \( a + b + c = 4 + 1 + 9 = 14 \) (Condition met).
  • \( b + c + d = 1 + 9 + 5 = 15 \) (Condition met).
  • \( c = d + 4 \Rightarrow 9 = 5 + 4 \) (Condition met).

Conclusion

The largest 4-digit number that satisfies all given conditions is 4195.

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