Question

For a 4-digit number (greater than 1000), sum of the digits in the thousands, hundreds, and tens places is 15. Sum of the digits in the hundreds, tens, and units places is 16. Also, the digit in the tens place is 6 more than the digit in the units place. The difference between the largest and smallest possible value of the number is

• A. \(40\)
• B. \(78\)
• C. \(811\)
• D. \(735\)

Hint:

When dealing with digit problems, always convert the verbal conditions into equations using place-value notation, and apply the digit constraints \(0 \leq \text{digit} \leq 9\) to narrow down possible values quickly.

Answer 1

The Correct Option is C

Let the four-digit number be denoted by \(\overline{abcd}\), where \[ a = \text{thousands digit}, \quad b = \text{hundreds digit}, \quad c = \text{tens digit}, \quad d = \text{units digit}, \] with \(a \neq 0\). Step 1: Form equations from the given conditions. According to the question, \[ a + b + c = 15 \tag{1} \] \[ b + c + d = 16 \tag{2} \] Also, it is given that \[ c = d + 6 \quad \Rightarrow \quad d = c - 6. \tag{3} \] Step 2: Substitute for \(d\) in equation (2). Using equation (3) in equation (2), \[ b + c + (c - 6) = 16, \] \[ b + 2c - 6 = 16, \] \[ b + 2c = 22, \] \[ b = 22 - 2c. \tag{4} \] Step 3: Express \(a\) in terms of \(c\). Substituting equation (4) into equation (1), \[ a + (22 - 2c) + c = 15, \] \[ a + 22 - c = 15, \] \[ a = c - 7. \tag{5} \] Step 4: Apply digit restrictions. Since \(a, b, c, d\) are digits, they must satisfy \(0 \leq a,b,c,d \leq 9\) and \(a \geq 1\). From equation (5), \[ c - 7 \geq 1 \Rightarrow c \geq 8. \] Also, \[ c - 7 \leq 9 \Rightarrow c \leq 16. \] Thus, possible values of \(c\) are \(8\) and \(9\). Case 1: \(c = 8\) \[ a = 1, \quad b = 22 - 16 = 6, \quad d = 8 - 6 = 2. \] The number formed is \(1682\), which satisfies all conditions. Case 2: \(c = 9\) \[ a = 2, \quad b = 22 - 18 = 4, \quad d = 9 - 6 = 3. \] The number formed is \(2493\), which also satisfies all conditions. Step 5: Compute the required difference. The largest possible number is \(2493\) and the smallest possible number is \(1682\). \[ \text{Difference} = 2493 - 1682 = 811. \] Hence, the required difference is \(811\).

Answer 2

Let the 4-digit number be \( \overline{abcd} \), where: \[ a = \text{thousands digit}, \quad b = \text{hundreds digit}, \quad c = \text{tens digit}, \quad d = \text{units digit} \] with \( a \neq 0 \). 
Step 1: Write equations from the given conditions. From the question: \[ a + b + c = 15 \quad \cdots (1) \] \[ b + c + d = 16 \quad \cdots (2) \] Also, \[ c = d + 6 \quad \Rightarrow \quad d = c - 6 \quad \cdots (3) \] 
Step 2: Substitute \( d = c - 6 \) in equation (2). From (2): \[ b + c + d = 16 \Rightarrow b + c + (c - 6) = 16 \Rightarrow b + 2c - 6 = 16 \Rightarrow b + 2c = 22 \Rightarrow b = 22 - 2c \quad \cdots (4) \] 
Step 3: Use equation (1) to express \( a \) in terms of \( c \). From (1): \[ a + b + c = 15 \Rightarrow a + (22 - 2c) + c = 15 \Rightarrow a + 22 - c = 15 \Rightarrow a = 15 - 22 + c = c - 7 \quad \cdots (5) \] 
Step 4: Use digit constraints. Digits must satisfy \( 0 \leq a,b,c,d \leq 9 \) and \( a \geq 1 \). From (5): \( a = c - 7 \geq 1 \Rightarrow c \geq 8 \). Also \( a \leq 9 \Rightarrow c - 7 \leq 9 \Rightarrow c \leq 16 \). Since \( c \) is a digit, \( c \in \{8,9\} \). 
\underline{Case 1:} \( c = 8 \) \[ a = c - 7 = 1, \quad b = 22 - 2c = 22 - 16 = 6, \quad d = c - 6 = 2 \] Number: \( \overline{abcd} = 1682 \). Check: \[ 1 + 6 + 8 = 15, \quad 6 + 8 + 2 = 16, \quad 8 = 2 + 6 \; \text{(OK)} \] 
Case 2: \( c = 9 \) \[ a = c - 7 = 2, \quad b = 22 - 2c = 22 - 18 = 4, \quad d = c - 6 = 3 \] Number: \( \overline{abcd} = 2493 \). Check: \[ 2 + 4 + 9 = 15, \quad 4 + 9 + 3 = 16, \quad 9 = 3 + 6 \; \text{(OK)} \] 
Step 5: Find the required difference. Largest possible number \( = 2493 \) 
Smallest possible number \( = 1682 \) \[ \text{Difference} = 2493 - 1682 = 811 \] So, the required difference is \( 811 \).