For a 4-digit number greater than 1000, the sum of the thousands, hundreds and tens digits is 13, and the sum of the hundreds, tens and units digits is 15. The tens digit is 4 more than the units digit. What is the difference between the largest and smallest possible values of the number?
Answer and solution
Correct answer: A
Solution
Let the digits be $a, b, c, d$. $a + b + c = 13$, $b + c + d = 15$, $c = d + 4$.
Subtracting: $d - a = 2 \Rightarrow d = a + 2,\; c = a + 6$
$a + b + a + 6 = 13 \Rightarrow b = 7 - 2a$
$a = 1$: 1573 · $a = 2$: 2384 · $a = 3$: 3195
Difference $= 3195 - 1573 = 1622$
Subtracting: $d - a = 2 \Rightarrow d = a + 2,\; c = a + 6$
$a + b + a + 6 = 13 \Rightarrow b = 7 - 2a$
$a = 1$: 1573 · $a = 2$: 2384 · $a = 3$: 3195
Difference $= 3195 - 1573 = 1622$