Number System Question 2 - CAT 2021
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:
Step-by-Step Explanation
Let the four digits of the number be a (thousands), b (hundreds), c (tens), and d (units).
- Write out the rules given in the problem:
- a + b + c = 14 --- (Equation 1)
- b + c + d = 15 --- (Equation 2)
- c = d + 4 --- (Equation 3)
- Connect the first and last digits:
Subtract Equation 1 from Equation 2. The digits b and c appear in both equations and cancel out completely:
(b + c + d) - (a + b + c) = 15 - 14
d - a = 1 ⇒ d = a + 1
- Maximize the number:
To make a 4-digit number as large as possible, we want the leading thousands digit (a) to be the highest possible value. Let's find the boundaries for our single digits (which can only go from 0 to 9):
- We know d = a + 1.
- From Equation 3, we know c = d + 4. Substituting d into this equation gives: c = (a + 1) + 4 = a + 5.
Since c is a single digit, it cannot be greater than 9:
a + 5 ≤ 9 ⇒ a ≤ 4
The largest value we can choose for the thousands digit is a = 4.
- Fill in the remaining digits:
- If a = 4, then d = 4 + 1 = 5.
- If d = 5, then c = 5 + 4 = 9.
- Now find b using Equation 1: 4 + b + 9 = 14 ⇒ 13 + b = 14 ⇒ b = 1.
Putting the digits a = 4,b = 1,c = 9,d = 5 together gives the highest possible number: 4195.
Correct Answer: 4195
đź’ˇ FundaMakers Tip:
When maximizing a number, focus entirely on the highest place value. By subtracting the two sum equations, you create a direct link between the first and last digits (d = a + 1). Combining this with the tens digit rule gives c = a + 5, which allows you to cap c at 9 and find the maximum value for a instantly.