2012 AMC 8 Problems/Problem 16
Contents
Problem
Each of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 is used only once to make two five-digit numbers so that they have the largest possible sum. Which of the following could be one of the numbers?
Solution 1
In order to maximize the sum of the numbers, the numbers must have their digits ordered in decreasing value. There are only two numbers from the answer choices with this property: 
and 
. To determine the answer we will have to use estimation and the first two digits of the numbers.
For the number that would maximize the sum would start with 
. The first two digits of (when rounded) are 
. Adding and , we find that the first three digits of the sum of the two numbers would be 
.
For the number that would maximize the sum would start with 
. The first two digits of (when rounded) are 
. Adding and , we find that the first three digits of the sum of the two numbers would be 
.
From the estimations, we can say that the answer to this problem is 
.
Solution 2
In order to determine the largest number possible, we have to evenly distribute the digits when adding. The two numbers that show an example of this are 
and 
. The digits can be interchangeable between numbers because we only care about the actual digits.
The first digit must be either 
or 
. This immediately knocks out 
.
The second digit must be either 
or 
. This doesn't cancel any choices.
The third digit must be either 
or 
. This knocks out 
and 
.
The fourth digit must be 
or 
. This cancels out 
.
This leaves us with .
See Also
| 2012 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 15 |
Followed by Problem 17 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AJHSME/AMC 8 Problems and Solutions | ||
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