Each of the positive integers a, b, and c is a three-digit integer. If

Question

Each of the positive integers a, b, and c is a three-digit integer. If each of the digits 1 through 9 appears in one of these three integers, what is the minimum possible value of the sum of a, b, and c?

A. 45
B. 666
C. 774
D. 801
E. 1368

A. 45
B. 666
C. 774
D. 801
E. 1368

Bunuel · Accepted Answer

According to the stem we should use the digits 1 through 9 to construct 3 three-digit integers, so that their sum is as small as possible.

To minimize the sum, minimize the hundreds digits of a, b, and c, so make them 1, 2, and 3.
Next, minimize tens digits. Make them 4, 5, and 6.
Use the remaining digits (7, 8, and 9) for units digits.

So, a would be 147, b would be 258 and c would be 369.

147 + 258 + 369 = 774.

Answer: C.

KarishmaB · Answer

Let me add a line to Bunuel's perfect solution here:

The highest digits should be accorded the lowest place value so that they add lowest value to the sum.

Given a 3 digit number such as xyz, to get the lowest number, you should given the lowest digit to the highest place value i.e. to hundreds and the highest digit to the lowest place value i.e. units digit.

clipea12 · Answer

crystal clear

thanks

TeamGMATIFY · Answer

We are given three 3-digit integers and told that each digit from 1-9 comes in the integers.
This means that there is no repetition of digits as there are 9 places to fill and we have 9 items with us.

Now coming to the next portion. We need to find the minimum sum of the three 3-digit integers.
To make a 3-digit number smallest, we need to give the largest digit to the units place.

Hence the numbers would look like this at this stage: _ _ 9, _ _ 8, _ _ 7
Placing the left out largest integers on the tens places: _ 69, _58, _47
Left ones in the hundreds places: 369, 258, 147
Sum = 774

Option C

U473 · Answer

solution A is 20% salt and B is 80% salt.if you have 30 ounces of solution A and 60% ounces of B, in what ratio could you mix solution A with solution B to produce 50 ounces of a 50% salt solution.

Posted from my mobile device

U473 · Answer

Solution A is 20% salt and B is 80% salt.if you have 30 ounces of solution A and 60% ounces of B, in what ratio could you mix solution A with solution B to produce 50 ounces of a 50% salt solution.

Posted from my mobile device

EgmatQuantExpert · Answer

Solution

Given  
 • a, b, and c are three positive three-digit integer
• Each of the digits 1 through 9 appears in one of these three integers.

To find  
 • Minimum possible value of the sum of a, b, and c

Approach and Working out  
 • We will get the minimum sum when a, b and c are minimum.
 o And, all 9 digits from 1 to 9, both inclusive, are appearing in these 3 numbers. So, each digit is appearing only once.

• Now, for a, b, and c to be minimum, the thousand digit has to be minimum – So, thousand digit can be-> 1, 2, 3
 o Tens digit -> 4, 5, 6
o Units digit -> 7, 8, 9
 
• It does not matter which digit is in which number as we will ultimately be adding all the digits of one particular place.
 
So, sum = 147 + 258 + 369 = 774

Thus, option C is the correct answer.

Correct Answer: Option C

Thelionking1234 · Answer

Since a, b, and c are each three-digit integers, they contain a total of nine digits among them. There are nine different digits from 1 through 9, so each of those digits must appear exactly once among all three numbers; therefore, there also cannot be any repeated digits.

Our goal is to minimize the sum (total) of all three numbers. The central principle is that the hundreds digits have a greater impact on the sum than do the tens digits, which, in turn, have a greater impact than do the units digits. For instance, 912 is larger than 192; while the two numbers contain the same digits, the hundreds digit has a much greater impact, and so the first number is much larger than the second. Therefore, the key is to make the hundreds digits as small as possible; then, from whatever digits remain, to choose the smallest tens digits; and, finally, to use the remaining digits for the units places.

The smallest digits are 1, 2, and 3, so those should be the hundreds digits for each of the three numbers. Now, we have 1__, 2__, and 3__. After these digits are in place, the smallest remaining digits are 4, 5, and 6, which become the tens digits. These can be paired with the hundreds digits in any order; one possibility is: 14_, 25_, 36_. Once these are used, the only remaining digits are 7, 8, and 9, which become the units digits of the three numbers; again, these can be paired in any order and one possibility is: 149, 258, 367. Add the three numbers to get 774. (It doesn't matter how we pair the numbers at each step; the final sum will still be 774.)

Alternatively we can make the addition step easier using our knowledge of place values. Because we want to use 1, 2, and 3 as the hundreds digits, we have 100, 200, and 300. Using 4, 5, and 6 as the tens digits gives us 40, 50, and 60. Finally, 7, 8, and 9 are the units digits. Add up these 9 numbers: 100 + 200 + 300 + 40 + 50 + 60 + 7 + 8 + 9 = 774. Note: this will give us the same sum as, for example, 149 + 258 + 367 because the number 149 consists of 100 + 40 + 9, the number 258 consists of 200 + 50 + 8 and the number 367 consists of 300 + 60 + 7. In other words, any combination that we choose at the first stage, as long as we are choosing the minimum options at each step, will ultimately result in 100 + 200 + 300 + 40 + 50 + 60 + 7 + 8 + 9 = 774.

The correct answer is C.

Anonymous · Answer

Hey Bunuel! I followed the same reasoning but I'd like your thoughts to clarify the "We are given three 3-digit integers and told that each digit from 1-9 comes in the integers. This means that there is no repetition of digits as there are 9 places to fill and we have 9 items with us." I interpreted at first that there's no repetition in either of the 3 numbers (we cannot have for instance 115)

Thanks in advance ☺️

Thanks in advance ☺️

Bunuel · Answer

Since there are nine digits to distribute among three three-digit numbers, there is no repetition of any digit within or across the numbers. That means each digit 1 through 9 must be used exactly once, ensuring that no number has a repeated digit (e.g., 115 would be invalid).

Anonymous · Answer

each digit from 1 to 9 appears, so no digit can be omitted. ok now i get it, thank you!

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