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Our first step was to find all the possible integers based on the 4 given digits (2,0,1,4)
In the end, we came up with 18 different integers. We double checked by arranging them from least to greatest and by using place value. Through this, we can identify whether or not we missed any other possible integers. The numbers were:
1024 is the smallest 4210 is the biggest 2617 is the average of the smallest(1024) and the biggest(4210) 1024,1042,1204,1240,1402,1420,2014,2041,2104,2140,2401,2410,4012,4021,4102,4120,4201,4210 The median is 2270.5
First I decided to find all the possible number?integers can be made with 2014. 1024 1042 1204 1240 1402 1420 2014 2041 2104 2140 2401 2410 4012 4021 4102 4120 4201 4210 2. With this I figured that all these numbers had a basic pattern of switching numbers. The smallest number was 1024 The largest number was 4210.
The average of the largest and smallest numbers was 1024+4210= 5234 5234/2=2617
The median of all the integers was 2270.5 rounded to 2271
(First I roganized this from the least to greatest and then I found the central number) 1024, 1042, 1204, 1240, 1402, 1420, 2014, 2041, 2104, 2140, 2401, 2410, 4012, 4021, 4102, 4120, 4201, 4210
Good job everyone. Remember to check-on on the site even after you've provided an answer because your peers may be asking you questions.
Solution First, we will determine the average of the largest and smallest numbers. With a given set of four digits, the largest possible integer that can be formed puts the largest digit in the thousands place, the second largest digit in the hundreds place, the third largest digit in the tens place, and the smallest digit in the units place. That is, the largest integer is formed by putting the digits in decreasing order from the thousands place to the units place. This is because the largest digit can make the largest contribution in the place with the most value. Thus, the largest integer that can be formed with the digits 2, 0, 1, 4 is 4210. Similarly, with a given set of digits the smallest integer is formed by putting the digits in increasing order from the thousands place to the units place. Here, there is an added wrinkle that the integer must be at least 1000. Therefore, the thousands digit is at least 1. The smallest integer of this type that can be made uses a thousands digit of 1, and then lists the remaining digits in increasing order. This integer is 1024. Therefore, average of the largest and smallest numbers is 2617. Now, let's determine the median of all of the numbers that can be created. Since the thousands digit is at least 1, the smallest numbers will begin with a 1. The remaining digits will be 0, 2 and 4, in some order. There are six possibilities for the remaining digits: 024, 042, 204, 240, 402, or 420. Therefore, there are six numbers that begin with a 1. Similarly, there are six numbers that will begin with a 2 and six numbers that will begin with a 4. Therefore, there are 18 di erent numbers that can be created this way. The smallest six numbers will begin with a 1 and the largest six numbers will begin with a 4. The middle six numbers will begin with a 2 and the median of all of the numbers will equal the median of the middle six numbers. The middle six numbers, from smallest to largest, are 2014; 2041; 2104; 2140; 2401; 2410 Since there is an even number of numbers, the median of these six numbers is the average of the middle two, which is 2122. Therefore, the median of all of the numbers is 2122. (Alternatively, we could have listed out all 18 numbers that can be created this way in order to determine the median.) The di erence between the average of the largest and smallest numbers and the median of all numbers is therefore 2617 - 2122 = 495.
Hello.
ReplyDeleteMy sister and I did this together.
Our first step was to find all the possible integers based on the 4 given digits (2,0,1,4)
In the end, we came up with 18 different integers. We double checked by arranging them from least to greatest and by using place value. Through this, we can identify whether or not we missed any other possible integers. The numbers were:
1024, 1042, 1204, 1240, 1402, 1420, 2014, 2041, 2104, 2140, 2401, 2410, 4012, 4021, 4102, 4120, 4201, 4210.
A pattern we noticed from this is that the tens and ones always switch with every upcoming thousands and hundreds pair.
e.g
1024,1042 (10 is the thousands/hundreds) *Note how the last two digits switch
We realized how this applied to the other integers. So, we used it to ensure we had all the right numbers.
Average of least and greatest = 2617
(4210 + 1024) / 2
Median of all possible integers = 2122
*note that when finding the median, there were two remaining numbers, 2104 and 2140.
(2104 + 2140) / 2
The difference = 495
2617 - 2122 = 495
1024 is the smallest
ReplyDelete4210 is the biggest
2617 is the average of the smallest(1024) and the biggest(4210)
1024,1042,1204,1240,1402,1420,2014,2041,2104,2140,2401,2410,4012,4021,4102,4120,4201,4210
The median is 2270.5
Can you please explain to me how 2270.5 is the median? I got a different answer.
DeleteFirst I decided to find all the possible number?integers can be made with 2014.
ReplyDelete1024
1042
1204
1240
1402
1420
2014
2041
2104
2140
2401
2410
4012
4021
4102
4120
4201
4210
2. With this I figured that all these numbers had a basic pattern of switching numbers.
The smallest number was 1024
The largest number was 4210.
The average of the largest and smallest numbers was
1024+4210= 5234
5234/2=2617
The median of all the integers was 2270.5 rounded to 2271
(First I roganized this from the least to greatest and then I found the central number)
1024, 1042, 1204, 1240, 1402, 1420, 2014, 2041, 2104, 2140, 2401, 2410, 4012, 4021, 4102, 4120, 4201, 4210
Good job everyone. Remember to check-on on the site even after you've provided an answer because your peers may be asking you questions.
ReplyDeleteSolution
First, we will determine the average of the largest and smallest numbers.
With a given set of four digits, the largest possible integer that can be formed puts the largest
digit in the thousands place, the second largest digit in the hundreds place, the third largest
digit in the tens place, and the smallest digit in the units place. That is, the largest integer is
formed by putting the digits in decreasing order from the thousands place to the units place.
This is because the largest digit can make the largest contribution in the place with the most
value. Thus, the largest integer that can be formed with the digits 2, 0, 1, 4 is 4210.
Similarly, with a given set of digits the smallest integer is formed by putting the digits in
increasing order from the thousands place to the units place. Here, there is an added wrinkle
that the integer must be at least 1000. Therefore, the thousands digit is at least 1. The
smallest integer of this type that can be made uses a thousands digit of 1, and then lists the
remaining digits in increasing order. This integer is 1024.
Therefore, average of the largest and smallest numbers is 2617.
Now, let's determine the median of all of the numbers that can be created.
Since the thousands digit is at least 1, the smallest numbers will begin with a 1. The
remaining digits will be 0, 2 and 4, in some order. There are six possibilities for the remaining
digits: 024, 042, 204, 240, 402, or 420. Therefore, there are six numbers that begin with a 1.
Similarly, there are six numbers that will begin with a 2 and six numbers that will begin with a
4. Therefore, there are 18 di erent numbers that can be created this way. The smallest six
numbers will begin with a 1 and the largest six numbers will begin with a 4. The middle six
numbers will begin with a 2 and the median of all of the numbers will equal the median of the
middle six numbers. The middle six numbers, from smallest to largest, are
2014; 2041; 2104; 2140; 2401; 2410
Since there is an even number of numbers, the median of these six numbers is the average of
the middle two, which is 2122. Therefore, the median of all of the numbers is
2122. (Alternatively, we could have listed out all 18 numbers that can be created this way in
order to determine the median.)
The di erence between the average of the largest and smallest numbers and the median of all
numbers is therefore 2617 - 2122 = 495.