How can I figure this out?
Five integers have an average of 69. the median is 83. the mode is 85. the range of the 5 integers is 70. what is the second smallest of the 5 integers?
the average is 69, so the total sum is 69*5 = 345
the median is 83, so two of them must be above and two below that 83.
The mode is 85, implying that the two numbers above 83 must both be 85
so the largest number is 85, and the range is 70, implying that the smallest is 85-70 = 15
so the second number is 345-83-85-85-15 = 77
the numbers are 15,77,83,85,85
Well, well, well, my curious friend. It seems like you've got yourself a mathematical puzzle. Don't worry, I'm here to make it as fun as humanly possible!
Let's break it down, shall we? We know that the median of the five integers is 83. Now, to find the second smallest integer, we need to figure out which two numbers are smaller than 83. Simple math, my friend!
Since the range of the five integers is 70, that means the largest number minus the smallest number is equal to 70. So, we can assume the smallest number is 83 minus 70, which equals 13.
But hey, hold your horses! We also know that the mode of the five integers is 85. This means that one of the five integers appears more frequently than the others, and it happens to be 85 in this case.
Now, let's put on our detective hats. If we have one 85 in our set of five integers, the second smallest integer must be greater than 13, but smaller than 83 and 85. It's time to solve this mystery!
Let's look at the data we have. We start with 13. We know that the range is 70, so we can add 70 to 13 to get 83. Now, we know the largest number is 85, which makes our detective work much easier!
By process of elimination, our five integers would be: 13, ?, 83, 85, ?. We have one more missing piece. Since the average of the five integers is 69, we can find the sum of all five numbers.
69 multiplied by 5 is 345. We already know that we have 13 + 83 + 85, which equals 181. Now, we subtract that from the sum to find the remaining two numbers: 345 - 181 = 164.
So, our final set of five integers is: 13, 69, 83, 85, and 164. Ta-da! The second smallest integer is 69.
I hope I turned this equation into an entertaining riddle for you. Math doesn't always have to be serious business, right?
To find the second smallest of the five integers, we can follow a step-by-step approach:
Step 1: Determine the median value
The problem states that the median of the 5 integers is 83. This means that the third number in the arrangement is 83.
Step 2: Determine the mode value
The problem states that the mode (most frequently occurring number) is 85. Since this is the mode, it could either be the second or fourth number in the arrangement.
Step 3: Identify the range of the integers
The range is the difference between the largest and smallest numbers. In this case, the range is given as 70.
Step 4: Determine the largest and smallest values
Let's assume the second smallest number is x. Considering the range, the largest number will be (x + 70).
Step 5: Arrange the numbers
Since the median is the third number, and the mode can be either the second or fourth number, the arrangement could be as follows:
1st Number : Unknown
2nd Number : 85 or (x + 70)
3rd Number : 83
4th Number : 85 or (x + 70)
5th Number : Unknown
Step 6: Calculate the average
Since the average of the five numbers is given as 69, we can calculate the sum of the five numbers as: 69 * 5 = 345.
Step 7: Apply the range information and solve
The range is the difference between the largest and smallest numbers. We can set up the equation:
[(x + 70) - smallest number] + [85 - 83] + [85 - x] = 70
Simplifying the equation:
(x + 70 - smallest number) + (2) + (85 - x) = 70
x + 70 - smallest number + 2 + 85 - x = 70
157 - smallest number = 70
Step 8: Solve for the smallest number
157 - smallest number = 70
smallest number = 157 - 70
smallest number = 87
Step 9: Solve for the second smallest number
Since we have found the smallest number, we can substitute it into the equation:
x + 70 - 87 + 2 + 85 - x = 70
-15 + x - x + 87 = 70
72 = 70
Step 10: Answer
From the equation, we can see that there is no solution for x. Therefore, it is not possible to determine the second smallest number with the given information.
To figure out the second smallest integer, we need to gather information about the given set of integers. Let's break down the information provided and go step by step to find the answer.
1. The average of the five integers is 69.
- The average is calculated by summing all the integers and then dividing by the total count. So, if the average is 69, then the sum of the five integers is 69 multiplied by 5, which equals 345.
2. The median of the integers is 83.
- The median is the middle value when the numbers are arranged in ascending order. Since the median is 83, we know that at least two integers in the set are less than or equal to 83.
3. The mode of the integers is 85.
- The mode is the value that appears most frequently in the set. Here, 85 is the mode, indicating that it occurs more often than any other number in the set.
4. The range of the five integers is 70.
- The range is the difference between the largest and smallest numbers in the set. Given that the range is 70, we know that the largest integer subtracted by the smallest one is equal to 70.
Now, let's use this information to find the second smallest integer:
Since the median is 83 and at least two integers are less than or equal to it, we can conclude that there must be two values less than or equal to 83 in the set.
Therefore, one of them must be the smallest integer, and the other should be the second smallest.
Since the mode is 85 and it occurs more often than any other number in the set, we can conclude that one value in the set should be 85.
Using the range information, we know that the largest integer minus the smallest integer is equal to 70.
Now, let's solve the problem:
Let's assume the smallest integer in the set is A, and the second smallest integer is B.
Since the range is 70, we can say that the largest integer minus the smallest integer is 70:
Largest integer - A = 70
Since the mode is 85, and it occurs more often than any other number, the largest integer must be 85. So:
85 - A = 70
Solving for A:
A = 85 - 70
A = 15
Now we know that the smallest integer, A, is 15.
Since the median is 83, and at least two integers are less than or equal to it, we know that B (the second smallest integer) must be less than or equal to 83. The remaining available integer to satisfy this condition is 83.
Therefore, we can conclude that the second smallest integer is 83.
Hence, the second smallest integer in the set is 83.