The mean of four numbers is 5 and the mean deviation is 3.Find the fourth number if the mean deviation of the first three numbers is 2?
The mean of four numbers is 5 and the mean deviation is 3. Find the fourth number if the mean deviation of the first 3 number is 2
15
To solve this problem, we need to first understand what mean and mean deviation are.
1. Mean: The mean of a set of numbers is the sum of all the numbers divided by the total count of numbers.
2. Mean Deviation: The mean deviation of a set of numbers measures how spread out the numbers are from the mean. It is the average of the absolute differences between each number and the mean.
Now, let's solve the problem step by step:
Step 1: Given that the mean of four numbers is 5 and the mean deviation is 3. We can assign variables to represent the four numbers. Let's call them a, b, c, and d.
Step 2: The mean value of the four numbers is 5. Since the mean is calculated by summing all the numbers and dividing by their count, we can form the equation:
(a+b+c+d)/4 = 5
Step 3: The mean deviation of the first three numbers is given as 2. This means the absolute differences between each of the first three numbers and the mean of the first three numbers, (a+b+c)/3, average to 2. We can form the equation:
(|a - (a+b+c)/3| + |b - (a+b+c)/3| + |c - (a+b+c)/3|)/3 = 2
Step 4: To simplify the second equation, we can substitute the value of the mean of the three numbers, (a+b+c)/3, with 2. This gives us:
(|a - 2| + |b - 2| + |c - 2|)/3 = 2
Step 5: Now we have a system of equations with two unknowns, a, b, c, and d. We can solve this system by substitution or elimination.
Step 6: Since we need to find the fourth number, let's solve for d. First, simplify the first equation (from step 2) by multiplying both sides by 4:
a + b + c + d = 20
Step 7: In the second equation (from step 4), we can do a bit of rearranging by multiplying both sides by 3:
|a - 2| + |b - 2| + |c - 2| = 6
Step 8: To solve for d, we can isolate it by subtracting the sum of the other variables from both sides of the equation in step 6:
d = (20 - a - b - c)
Step 9: Substitute this value of d into the first equation:
a + b + c + (20 - a - b - c) = 20
Step 10: Simplify and combine like terms:
a + b + c - a - b - c + 20 = 20
Step 11: Simplify further:
0 + 20 = 20
Step 12: Therefore, the fourth number, d, is 0.
So, the fourth number is 0.