Question

If the mean, median, and mode are all equal for the set (10, 80, 70, 120, x}, find the value of x.

X
(Simplify your answer. Type an integer or a decimal.)
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Answer 1

The value of x that makes the mean, median, and mode of the set equal for the set (10, 80, 70, 120, x) is 70.

Step-by-step explanation:

To find the value of x in the set (10, 80, 70, 120, x) where the mean, median, and mode are all equal, we need to arrange the numbers in ascending order and solve for x using the criteria that all three measures are the same.

Step 1: Arrange the data

Ascending order without x: (10, 70, 80, 120)

Step 2: Calculate the median

The median is the middle number, so with x included, it would be the third number in the ordered list. To make the median equal to the mean, x needs to be placed between 70 and 80.

Step 3: Solve for x

If the mean is equal to the median and there is no mode since all numbers would be different, the sum of all numbers must be divisible by 5. Since we know the median is between 70 and 80, x must be 70 or 80 for it to be equal to the mean and median. In this case, x must be 70 to make the set symmetric around the median.

Now, let's calculate the sum needed for the mean: (10 + 70 + 70 + 80 + 120)/5 = 350/5 = 70. So the mean is also 70.

Therefore, the value of x is 70.

Answer 2

x=70

Step-by-step explanation:

First, we know that the mode is the number that is the most common. As each value in the set so far only has one of each number, we know that x must be one of the current numbers, making that the mode.

Next, because x is the mode and has to be the median as well, and our number line so far is

(10, 70, 80, 120), x must be either 70 or 80 to make it the median. This is because if x is 10 or 120, we would end up with (10, 10, 70, 80, 120) with 70 as the median or (10, 70, 80, 120, 120) with 80 as the median.

Finally, to calculate the mean, we have

mean = sum / count

The mean must be x, as it is equal to the mode, so we have

x = (10+70+80+120 + x)/5 (as there are 5 numbers including x)

multiply both sides by 5 to remove the denominator

5 * x = 10+70+80+120+x

5 * x = 280 + x

subtract x from both sides to isolate the x and the coefficient

4 * x = 280

divide both sides by 4 to get x

x= 70

We see that x is 70 or 80 and is one of the current numbers, checking off all boxes.