The twelve numbers 3,6,9,12,4,6,8,10,12,14,x,y have a mode of 6 and a mean of 8.

Find the values of x and y and the median of the above twelve numbers.

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The twelve numbers 3,6,9,12,4,6,8,10,12,14,x,y have a

3,4,6,6,8,9,x,y,10,12,12,14

It is given that mode is 6. Therefore either x=6 or y=6.

Mode=3 x median-2 x mean

6=3 x median - 2 x 8

3 median= 6+16

median=22/3

Mean of these twelve numbers is 8. Therefore

8=(3+6+9+12+4+6+8+10+12+14+x+y)/12

96=84+x+y

x+y=12 (i)

Then from (i) if x =6 then y =6.

Thus median=22/3 and x=y=6.

17/3+y=12

y=19/3

The twelve numbers are 3,6,9,12,4,6,8,10,12,14,x,y .

It is given that mode is 6. Therefore either x=6 or y=6.

Mode=3 * median-2 * mean

6=3 * median - 2 * 8

3 median= 6+16

median=22/3

Mean of these twelve numbers is 8. Therefore

8=(3+6+9+12+4+6+8+10+12+14+x+y)/12

96=84+x+y

x+y=12 (i)

Then from (i) if x =6 then y =6.

Thus median=22/3 and x=y=6.

The number 3,6,9,12,4,6,8,10,12,14 have a sum of 84. If 12 numbers have an mean if 8, they have to have a sum of 96. So, the final 2 numbers have to total 12. Since the mode has to be 6 with these numbers, and right now the numbers are bi-modal with 6 and 12, then one of these numbers has to be 6. Given that and their sum as to be 12, **both numbers have to be 6**. So, we would have:

3,6,6, 6, 9,12,4,6,8,10,12,14

For the 12 numbers.

For the median, we are looking for the middle of the 12 numbers. First, we have to put them in numerical order:

3, 4, 6, 6, 6, 6, 8, 9, 10, 12, 12, 14

The median is the middle value of these numbers. However, if you consider, there isn't one single "middle value". 6 and 8 are the middle values. In this case, we take the average of the middle values:

(6+8)/2 = 7

**So, 7 is the median value for these numbers.**

Since the mode is 6, 6 needs to be the number that appears the most. In the string of numbers that you gave, both 6 and 12 appear twice, so in order for the mode to be 6, either x or y need to be 6.

When you add the ten numbers you provided, plus the two variables, then the sum is 84+6+y=90+y. For the twelve numbers to have a mean of 8, the total must be 96 (8x12). Therefore:

90+y=96

y=96-90=6

Both variables are 6!

To find the median, arrange the numbers in numerical order:

3,4,6,6,6,6,8,9,10,12,12,14

Since there are twelves numbers, the median would be the average of the middle two numbers, which are 6 and 8.

(6+8)/2=7

**In conclusion, x and y are both 6 and the median is 7.**

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