Question

Could someone please help me?

A set of five natural numbers has a mean, median and mode. The mean is 100. Two of the numbers are 19 and 96. If the median is two less than the mode, what is the value of the mode?

Answer 1

You have five natural numbers that sum to 500:

a + b + c + d + e = 500

You're told two are 19 and 96, leaving c, d and e, say:

19 + 96 + c + d + e = 500

The median, c, say, is two less than the mode:

c = d - 2 = e - 2

e and d will be the mode (i.e. d = e) since 19, 96 and c will be three distinct numbers and the mode is the score with the highest frequency (the other scores have frequency 1, and since there are only two numbers left, these two scores (i.e. frequency two) must be the mode).

So we have,

19 + 96 + c + d + e = 19 + 96 + (d-2) + d + d = 19 + 96 + 3d - 2 = 500

Hence,

3d = 387, which means

d = 129.

Since d was one of the scores that comprised the mode, the mode is 129.

Answer 2

To check, the scores are:

19, 96, 127, 129, 129

They sum to 500, 127 is indeed the median, 129 the mode.

Answer 3

A stupid question I'm sure, but how did you know the sum was 500?

Answer 4

Oh, because the mean is 100 and there are five numbers:
\[\frac{a+b+c+d+e}{5}=100 \iff a+b+c+d+e=500\]

Answer 5

AAHHHH so. thanks, you are already a fan.
Like the new icon for your profile.

Answer 6

Thank you :)

At least brick96 appreciated the help.

Answer 7

I'm sure he did, and I learned something to. Have a good day.

Answer 8

You too.