# Properties Questions on Arithmetic Mean

Here using the properties we will solve the different properties questions on arithmetic mean. Follow the steps to understand the explanations on how to solve the problems based on properties.

Worked-out examples on properties questions on arithmetic mean:

1. The mean of eight numbers is 25. If five is subtracted from each number, what will be the new mean?

Solution:

Let the given numbers be x1, x2, . . ., x8.

Then, the mean of these numbers = (x1 + x2 + ...+ x8)/8.

Therefore, (x1 + x2+...+x8)/8 = 25

⇒ (x1 + x2 + ... + x8) = 200 ……. (A)

The new numbers are (x1 - 5), (x2 - 5), …… ,(x8 - 5)

Mean of the new numbers = {(x1 - 5) + (x1 - 5) + …… + (x8 - 5)}/8

= [(x1 + x2 + ... + x8) - 40]/8

= (200 - 40)/8, [using (A)]

= 160/8

= 20

Hence, the new mean is 20.

2. The mean of 14 numbers is 6. If 3 is added to every number, what will be the new mean?

Solution:

Let the given numbers be x1, x2, x3, ….. x14.

Then, the mean of these numbers = x1 + x2 + x3+ ….. x14/14

Therefore, (x1 + x2 + x3 + ….. x14)/14 = 6

⇒ (x1 + x2 + x3 + ….. x14) = 84 ………………. (A)

The new numbers are (x1 + 3), (x2 + 3), (x3 + 3), …. ,(x14 + 3)

Mean of the new numbers

= (x1 + 3), (x2 + 3), (x3 + 3), …. ,(x14 + 3)/14

= (x1 + x2 + x3 + ….. x14) + 42

= (84 + 42)/14, [Using (A)]

= 126/14

= 9

Hence, the new mean is 9.

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