Let the mean and variance of four numbers 3, 7, $$x$$ and $$y$$ ($$x > y$$) be 5 and 10 respectively. Then the mean of four numbers 3 + 2x, 7 + 2y, $$x + y$$ and $$x - y$$ is _________

We have four original numbers $$3,\,7,\,x,\,y$$ whose mean is given to be $$5$$. The definition of mean says

$$\bar{x}= \frac{3+7+x+y}{4}=5.$$

Multiplying by $$4$$ we get

$$3+7+x+y = 20.$$

Simplifying,

$$x+y = 10.$$

Now the variance of these four numbers is given as $$10$$. For a population of four numbers, variance is defined by the formula

$$\sigma^{2}= \frac{1}{4}\left[(3-\bar{x})^{2}+(7-\bar{x})^{2}+(x-\bar{x})^{2}+(y-\bar{x})^{2}\right].$$

Here $$\bar{x}=5$$, so we substitute:

$$10 = \frac{1}{4}\Big[(3-5)^{2} + (7-5)^{2} + (x-5)^{2} + (y-5)^{2}\Big].$$

Calculating the first two squares,

$$(3-5)^{2}=(-2)^{2}=4,\qquad (7-5)^{2}=2^{2}=4.$$

Hence

$$10 = \frac{1}{4}\Big[4 + 4 + (x-5)^{2} + (y-5)^{2}\Big].$$

Multiplying both sides by $$4$$,

$$40 = 8 + (x-5)^{2} + (y-5)^{2}.$$

So

$$(x-5)^{2} + (y-5)^{2} = 32.$$

From the mean calculation we already have $$y = 10 - x$$. Substituting this into the variance equation,

$$(x-5)^{2} + \big[(10 - x) - 5\big]^{2} = 32.$$

The second square simplifies as

$$(10 - x) - 5 = 5 - x,$$

and $$\big(5 - x\big)^{2} = (x - 5)^{2}$$ (since squaring removes the sign). Therefore

$$(x-5)^{2} + (x-5)^{2} = 32,$$

or

$$2(x-5)^{2} = 32.$$

Dividing by $$2$$,

$$(x-5)^{2} = 16.$$

Taking square roots,

$$x-5 = \pm 4 \;\;\Longrightarrow\;\; x = 5 \pm 4.$$

Thus $$x = 9$$ or $$x = 1$$. We are told $$x > y$$, and recall $$x + y = 10$$, so

• If $$x = 9$$, then $$y = 1$$, giving $$x > y$$ (acceptable).
• If $$x = 1$$, then $$y = 9$$, which violates $$x > y$$.

Hence the only valid pair is $$x = 9,\,y = 1$$.

Now we form the new set of numbers: $$3 + 2x,\; 7 + 2y,\; x + y,\; x - y.$$

Substituting $$x = 9,\,y = 1$$:

$$3 + 2x = 3 + 2\cdot9 = 21,$$ $$7 + 2y = 7 + 2\cdot1 = 9,$$ $$x + y = 9 + 1 = 10,$$ $$x - y = 9 - 1 = 8.$$

The mean of these four numbers is

$$\text{Mean} = \frac{21 + 9 + 10 + 8}{4} = \frac{48}{4} = 12.$$

So, the answer is $$12$$.