Physics and Maths Tutoring from Michael A Smith.

Preamble

Calculating the mean of a set of numbers involves adding up the numbers and dividing by how many there are, but can we calculate the mean if we don't know the numbers?

Warm-up:

Calculating the mean

The Problem:

$\frac{1}{x} + \frac{1}{y} = 10$

and

$x y = 10$

What is the mean of $x$ and $y$ ?

-/|\-

Solution:

The mean

$\overline{x} = \frac{x + y}{2}$

Multiplying both parts of the quotient by $x y$ gives

$\overline{x} = \frac{(x + y) x y}{2 x y} = \frac{1}{2} (\frac{1}{x} + \frac{1}{y}) x y$

Substituting the values given we get

$\overline{x} = \frac{10 \times 10}{2} = 50$

-oOo-

Maths Problem of the Week
the Mean of Two Unknowns

Preamble

Warm-up:

The Problem:

$\frac{1}{x} + \frac{1}{y} = 10$

$x y = 10$

Solution:

$\overline{x} = \frac{x + y}{2}$

$\overline{x} = \frac{(x + y) x y}{2 x y} = \frac{1}{2} (\frac{1}{x} + \frac{1}{y}) x y$

$\overline{x} = \frac{10 \times 10}{2} = 50$

Further Reading:

Preamble

Warm-up:

The Problem:

1x+1y=10

x⁢y=10

Solution:

x_=x+y2

x_ = (x+y)⁢x⁢y 2⁢x⁢y =12 ⁢(1x +1y) ⁢x⁢y

x_ =10×102 =50

Further Reading:

$\frac{1}{x} + \frac{1}{y} = 10$

$x y = 10$

$\overline{x} = \frac{x + y}{2}$

$\overline{x} = \frac{(x + y) x y}{2 x y} = \frac{1}{2} (\frac{1}{x} + \frac{1}{y}) x y$

$\overline{x} = \frac{10 \times 10}{2} = 50$