Show that the rational number a+b/2 lies between the rational numbers a and b. Class 9

Show that the rational number a+b/2 lies between the rational numbers a and b. Class 9

Question 1.

Show that the rational number $\left(\frac{a+b}{2}\right)$ lies between the rational numbers a and b. Class 9

Solution:

Case 1 : a < b

Adding a on both sides, we get

a + a < a + b

∴ 2a < a + b

Dividing both sides by 2, we get

a < $\frac{a+b}{2}$ ........(i)

Again, from a < b

Adding b on both sides, we get

a + b < b + b

∴ a + b < 2b

Dividing both sides by 2, we get

$\frac{a+b}{2}$ < b ...(ii)

∴ a < $\frac{a+b}{2}$ < b ....[From (i) and (ii)]

Hence, $\frac{a+b}{2}$ lies between a and b.

Case 2: b < a

Adding b on both sides, we get

b + b < a + b

∴ 2b < a + b

Dividing both sides by 2, we get

b < $\frac{a+b}{2}$ ...(iii)

Again, b < a

Adding a on both sides, we get

a + b < a + a

∴ a + b < 2a

Dividing both sides by 2, we get

$\frac{a+b}{2}$ < a ...(iv)

∴ b < $\frac{a+b}{2}$ < a ...[From (iii) and (iv)]

∴ The rational number $\frac{a+b}{2}$ lies between the rational numbers a and b.