Question

If a + c = mb and `1/b + 1/d = m/c`, prove that a, b, c and d are in proportion.

Answer 1

a + c = mb and `1/b + 1/d = m/c`

a + c = mb   ...(1)

`1/b + 1/d = m/c`   ...(2)

Step 1: Simplify the second condition

`1/b + 1/d = m/c`

Take LCM of b and d:

`(d + b)/(bd) = m/c`

c(d + b) = mbd

cd + cb = mbd   ...(3)

Step 2: Use the first condition

a + c = mb

Multiply both sides by d:

d(a + c) = mbd

ad + cd = mbd   ...(4)

Step 3: Compare equations (3) and (4)

cd + cb = mbd

ad + cd = mbd

ad + cd = cd + cb

Subtract cdcdcd from both sides:

ad = cb

Step 4: Convert to ratio form

ad = bc

Divide both sides by bd:

`a/b = c/d`

Thus,

a : b = c : d

Hence, a, b, c and d are proportional.