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