Question

If a, b, c are in continued proportion and if a(b-c)=2b then (2(a+b))/a is equal to___.
Answer with full steps.​

Answer 1

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Answer 2

SOLUTION:

Given That: a, b, c are in continued proportion.

This implies that:

$\rm \longrightarrow \dfrac{a}{b} = \dfrac{b}{c}$

$\rm \longrightarrow b \cdot b = a \cdot c$

$\rm \longrightarrow {b}^{2} = ac$

Now, we have:

$\rm \longrightarrow a \cdot(b - c) = 2 \cdot b$

$\rm \longrightarrow ab - ac = 2b$

$\rm \longrightarrow ab - {b}^{2} = 2b \: \: \: \: \: \: \: \: ( {b}^{2} = ac)$

$\rm \longrightarrow b(a -b) = 2b$

$\rm \longrightarrow a -b = 2$

Now, we have:

$\rm = \dfrac{2(a + b)}{a}$

Can be written as:

$\rm = \dfrac{(a - b)(a + b)}{a} \: \: \: \: \: (a - b = 2)$

$\rm = \dfrac{ {a}^{2} - {b}^{2} }{a}$

$\rm = \dfrac{ {a}^{2} -ac}{a} \: \: \: \: \: \: ( {b}^{2} = ac)$

$\rm = \dfrac{a(a -c)}{a}$

$\rm =a - c$

Therefore:

$\rm \longrightarrow \dfrac{2(a + b)}{a} = a - c$

★ Which is our required answer.

LEARN MORE:

If a : b and c : d are two ratios such that a : b : : c : d. Then the following results hold true.

1. Invertendo.

$\rm\longrightarrow b : a : : d : c$

2. Alternendo.

$\rm\longrightarrow a : c : : b : d$

3. Componendo.

$\rm\longrightarrow (a + b) : b: : (c + d) : d$

4. Dividendo.

$\rm\longrightarrow (a - b) : b: : (c - d) : d$

5. Componendo and dividendo.

$\rm\longrightarrow (a + b) : (a - b): : (c + d) :(c - d)$

6. Convertendo.

$\rm\longrightarrow a : (a - b): :c:(c - d)$