carcass wrote:

If \(\frac{a}{b}=\frac{2}{3}\) which of the following is not true ?

A) \(\frac{a+b}{b}=\frac{5}{3}\)

B) \(\frac{b}{b-a}=3\)

C) \(\frac{a-b}{b}=\frac{1}{3}\)

D) \(\frac{2a}{3b}=\frac{4}{9}\)

E) \(\frac{a+3b}{a}=\frac{11}{2}\)

We are given that a/b = 2/3 or 3a = 2b.

Let’s simplify each answer choice to determine which one cannot be manipulated to yield 3a = 2b.

A) (a + b)/b = 5/3

3(a + b) = 5b

3a + 3b = 5b

3a = 2b

Since we have 3a = 2b, answer choice A is not correct.

B) b/(b - a) = 3

b = 3(b - a)

b = 3b - 3a

3a = 2b

Since we have 3a = 2b, answer choice B is not correct.

C) (a - b)/b = 1/3

3(a - b) = b

3a - 3b = b

3a = 4b

Since we have 3a = 4b, answer choice C is correct.

Alternative solution:

Let’s test each answer choice to determine which one is not true (keep in mind that a/b = 2/3).

A) (a + b)/b = 5/3

(a + b)/b = a/b + b/b = 2/3 + 1 = 5/3 → This is true.

B) b/(b - a) = 3

Notice that b/(b - a) = 3 means (b - a)/b = 1/3 when we reciprocate both sides of the equation.

(b - a)/b = b/b - a/b = 1 - 2/3 = ⅓ → This is true.

C) (a - b)/b = 1/3

(a - b)/b = a/b - b/b = 2/3 - 1 = -1/3 → So (a - b)/b = 1/3 is not true. This is the correct answer choice.

Answer: C

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