In the following question, two statements (i) and (ii) are given. Choose the valid statement. i. If (4a + 3b) : (6a +7b) = 16 : 25 then a : b = 31 : 4. ii. If (1 + x) : (3x + 1) is the duplicate ratio of 3 : 5 then x = 8.

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Only (ii) Explanation: i. (4a + 3b)/(6a +7b) = $\frac{16}{25}$ $\Rightarrow$ 25(4a + 3b) = 16(6a +7b) $\Rightarrow$ 100a + 75b = 96a + 112b $\Rightarrow$ 4a = 37b $\Rightarrow$ a/b = $\frac{37}{4}$ $\Rightarrow$ a : b = 37 : 4 Therefore, statement (i) is invalid. ii. The duplicate ratio of 3 : 5 is 33 : 52 9 : 25 Set up the equation, $\Rightarrow$ (1 + x)/(3x + 1) = $\frac{9}{25}$ $\Rightarrow$ 25(1 + x) = 9(3x + 1) $\Rightarrow$ 25 + 25x = 27x + 9 $\Rightarrow$ 25 − 9 = 27x − 25x $\Rightarrow$ 16 = 2x $\Rightarrow$ x = 8

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