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Q.

The denominator of a rational number is greater than its numerator by 7. If the numerator is increased by 17 and the denominator decreased by 6, the new number becomes 2. Find the original number and choose the option which represents the same.

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a

\frac{15}{22}

b

\frac{16}{23}

c

\frac{17}{22}

d

\frac{18}{25}

answer is A.

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Detailed Solution

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It is given that the denominator of a rational number is greater than its numerator by 7 and when the numerator is increased by 17 and the denominator decreased by 6, the new number becomes 2.
Then, the original rational number can be expressed as

\frac{x}{x + 7}

.
Next, it is given that the numerator is increased by 17 and the denominator decreased by 6.
Using this condition, then, numerator of the new rational number becomes

x + 17

and its denominator becomes

x + 7 - 6 = x + 1

.
Hence, we can set up the new equation as follows,

\frac{x + 17}{x + 1} = \frac{2}{1}

Using the method of cross multiplication, we will multiply 1 with x+17 and 2 with x+1 and equate them as follows,

\begin{matrix} 1 (x + 17) = 2 (x + 1) \\ x + 17 = 2 (x + 1) \\ x + 17 = 2 x + 2 \end{matrix}

x + 17 - 17 = 2 x + 2 - 17

x = 2 x - 15

x - 2 x = 2 x - 15 - 2 x

- x = - 15

\frac{- x}{- 1} = \frac{- 15}{- 1}

x = 15

Now, we will substitute this value in the expression for the original rational number and simplify to get the same as follows,

\begin{matrix} \frac{x}{x + 7} = \frac{15}{15 + 7} \\ = \frac{15}{22} \end{matrix}

So, the original rational number is

\frac{15}{22}

.
Hence, the correct option is (1).

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