A closed and an open organ pipe have same lengths. If the ratio of frequencies of their seventh overtones is $\left(\frac{a-1}{a}\right)$ then the value of $a$ is _________.

<p><b>Step 1: Write the formulas for the frequencies of open and closed organ pipes.</b></p> <p>The frequency of an open organ pipe is $f_o = \frac{v}{2l}$, where <i>v</i> is the speed of sound and <i>l</i> is the length of the pipe.</p> <p>The frequency of a closed organ pipe is $f_c = \frac{v}{4l}$.</p> <p><b>Step 2: Find the frequency of the seventh overtone for each pipe.</b></p> <p>The seventh overtone in an open pipe is the 8th harmonic (since the first overtone is the 2nd harmonic). So, its frequency is: $f_{o_7} = 8 \times \frac{v}{2l}$.</p> <p>In a closed pipe, only odd harmonics are present. The seventh overtone is the 15th harmonic, so: $f_{c_7} = 15 \times \frac{v}{4l}$.</p> <p><b>Step 3: Find the ratio of the seventh overtone frequencies.</b></p> <p>The ratio of the frequencies is: $\frac{f_{c_7}}{f_{o_7}} = \frac{15 \frac{v}{4l}}{8 \frac{v}{2l}}$.</p> <p>Simplify the ratio: $$\frac{15}{4l} \div \frac{8}{2l} = \frac{15}{4l} \times \frac{2l}{8} = \frac{30}{32} = \frac{15}{16}$$.</p> <p><b>Step 4: Connect the answer to the value of <i>a</i></b></p> <p>The given ratio is $\frac{a-1}{a}$, so: $\frac{a-1}{a} = \frac{15}{16}$.</p> <p>Solve for $a$: $a = 16$.</p>