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(C) Let the true length of the scale be $100 \, cm$ and the cost price of $100 \, cm$ of cloth be $100$ units.Since the merchant sells at a $10 \%$ lo

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$78.25$

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(C) Let the true length of the scale be $100 \, cm$ and the cost price of $100 \, cm$ of cloth be $100$ units.Since the merchant sells at a $10 \%$ loss,the selling price of $100 \, cm$ of cloth is $90$ units.Let the actual length of the inaccurate scale be $y \, cm$.Since the merchant uses a scale of length $y \, cm$ to sell the cloth,he is effectively selling $y \, cm$ of cloth for the price of $100 \, cm$ of cloth (which is $90$ units).The cost price of $y \, cm$ of cloth is $y$ units.Given that he gains $15 \%$,the selling price is $1.15 	imes y$ units.Equating the selling prices: $1.15 	imes y = 90$.$y = \frac{90}{1.15} \approx 78.26 \, cm$.Using the standard formula $\frac{100+g}{100-l} = \frac{	ext{True length}}{	ext{False length}}$:$\frac{100+15}{100-10} = \frac{100}{y} \Rightarrow \frac{115}{90} = \frac{100}{y}$.$y = \frac{100 	imes 90}{115} = \frac{9000}{115} \approx 78.26 \, cm$.

$A$ cloth merchant says that due to a slump in the market,he sells cloth at a $10 \%$ loss,but he uses an inaccurate metre scale and actually gains $15 \%$. Find out the actual length of the scale (in $cm$).

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