The average height of $35$ girls in a class was calculated as $160$ cm. It was later found that the height of one of the girls in the class was wrongly written as $144$ cm, whereas her actual height was $104$ cm. What is the actual average height of the girls in the class? (rounded off to 2 digits after decimal)

Concept & Strategy When an observation is recorded incorrectly, we do not need to recalculate the entire sum from scratch. We can simply calculate the net error (Actual Value - Wrong Value) and distribute this error evenly across all observations to find the change in the average. $$ \text{New Average} = \text{Old Average} + \frac{\text{Actual Value} - \text{Wrong Value}}{\text{Total Number of Observations}} $$ Step-by-Step Solution * Given: Total girls ($n$) = $35$. Initial incorrect average = $160$ cm. * Wrong observation recorded = $144$ cm. * Actual correct observation = $104$ cm. * Calculate the net error in the total sum: $104 - 144 = -40$ cm. (The total sum was mistakenly $40$ cm too high). * Calculate how this error affects the average: Change in average = $-40 / 35$. * Simplify the fraction: $-40 / 35 = -8 / 7 \approx -1.1428$ cm. * Calculate the actual corrected average: $160 - 1.1428 = 158.8572$ cm. * Rounding to two decimal places, we get $158.86$ cm. * Comparing this to the given options, $158.86$ cm is not present in options (a) through (d). Exam Strategy & Shortcut Use the Deviation Method. The difference is $-40$. Divide this quickly by $35$: $40/35 = 8/7$. Knowing your basic fractions, $1/7 \approx 0.1428$. So $8/7 = 1 \frac{1}{7} \approx 1.14$. Subtract this from $160$: $160 - 1.14 = 158.86$. None of the numerical options perfectly match $158.86$ cm, so you can confidently mark "None of these". Common Pitfall A major pitfall is carelessly subtracting the wrong way: $144 - 104 = +40$. If you mistakenly think the sum needs to *increase* by $40$, you will add the deviation ($160 + 1.14$) instead of subtracting it, leading to a completely wrong baseline. Always use the formula: (True Value - False Value). Final Answer **Therefore, the correct answer is None of these.**