The average height of 50 pupils in a class is $ 150\text{ cm}$... | Mathematics MCQ | Examoogle

MathematicsPrevious Year Paper 2024

The average height of 50 pupils in a class is $150\text{ cm}$ . Five of them whose average height is $146\text{ cm}$ leave the class, and five others whose average height is $156\text{ cm}$ join. The new average height of the pupils of the class (in cm) is _________.

152

153

151

149

Correct Answer

📐 MA • Math Direct FormulaMathematicsPrevious Year Paper 2024

Option C

151

Quick Summary:

Since the number of leaving and joining pupils is the same (5), we can directly find the net change in total height: $(156 - 146) \times 5 = +50$ cm. Divide this by the total 50 pupils to get a $+1$ cm change in the average, making the new average $150 + 1 = 151$ cm.

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📐MAMath SolutionDirect Formula

📋 Given

Total number of pupils = 50, Initial average height = 150 cm, 5 pupils with average height 146 cm leave, 5 other pupils with average height 156 cm join.

🔢 Formula Used

New Average=Initial Average−(Avg of leaving−Avg of joining)×nTotal pupils\text{New Average} = \text{Initial Average} - \frac{(\text{Avg of leaving} - \text{Avg of joining}) \times n}{\text{Total pupils}}New Average=Initial Average−Total pupils(Avg of leaving−Avg of joining)×n​

⚡ Exam Hall Shortcut / Speed Trick

⚠️ Common Student Trap / Pitfall

Recalculating the total sum from scratch by multiplying all 50 pupils by 150 is time-consuming and prone to arithmetic errors in the exam hall.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Calculate Initial Total Height

Multiplying the initial number of pupils by the given average height gives the total sum of heights.

Initial Sum=50×150=7500 cm\text{Initial Sum} = 50 \times 150 = 7500 \text{ cm}Initial Sum=50×150=7500 cm

Calculate Net Change in Height

Five pupils of average height 146 cm leave (loss of $5 \times 146 = 730$ cm) and five pupils of average height 156 cm join (gain of $5 \times 156 = 780$ cm).

Net Change=780−730=+50 cm\text{Net Change} = 780 - 730 = +50 \text{ cm}Net Change=780−730=+50 cm

Find New Total and Average Height

Adding the net change to the initial sum and dividing by the total number of pupils (which remains 50):

New Average=7500+5050=755050=151 cm\text{New Average} = \frac{7500 + 50}{50} = \frac{7550}{50} = 151 \text{ cm}New Average=507500+50​=507550​=151 cm

✅

C is correct because the new average height of the class after the replacement of 5 pupils is calculated to be 151 cm.

Core Concepts Used

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Arithmetic Mean Sum of Observations Weighted Replacements

💡 EXAM TIP

This concept of net deviation is extremely useful in weighted average problems across mixtures, alligations, and profit-loss quantities.

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