Question 18/20: Set 2 - Try Out - Mathematical Reasoning (SNBT)

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Number 18

A written selection test is taken by $60$ participants. The score obtainable by participants ranges from $0$ to $100$ . It is known that only $21$ participants received a score greater than or equal to $80$ . The smallest possible average score for the $60$ participants is...

$70$

$36$

$34$

$28$

$25$

Explanation

Strategy to Minimize Average

To find the smallest possible average score, we must make every participant's score as low as possible while strictly adhering to the constraints given in the problem.

We have a total of $60$ participants divided into two groups based on the information provided:

First group: $21$ participants with scores $\geq 80$ .
Second group: The remaining $60 - 21 = 39$ participants with scores $< 80$ .

Determining Minimum Scores for Each Group

To minimize the total sum (and consequently the average), we assign the lowest possible scores to each group:

For the 21 participants (scores $\geq 80$ ): The smallest value satisfying this condition is exactly $80$ . We assume all $21$ people scored exactly $80$ .
For the other 39 participants (scores $< 80$ ): Since the score range is $0$ to $100$ , the lowest possible score they can get is $0$ . We assume all $39$ people scored $0$ .

Calculating the Smallest Average

Now we calculate the average based on this minimum score scenario:

\bar{x}_{\text{min}} = \frac{(21 \times 80) + (39 \times 0)}{60}

\bar{x}_{\text{min}} = \frac{1,680 + 0}{60}

\bar{x}_{\text{min}} = \frac{1,680}{60}

\bar{x}_{\text{min}} = 28

Thus, the smallest possible average score for the 60 participants is $28$ .

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Number 18

Explanation