Fill in the blanks such that the median of the collection is : . How many possibilities exist if only counting numbers are allowed?

Data Handling (Mean and Median) | FIO

Question 17

Fill in the blanks such that the median of the collection is $13$ : $5, 21, 14, \underline{\quad}, \underline{\quad}, \underline{\quad}$ . How many possibilities exist if only counting numbers are allowed?

Solution

To find the median of a collection of numbers, we first arrange them in increasing order.

Step 1 — Understand the median for an even number of items

We have a collection of 6 numbers: $5, 21, 14$ , and three blank numbers. Since there are 6 numbers (an even count), the median is the average of the two middle numbers. These middle numbers are the (6/2)th term and the (6/2 + 1)th term. So, the median is the average of the 3rd and 4th terms when the numbers are arranged in increasing order. We are told the median is 13. Let the 3rd term be $n_3$ and the 4th term be $n_4$ .

$\frac{n_3 + n_4}{2} = 13$

To find the sum of these two middle terms, we multiply both sides by 2.

$n_3 + n_4 = 13 \times 2$

$\boxed{n_3 + n_4 = 26}$

Step 2 — Identify the middle terms

We know the sum of the 3rd and 4th terms is 26. Also, the 3rd term must be less than or equal to the 4th term ( $n_3 \le n_4$ ). The given numbers are $5, 14, 21$ . Let us think about where 14 fits in the sorted list. If $n_3$ were 14, then $n_4$ would be $26 - 14 = 12$ . But this is not possible because $n_3$ cannot be greater than $n_4$ . So, 14 must be the 4th term ( $n_4$ ) or a later term. If $n_4 = 14$ , then we can find $n_3$ .

$n_3 + 14 = 26$

$n_3 = 26 - 14$

$\boxed{n_3 = 12}$

So, the two middle terms in the sorted collection are 12 and 14.

Step 3 — Fill in the blanks for one possibility

The sorted collection of 6 numbers looks like this: $n_1, n_2, \mathbf{12}, \mathbf{14}, n_5, n_6$ . The original numbers given are $5, 14, 21$ . We have already placed 14 as $n_4$ . Since 12 is not one of the given numbers ( $5, 14, 21$ ), one of the three blank numbers must be 12. Let the three blank numbers be $a, b, c$ . We can say $a=12$ . Now the collection is $\{5, 14, 21, 12, b, c\}$ . The numbers $5, 21, b, c$ must fill the remaining positions $n_1, n_2, n_5, n_6$ . We need to make sure that $n_1 \le n_2 \le 12$ and $14 \le n_5 \le n_6$ . The number 5 must be $n_1$ or $n_2$ (since $5 \le 12$ ). The number 21 must be $n_5$ or $n_6$ (since $21 \ge 14$ ).

Let's choose one way to fill the blanks: Let $n_1 = 5$ . Let $n_2$ be one of the blank numbers, say $b$ . We need $5 \le b \le 12$ . Let's pick $b=8$ . Let $n_5$ be 21. Let $n_6$ be the last blank number, say $c$ . We need $c \ge 21$ . Let's pick $c=28$ . So, the three blank numbers are 8, 12, 28. The complete collection is $5, 21, 14, 8, 12, 28$ . When sorted, this becomes: $5, 8, 12, 14, 21, 28$ . The median is $(12+14)/2 = 13$ . This works!

Step 4 — Count the number of possibilities

We found that one of the blank numbers must be 12. Let the other two blank numbers be $x$ and $y$ . The sorted list is $n_1, n_2, 12, 14, n_5, n_6$ . The numbers $5, 21, x, y$ must fill the positions $n_1, n_2, n_5, n_6$ . We must maintain the order: $n_1 \le n_2 \le 12$ and $14 \le n_5 \le n_6$ . Let's consider a case where $n_1=5$ , $n_2=x$ , $n_5=21$ , and $n_6=y$ . For this to be true:

$x$ must be a counting number such that $5 \le x \le 12$ . There are 8 choices for $x$ (5, 6, 7, 8, 9, 10, 11, 12).
$y$ must be a counting number such that $y \ge 21$ . This means $y$ can be $21, 22, 23, 24, \dots$ . Since there are infinitely many counting numbers greater than or equal to 21, there are infinitely many choices for $y$ . Because there are infinite choices for $y$ , there are an infinite number of ways to fill the blanks.

Answer

(i) One possible set of numbers to fill the blanks is 8, 12, 28. (ii) If only counting numbers are allowed, infinite possibilities exist.

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Q10

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(i) Find the average number of times students rode their cycles. (ii) Find the median number of times students rode their cycles. (iii) Which of the following statements are valid? Why? (a) Everyone used their cycle at least once. (b) Almost everyone used their cycle a few times. (c) There are some students who cycled more than once on some days. (d) Exactly 5 students have used their cycles more than once on some days. (e) The following week, if all of them cycled 1 more time than they did the previous week, what would be the average and median of the next week's data?

Q11

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Q12

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Q13

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(iii) Based on the line for New Delhi in the graph fill the data in the table.

(iv) Which city among these receives the most number of days of rainfall per year? Which city gets the least number of days of rainfall per year?

(v) Looking at the table, when is the rainy season in New Delhi and Rameswaram?

Q14

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(v) Estimate the number of births in the year 2019.

Q15

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Q16

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Q17

Q18

Fill in the blanks such that the mean of the collection is $6.5$ : $3$ , $11$ , $\underline{\quad}$ , $\underline{\quad}$ , $15$ , $6$ . How many possibilities exist if only counting numbers are allowed?

Q19

Check whether each of the statements below is true. Justify your reasoning. Use algebra, if necessary, to justify.

(i) The average of two even numbers is even.

(ii) The average of any two multiples of 5 will be a multiple of 5.

(iii) The average of any 5 multiples of 5 will also be a multiple of 5.

Q20

There were 2 new admissions to Sudhakar’s class just a couple of days after the class average height was found to be 150.2 cm.

(i) Which of the following statements are correct? Why? (a) The average height of the class will increase as there are 2 new values. (b) The average height of the class will remain the same. (c) The heights of the new students have to be measured to find out the new average height. (d) The heights of everyone in the class has to be measured again to calculate the new average height.

(ii) The heights of the two new joinees are 149 cm and 152 cm. Which of the following statements about the class’ average height are correct? Why? (a) The average will remain the same. (b) The average will increase. (c) The average will decrease. (d) The information is not sufficient to make a claim about the average.

(iii) Which of the following statements about the new class average height are correct? Why? (a) The median will remain the same. (b) The median will increase. (c) The median will decrease. (d) The information is not sufficient to make a claim about median.

Q21

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Q22

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Q23

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Q24

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Answer the following questions based on the graphs:

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(ii) Which place has the longest day length over the year?

(iii) Share your observations—what do you find interesting? What are you curious to find out?