Expressions Using Letter-Numbers | IT

Question 23

Context: Design C appears at positions that are multiples of 3 (remainder 0). Design B appears at positions that are 1 less than a multiple of 3 (remainder 2). Design A appears at positions that are 2 less than a multiple of 3 (remainder 1).

(a) Can the remainder obtained by dividing the position number by 3 be used for this? Observe the table below.

b) Use this to find what design appears at positions 99, 122, and 148.

Solution

We can use the remainder when a position number is divided by 3 to find the design type.

Step 1 — Verifying the table

We need to check if the given table values are correct. We will divide each position number by 3. We will find the quotient and the remainder for each.

Let us take the first position number, 99. We divide 99 by 3. $99 \div 3 = 33$ The quotient is 33. The remainder is 0. This matches the table entry for position 99.

Let us take the second position number, 122. We divide 122 by 3. $122 = 3 \times 40 + 2$ The quotient is 40. The remainder is 2. This matches the table entry for position 122.

Let us take the third position number, 148. We divide 148 by 3. $148 = 3 \times 49 + 1$ The quotient is 49. The remainder is 1. This matches the table entry for position 148.

All entries in the table are correct. So, the remainder obtained by dividing the position number by 3 can be used.

Diagram 1

Step 2 — Determining the designs

We will use the remainders we found to identify the design for each position. The problem gives us rules for each design. Design C appears when the remainder is 0. Design B appears when the remainder is 2. Design A appears when the remainder is 1.

For position 99: The remainder is 0. According to the rule, Design C appears.

For position 122: The remainder is 2. According to the rule, Design B appears.

For position 148: The remainder is 1. According to the rule, Design A appears.

Answer

(a) Yes, the remainder obtained by dividing the position number by 3 can be used to determine the design. (b) (i) At position 99, Design C appears. (ii) At position 122, Design B appears. (iii) At position 148, Design A appears.

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Q14

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Q15

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Q18

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Q19

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Q22

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Q23

(a) Can the remainder obtained by dividing the position number by 3 be used for this? Observe the table below.

b) Use this to find what design appears at positions 99, 122, and 148.

Q24

Context: Let us take the marked $2 \times 2$ square, and consider the numbers lying on the diagonals; 12 and 20; 13 and 19. Find their sums; $12 + 20$ , $13 + 19$ . What do you observe? They are equal. Let us extend the numbers in the calendar beyond 30, creating endless rows.

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Q25

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Q26

Verify this expression for diagonal sums by considering any $2 \times 2$ square and taking its top left number to be 'a'.

Q27

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Q28

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Q30

How many matchsticks will there be in Step 33, Step 84, and Step 108? Of course, we can draw and count, but is there a quicker way to find the answers using the pattern present here?

Q31

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Q33

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Q34

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Q35

Numbers are written in a particular sequence in this endless 4-column grid.

(a) Give expressions to generate all the numbers in a given column (1, 2, 3, 4).

(b) In which row and column will the following numbers appear:

(i) 124

(ii) 147

(iii) 201

(d) Observe the positions of multiples of 3.

Do you see any pattern in it? List other patterns that you see.