Example 12 - Chapter 4 Class 7 - Expressions using Letter-Numbers (Ganita Prakash)

Transcript

Example 12 Somjit noticed a repeating pattern along the border of a saree. Somjit wonders if there is a way to describe all the positions where the (i) Design A occurs, (ii) Design B occurs, and (iii) Design C occurs.Notice that Design D is the same as Design A. Design E is the same as Design B. Design F is the same as Design C. This means the basic pattern that repeats over and over is just A, B, C. This is a repeating cycle of 3 designs. Finding the Formula for Each Design's Position The problem asks for a formula to find the position of the nth time a specific design appears. (For example, where will the 10th "A" design be?) Design C Design C appears at the end of each 3-design cycle. It's at positions 3, 6, 9, and so on. These are all multiples of 3. Answer: The formula for the position of the nth occurrence of Design C is 3n. Design B Design B appears one spot before Design C. Its positions are 2, 5, 8, 11, and so on. Each of these positions is one less than the corresponding position of Design C (e.g., 3−1=2, 6−1=5). Answer: The formula for the position of the nth occurrence of Design B is 3n − 1. Design A Design A is the first in the cycle, so it is two spots before Design C. Its positions are 1, 4, 7, and so on. Each position is two less than the corresponding position of Design C (e.g., 3−2=1, 6−2=4). Answer: The formula for the position of the nth occurrence of Design A is 3n − 2. Finding the Design at a Specific Position To find the design at any given position, we can use the remainder when we divide the position number by the cycle length, which is 3. If the remainder is 1, it's the first design in the cycle: Design A. If the remainder is 2, it's the second design in the cycle: Design B. If the remainder is 0, it means the number is a multiple of 3, so it's the last design in the cycle: Design C. Using the table provided in the image: Which design appears at position 99? 99÷3=33 with a remainder of 0. The design is C. Which design appears at position 122? 122÷3=40 with a remainder of 2. The design is B. Which design appears at position 148? 148÷3=49 with a remainder of 1. The design is A. Using the table provided in the image: Which design appears at position 99? 99/3 = 33 with a remainder of 0. ∴ The design is C. Which design appears at position 122? 122/3=40 2/3 with a remainder of 2. ∴ The design is B. Which design appears at position 148? 148/3=49 1/3 with a remainder of 1. ∴ The design is A.