Question 15 - Figure it out - Page 102 to 105 - Chapter 4 Class 7 - Expressions using Letter-Numbers (Ganita Prakash)

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Question 15 (a) 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).Let’s look at the pattern Column 1: Contains numbers 1, 5, 9, ... The expression is 4n + 1 Column 2: Contains numbers 2, 6, 10, ... The expression is 4n + 2 Column 3: Contains numbers 3, 7, 11, ... The expression is 4n + 3 Column 4: Contains numbers 4, 8, 12, ... The expression is 4n + 4 Question 15 (b) (b) In which row and column will the following numbers appear: (i) 124 (ii) 147 (iii) 201 To find the position of any number 'N': Column: The column is determined by Remainder when N is divided by 4 (if the remainder is 0, it's in Column 4). Row: The row is determined by Quotient + 1 when N is divided by 4 and remainder is not 0 Quotient when N is divided by 4 and remainder is 0 Let’s take examples from figure For number 6 6/4=1 2/4 Remainder = 2 Quotient = 1 Thus, Column = Remainder = 2 Row = Quotient + 1 = 1 + 1 = 2 For number 12 12/4=3 Since remainder is 0, Column = 4 Row = Quotient = 31 For number 147 147/4=36 3/4 Remainder = 3 Quotient = 36 Thus, Column = Remainder = 3 Row = Quotient + 1 = 36 + 1 = 37 Remainder = 0 Quotient = 3 Since remainder is 0, Column = 4 Row = Quotient = 3 Now, let’s look at our numbers For number 124 124/4=31 Remainder = 0 Quotient = 31 For number 201 201/4=50 1/4 Remainder = 1 Quotient = 50 Thus, Column = Remainder = 1 Row = Quotient + 1 = 50 + 1 = 51 Question 15 (c) (c) What number appears in row r and column c? Okay, so we have two variables here Row r Column c Let’s look at our current numbers and find a pattern For r = 1, c = 1: Number = 1 For r = 1, c = 2 : Number = 2 So number increases by 1 if c is increased by 1 For r = 2, c = 1, Number = 5, So number increases by 4 if r is increased by 1 Thus, We multiply 4 by r, i.e. 4r And, we multiply 1 by c, i.e. c Now, let’s try to form the expression using hit-and-trial 4r + c doesn’t work for Row 1 4r – 4 + c works for Row 1 Checking for Row 2, it works for r = 2, c = 3 Checking for Row 3, and 4 – it also works Therefore, Number = 4r – 4 + c Question 15 (d) (d) Observe the positions of multiples of 3. Do you see any pattern in it? List other patterns that you see.Marking multiples of 3 The pattern is The columns in which multiples of 3 appear follow a repeating sequence: 3, 2, 1, 4, 3, 2, 1, 4, ... Other Patterns: All numbers in Column 4 are multiples of 4. The numbers in any diagonal from top-left to bottom-right increase by 5 (e.g., 1, 6, 11, 16). The numbers in any diagonal from top-right to bottom-left increase by 3 (e.g., 4, 7, 10, 13).