JESSEL O. GESALTA BACHELOR OF ARTS IN HISTORY 1-3 MATHEMATICS IN THE MODERN WORLD ASSESSMENT 4 1. Explain why you can never be sure that a conclusion you arrived at using inductive reasoning is true. Inductive reasoning is based on recognizing patterns in data and drawing likely conclusions based on those patterns. Its process arrives at a general conclusion based on the observation of specific examples. Therefore, the conclusion we get from inductive reasoning could be wrong because we can never be sure that what is true in a specific case will be true in general. 2. Select any two-digit number. Multiply it by 9. Then add the digits. Keep adding the digits in the answer until you get a single-digit answer. Using inductive reasoning, what can you conjecture about any whole number multiplied by 9? Use deductive reasoning to prove that your conjecture is true. 21 x 9 = 189 1+8+9 = 18 1+8 = 9 Inductive Reasoning: We may conjecture that the sum of the digits of any whole number multiplied by 9 is also 9. Deductive Reasoning: This is because by multiplying any number with 9, we are making it a multiple of 9; and the sum of the digits of all multiples of 9 is also a multiple of 9. Since we keep adding the digits in the given example, we established a chain of decreasing multiples of 9 and eventually end up with 9. Therefore, the sum of the digits of any whole number multiplied by 9 is also 9. The reason why 9 have this “magic property” is because we use base-10 system. 3. Use Polya’s Four Steps to solve the following problems. (a) Susie’s age this year is a multiple of 5. Next year, her age is a multiple of 7. What is her present age? Understand the problem: Susie’s present age is a multiple of 5 and her age next year will be a multiple of 7. Solve for Susie’s present age. Devise a plan: Make a table and list down multiples of 5 and multiples of 7. Then identify which multiple of 5 becomes a multiple of 7 when we add 1 to it. Carry out the plan: Multiples of 5 5 10 15 20 25 Multiples of 7 7 14 21 28 35 Look back: 20 is a multiple of 5 that becomes a multiple of 7 when we add 1 to it. Therefore, if Susie’s age is multiple of 5 and her age next year will be a multiple of 7, Susie’s present age is 20. (b) Consider a square whose side is 1 unit. If the measure of its side is doubled, what will be its new area as compared to the smaller square? How about if the side of the smaller square was tripled, what will be its new area? Understand the problem: If the side of a square is 1 unit, what will be its area compared to the smaller square if its side will be doubled? How about if it is tripled? Devise a plan: Let: Sqr1 be the square with a side of one unit, Sqr2 be the square whose side is double the side of Sqr1 and Sqr3 be the square whose side is tripled the side of Sqr1 to get the area of each square. Use the formula A = s2 Carry out the plan: Sqr1: S = 1 Unit Sqr2: S = (1)(2) Units Sqr3: S = (1)(3) Units A = s2 S = 2 units S = 3 units A =12 A = s2 A = s2 A = 1 unit2 A = 22 A = 32 A = 4 units2 A = 9 units2 Look back: If the smaller square has a side of 1 unit, then the second square whose side was doubled has a new area of 4 units2 and the third square whose side was tripled has a new area of 9 units2. (c) How many perfect squares are there between 1,000,000 and 9,000,000? Understand the problem: a = square root of 9,000,000 b = square root of 1,000,000 N - number of perfect squares between a and b Devise a plan: a = 3000 b = 1000 N=a-b Carry out the plan: N = 3000 - 1000 = 2000 + 1 perfect square N = 2001 perfect square Look back: There are 2001 perfect squares if we include 1,000,000 and 9,000,000 and 1,999 perfect squares if we exclude them. (d) Determine the number of different triangles that can be drawn given eight noncollinear points? Understand the problem: This is a combination problem where repeats are not accepted and selection order does not matter. We have 8 points and 3 must be chosen to create a triangle. Devise a plan: Use the formula: nCr Carry out the plan: 8C3 = 8! / (8-3)! * 3! = 8! / 5!*3! = 8*7*6 / 3*2 = 56 Look back: The number of different triangles that can be drawn using eight noncollinear points is 56. (e) There are 25 students asked by their literature instructor regarding with the type of literary works they prefer to read. He found out that 10 prefer to read novels, 11 prefer to read short stories, 15 prefer to read poems, 5 for both novels and short stories, 4 both short stories and poems, 7 for both novels and poems, and 3 prefer all. How many students prefer none of the given types of literary works? Understand the problem: 25 students 10 prefer to read novels 11 prefer to read short stories 15 prefer to read poems 5 for both novels and short stories 4 both short stories and poems 7 for both novels and poems 3 prefer all X students prefer none of the 3 Devise a plan: Use Venn diagram to solve the problem Carry out the plan: Look back: There are 2 students who prefer none of the given types of literary works.
