ROUTERA

Chemical Real Numbers

Class 10th Maths Chapter Case Study Questions

Case Study 1: Euclid’s Division Lemma in Real-Life Situations

Case Description:
A school principal wants to divide a group of 560 students into equal teams for a sports event. The principal finds that if each team has 15 students, there will be some students left out. To determine the exact number of leftover students, the principal uses Euclid’s Division Lemma. According to the lemma, for any two positive integers aa and bb, there exist unique integers qq and rr such that a=bq+ra = bq + r, where 0≤r<b0 \leq r < b. Here, aa represents the total number of students, bb represents the number of students per team, qq represents the number of teams, and rr represents the remaining students.

MCQs:

  1. If the total number of students is 560 and each team has 15 students, what is the quotient qq?

    • A) 37
    • B) 36
    • C) 38
    • D) 40

  2. What is the remainder rr when 560 students are divided into teams of 15?

    • A) 10
    • B) 5
    • C) 7
    • D) 12

  3. If the number of students per team is increased to 20, how many teams can be formed?

    • A) 28
    • B) 25
    • C) 30
    • D) 32

  4. What would be the remainder when 560 students are divided into teams of 18?

    • A) 6
    • B) 7
    • C) 5
    • D) 8

Case Study 2: Finding the HCF and LCM of Two Numbers

Case Description:
Two students, Priya and Rohit, decide to participate in a marathon. They plan their training schedule based on how many laps they can complete in a week. Priya completes 24 laps a week, while Rohit completes 36 laps. To synchronize their training, they want to know after how many weeks they will complete the same number of laps on the same day. This problem involves finding the Least Common Multiple (LCM) of their weekly laps, while Euclid’s algorithm can help determine their Highest Common Factor (HCF). By understanding these concepts, they can adjust their training schedules for a more coordinated workout.

MCQs:

  1. What is the HCF of 24 and 36?

    • A) 6
    • B) 8
    • C) 12
    • D) 18

  2. What is the LCM of 24 and 36?

    • A) 96
    • B) 72
    • C) 108
    • D) 84

  3. After how many weeks will Priya and Rohit complete the same number of laps on the same day?

    • A) 3
    • B) 6
    • C) 9
    • D) 12

  4. If Priya’s weekly lap count is changed to 30 laps, what will the new LCM be between 30 and 36?

    • A) 90
    • B) 180
    • C) 120
    • D) 60

Case Study 3: Decimal Expansions and Rational Numbers

Case Description:
A teacher conducts an experiment in class by dividing different pairs of numbers and observing the decimal expansions. The teacher divides 1 by 3, 1 by 4, 1 by 6, and 1 by 7 and asks the students to determine whether each decimal expansion is terminating or non-terminating. The students learn that if the denominator of a fraction (when expressed in its simplest form) has only 2 and/or 5 as its prime factors, the decimal expansion is terminating. Otherwise, the decimal is non-terminating and repeating. This exercise helps students understand how the properties of prime factors in the denominator influence the nature of decimal expansions.

MCQs:

  1. Which of the following fractions has a terminating decimal expansion?

    • A) 13\frac{1}{3}
    • B) 14\frac{1}{4}
    • C) 16\frac{1}{6}
    • D) 17\frac{1}{7}

  2. What type of decimal expansion does 16\frac{1}{6} have?

    • A) Terminating
    • B) Non-terminating and repeating
    • C) Non-terminating and non-repeating
    • D) None of the above

  3. Which of the following numbers will not have a terminating decimal expansion?

    • A) 38\frac{3}{8}
    • B) 720\frac{7}{20}
    • C) 1114\frac{11}{14}
    • D) 925\frac{9}{25}

  4. If a fraction has a denominator with prime factors other than 2 and 5, its decimal expansion will be:

    • A) Terminating
    • B) Non-terminating and repeating
    • C) Non-terminating and non-repeating
    • D) Constant

Case Study 4: Repeating Events Using the LCM

Case Description:
Two friends, Arjun and Bhavya, visit a community library on different schedules. Arjun visits the library every 12 days, while Bhavya visits every 18 days. They want to know when they will both visit the library on the same day. To solve this, they need to find the Least Common Multiple (LCM) of their visit intervals. This concept is useful in various real-life applications where events repeat over different cycles, such as scheduling or planning events with different periodicities.

MCQs:

  1. What is the LCM of 12 and 18?

    • A) 72
    • B) 36
    • C) 48
    • D) 54

  2. After how many days will Arjun and Bhavya both visit the library on the same day?

    • A) 24
    • B) 36
    • C) 48
    • D) 60

  3. If Bhavya’s visit frequency changes to every 24 days, what will be the new LCM of their visit schedules?

    • A) 48
    • B) 60
    • C) 72
    • D) 96

  4. Which mathematical concept is primarily used to solve this problem?

    • A) HCF
    • B) LCM
    • C) Euclid’s Division Algorithm
    • D) Prime Factorization

Case Study 5: Representing Rational Numbers on a Number Line

Case Description:
In a math activity, students are given several fractions, including 34\frac{3}{4}, 58\frac{5}{8}, 710\frac{7}{10}, and 1116\frac{11}{16}. They are tasked with converting these fractions to decimal form and placing them on a number line. The students also discuss whether each fraction has a terminating or non-terminating decimal expansion. This exercise helps them visualize how rational numbers can be represented on a number line and reinforces the concept of decimal expansions.

MCQs:

  1. Which of the following fractions has a non-terminating decimal expansion?

    • A) 34\frac{3}{4}
    • B) 58\frac{5}{8}
    • C) 710\frac{7}{10}
    • D) None of the above

  2. What is the decimal form of 1116\frac{11}{16}?

    • A) 0.75
    • B) 0.6875
    • C) 0.85
    • D) 0.625

  3. Which of the following statements is true about the number line representation of these fractions?

    • A) They all lie between 0 and 1.
    • B) They all lie between 0 and 2.
    • C) They all lie between 0.5 and 1.
    • D) They all lie between 0.25 and 0.75.

  4. What type of decimal expansion will 710\frac{7}{10} have?

    • A) Terminating
    • B) Non-terminating and repeating
    • C) Non-terminating and non-repeating
    • D) Constant