This case-based study test on Chapter 1: Real Numbers is designed according to the latest CBSE pattern to help students develop conceptual understanding and problem-solving skills. The test includes real-life situations, logical reasoning questions, and application-based problems related to Euclid’s Division Algorithm, HCF & LCM, prime numbers, irrational numbers, and decimal expansions. These competency-focused questions improve analytical thinking and prepare students for school exams, board exams, and competitive-level practice in an engaging and practical way.

Case Study 1: School Sports Ground Measurement

A school is preparing a rectangular sports ground for an annual event. The length of the ground is 96 m and the breadth is 60 m. The school wants to divide the ground into identical square sections for different sports activities such that no space is left unused.

To find the largest possible square section, students use the concept of HCF (Highest Common Factor) based on Euclid’s Division Algorithm.

Questions

Q1. What will be the side length of the largest square section?

A) 6 m
B) 12 m
C) 18 m
D) 24 m

Q2. How many square sections will be formed?

A) 20
B) 30
C) 40
D) 48

Q3. Which mathematical concept is used here?

A) Prime Factorisation
B) Euclid’s Division Lemma
C) Rational Numbers
D) Decimal Expansion

Q4. Find the HCF of 96 and 60 using Euclid’s algorithm.

Q5. Is the decimal expansion of $\frac{96}{60}$6096​ terminating or non-terminating?

---

Case Study 2: Online Shopping Discount

Riya purchased two items online costing ₹625 and ₹875. She wants to pack them into gift boxes such that each box contains the same value of items and no item is left unpacked.

She decides to use the concept of HCF to determine the greatest value possible for each box.

Questions

Q1. Find the HCF of 625 and 875.

A) 25
B) 50
C) 125
D) 175

Q2. The prime factorisation of 625 is:

A) $5^2 \times 25$52×25
B) $5^3 \times 5$53×5
C) $5^4$54
D) $25^2$252

Q3. Which method can also be used to find HCF?

A) Coordinate Geometry
B) Prime Factorisation
C) Algebraic Identities
D) Linear Equations

Q4. Express $\frac{625}{875}$875625​ in simplest form.

Q5. Will the decimal expansion of $\frac{5}{7}$75​ terminate?

---

Case Study 3: Water Tank Capacity

Two water tankers carry 540 litres and 648 litres of water respectively. Water is to be transferred into smaller containers of equal capacity such that each container is completely filled.

Students use Real Numbers concepts to determine the greatest possible capacity of each container.

Questions

Q1. Find the HCF of 540 and 648.

A) 18
B) 36
C) 54
D) 72

Q2. Using prime factorisation, 540 can be written as:

A) $2^2 \times 3^3 \times 5$22×33×5
B) $2^2 \times 3^2 \times 5$22×32×5
C) $2 \times 3^3 \times 5$2×33×5
D) $2^3 \times 3^2 \times 5$23×32×5

Q3. What is the greatest container capacity possible?

A) 18 L
B) 36 L
C) 54 L
D) 72 L

Q4. How many containers are needed for 540 litres?

Q5. Is $\sqrt{5}$5​ a rational number or irrational number?

---

Case Study 4: Music Playlist Arrangement

A DJ has two playlists containing 84 songs and 126 songs. He wants to divide the songs into smaller playlists with the same number of songs in each playlist.

Questions

Q1. Find the HCF of 84 and 126.

A) 14
B) 21
C) 28
D) 42

Q2. The LCM of 84 and 126 is:

A) 252
B) 504
C) 756
D) 1008

Q3. Which relation between HCF and LCM is correct?

$\text{HCF} \times \text{LCM} = \text{Product of the numbers}$HCF×LCM=Product of the numbers

A) HCF + LCM = Product
B) HCF × LCM = Product of numbers
C) HCF − LCM = Product
D) HCF ÷ LCM = Product

Q4. Verify the relation between HCF and LCM for 84 and 126.

Q5. Is $0.272727…$0.272727… rational or irrational?

Answers

Case Study 1 Answers

1. B) 12 m
2. C) 40
3. B) Euclid’s Division Lemma
4. HCF = 12
5. Terminating decimal

---

Case Study 2 Answers

1. C) 125
2. C) $5^4$54
3. B) Prime Factorisation
4. $\frac{5}{7}$75​
5. Non-terminating recurring decimal

---

Case Study 3 Answers

1. C) 54
2. A) $2^2 \times 3^3 \times 5$22×33×5
3. C) 54 L
4. 10 containers
5. Irrational number

---

Case Study 4 Answers

1. D) 42
2. A) 252
3. B) HCF × LCM = Product of numbers
4. $42 \times 252 = 84 \times 126$42×252=84×126
5. Rational number