Q. If the LCM of two numbers is 60 and their HCF is 5, what is the sum of the two numbers if they are both less than 30? (2023)
A.
25
B.
35
C.
40
D.
30
Solution
Let the two numbers be 5a and 5b. Then, LCM(5a, 5b) = 5 * LCM(a, b) = 60, which gives LCM(a, b) = 12. The pairs (3, 4) work, giving numbers 15 and 20, which sum to 35.
Q. In a class of 60 students, the ratio of boys to girls is 3:2. If the number of boys is increased by 10 and the number of girls is decreased by 5, what will be the new ratio of boys to girls? (2023)
A.
2:3
B.
3:2
C.
5:3
D.
4:3
Solution
Initially, there are 36 boys and 24 girls. After the changes, there will be 46 boys and 19 girls. The new ratio is 46:19, which simplifies to approximately 4:3.
Q. In a class of 60 students, the ratio of boys to girls is 3:2. If the number of boys is increased by 10, what will be the new ratio of boys to girls? (2023)
A.
4:3
B.
5:2
C.
3:2
D.
2:3
Solution
Initially, there are 36 boys and 24 girls. After increasing the boys by 10, there will be 46 boys and 24 girls, giving a new ratio of 46:24, which simplifies to 4:3.
Q. In a class of 60 students, the ratio of boys to girls is 3:2. If the number of boys is increased by 5, what will be the new ratio of boys to girls? (2023)
A.
4:3
B.
5:2
C.
3:2
D.
2:3
Solution
Initially, there are 36 boys and 24 girls. After increasing boys by 5, there will be 41 boys and 24 girls, giving a new ratio of 41:24, which simplifies to approximately 4:3.
Q. The LCM of two numbers is 84 and their HCF is 12. What are the two numbers? (2023)
A.
24 and 42
B.
12 and 84
C.
28 and 36
D.
21 and 48
Solution
Let the two numbers be 12x and 12y. Then, LCM(12x, 12y) = 12 * LCM(x, y) = 84, which gives LCM(x, y) = 7. The pairs (x, y) that satisfy this are (3, 4) or (4, 3), leading to 24 and 42.
Understanding HCF (Highest Common Factor) and LCM (Lowest Common Multiple) is crucial for students preparing for school and competitive exams. These concepts not only form the foundation of number theory but also frequently appear in various exam formats. Practicing HCF & LCM MCQs and objective questions helps students enhance their problem-solving skills and boosts their confidence, ensuring they are well-prepared for important questions in their exams.
What You Will Practise Here
Definition and significance of HCF and LCM
Methods to calculate HCF and LCM, including prime factorization and listing multiples
Applications of HCF and LCM in real-life scenarios
Formulas related to HCF and LCM
Common problems and practice questions on HCF and LCM
Visual aids and diagrams to understand concepts better
Tips and tricks for quick calculations
Exam Relevance
The concepts of HCF and LCM are integral to the mathematics syllabus across various educational boards in India, including CBSE and State Boards. These topics are frequently tested in school exams and competitive exams like NEET and JEE. Students can expect questions that require them to find the HCF or LCM of given numbers, often in multiple-choice formats. Understanding the common question patterns can significantly enhance exam performance.
Common Mistakes Students Make
Confusing HCF with LCM and vice versa
Incorrect application of formulas, especially in word problems
Overlooking the importance of prime factorization in calculations
Rushing through calculations, leading to simple arithmetic errors
Failing to check if the answer is reasonable based on the problem context
FAQs
Question: What is the difference between HCF and LCM? Answer: HCF is the largest number that divides two or more numbers, while LCM is the smallest number that is a multiple of two or more numbers.
Question: How can I quickly find the HCF of two numbers? Answer: You can find the HCF using the prime factorization method or by using the Euclidean algorithm for faster calculations.
Question: Are there any shortcuts for calculating LCM? Answer: Yes, one effective shortcut is to use the formula: LCM(a, b) = (a * b) / HCF(a, b).
Now that you have a clear understanding of HCF and LCM, it's time to put your knowledge to the test! Dive into our practice MCQs and challenge yourself to solve important HCF & LCM questions for exams. Your preparation starts here!
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