LCM HCF Percentage Average

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🔹 What is LCM?

The Least Common Multiple (LCM) of two or more numbers is the smallest number that is a multiple of each of the numbers.

🔹 Methods to Find LCM:

1. Listing Multiples

2. Prime Factorization

3. Division Method

🟩  LCM Questions with Answers

1. What is the LCM of 4 and 6?

Multiples of 4: 4, 8, 12, 16, 20...

Multiples of 6: 6, 12, 18, 24...

Common Multiple: 12

✅ Answer: 12

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2. Find the LCM of 5, 10, and 15.

Multiples of 5: 5, 10, 15, 20, 25, 30...

Multiples of 10: 10, 20, 30...

Multiples of 15: 15, 30, 45...

Common Multiple: 30

✅ Answer: 30

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3. Find the LCM of 12 and 18 using Prime Factorization.

12 = 2² × 3

18 = 2 × 3²

LCM = 2² × 3² = 4 × 9 = 36

✅ Answer: 36

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4. Two traffic lights blink every 20 and 30 seconds. When will they blink together again?

LCM of 20 and 30:

20 = 2² × 5

30 = 2 × 3 × 5

LCM = 2² × 3 × 5 = 60

✅ Answer: 60 seconds later

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5. Ramesh visits the library every 12 days, and Suresh every 16 days. When will they meet again?

LCM of 12 and 16:

12 = 2² × 3

16 = 2⁴

LCM = 2⁴ × 3 = 48

✅ Answer: After 48 days

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6. Find the LCM of 9, 15, and 21.

9 = 3²

15 = 3 × 5

21 = 3 × 7

LCM = 3² × 5 × 7 = 315

✅ Answer: 315

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7. Is LCM of two prime numbers always their product?

Yes.

Example: 5 and 7 are prime.

LCM = 5 × 7 = 35

✅ Answer: Yes, it is always their product.

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8. Find the smallest 4-digit number divisible by 12, 15, and 18.

Step 1: Find LCM

12 = 2² × 3

15 = 3 × 5

18 = 2 × 3²

LCM = 2² × 3² × 5 = 180

Now, find smallest 4-digit number divisible by 180:

1000 ÷ 180 ≈ 5.55 ⇒ 6 × 180 = 1080

✅ Answer: 1080

9. Find the LCM of 2, 3, 4, 5, and 6.

2 = 2

3 = 3

4 = 2²

5 = 5

6 = 2 × 3

LCM = 2² × 3 × 5 = 60

✅ Answer: 60

10. A bell rings every 45 minutes and another every 60 minutes. When will they ring together next?

LCM of 45 and 60:

45 = 3² × 5

60 = 2² × 3 × 5

LCM = 2² × 3² × 5 = 180

✅ Answer: After 180 minutes (or 3 hours)

Higher level studies questions are useful in preparation of higher level exams like UPSC PSC SSC railway Bank etc. this questions clear your basics and improve your questions solving speed to get 100% marks in exams.

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Machine Maintenance Challenge

Q1:

Machine A requires maintenance every 18 days, Machine B every 24 days, and Machine C every 30 days. If all three are serviced together today, after how many days will they be serviced together again?

Solution:

LCM of 18, 24, and 30

18 = 2 × 3²

24 = 2³ × 3

30 = 2 × 3 × 5

LCM = 2³ × 3² × 5 = 360 days

✅ Answer: 360 days

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🔧 2. Machine Downtime Overlap

Q2:

A machine stops for a 10-minute break every 45 minutes, and another machine stops every 75 minutes. Both machines were turned on at 9:00 AM. When will they both stop at the same time for the first time after 9:00 AM?

Solution:

LCM of 45 and 75

45 = 3² × 5

75 = 3 × 5²

LCM = 3² × 5² = 225 minutes

9:00 AM + 225 minutes = 12:45 PM

✅ Answer: 12:45 PM

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🔧 3. Industrial Gear Rotation

Q3:

Three industrial gears rotate at intervals of 36 seconds, 54 seconds, and 81 seconds. If they start rotating together at 10:00 AM, when will they rotate together next?

Solution:

Find LCM of 36, 54, 81

36 = 2² × 3²

54 = 2 × 3³

81 = 3⁴

LCM = 2² × 3⁴ = 4 × 81 = 324 seconds

Convert: 324 seconds = 5 minutes 24 seconds

✅ Answer: 10:05:24 AM

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🔧 4. Oil Filter Replacement

Q4:

An oil filter in Machine X needs to be replaced every 21 days, and a coolant in Machine Y needs changing every 28 days. The factory wants to schedule a combined maintenance day. What is the minimum number of days after which both items can be changed together?

Solution:

LCM of 21 and 28

21 = 3 × 7

28 = 2² × 7

LCM = 2² × 3 × 7 = 84 days

✅ Answer: 84 days

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🔧 5. Large-Scale Factory Sync

Q5:

A factory has four machines that require maintenance every 25, 30, 40, and 60 days respectively. What is the minimum number of days after which all four machines will be serviced together?

Solution:

LCM of 25, 30, 40, and 60

25 = 5²

30 = 2 × 3 × 5

40 = 2³ × 5

60 = 2² × 3 × 5

LCM = 2³ × 3 × 5² = 600

✅ Answer: 600 days

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HCF is the greatest number that divides two or more numbers exactly.  HCF stands for Highest Common Factor. It's the biggest number that divides all the given numbers without leaving a remainder.

We use HCF in daily life—like when we need to group items equally or simplify fractions. It’s very practical!

1. Find the HCF of 12 and 18.
Solution:
Factors of 12 = 1, 2, 3, 4, 6, 12
Factors of 18 = 1, 2, 3, 6, 9, 18
Common factors = 1, 2, 3, 6
👉 Answer AnswHCF = 6


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2. Find the HCF of 8 and 32.
Solution:
Factors of 8 = 1, 2, 4, 8
Factors of 32 = 1, 2, 4, 8, 16, 32
Common factors = 1, 2, 4, 8
👉 HCF = 8


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3. Find the HCF of 9, 12, and 15.
Solution:
Factors of 9 = 1, 3, 9
Factors of 12 = 1, 2, 3, 4, 6, 12
Factors of 15 = 1, 3, 5, 15
Common factor = 1, 3
👉 HCF = 3


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4. Find the HCF of 20 and 28 using prime factorization.
Solution:
20 = 2 × 2 × 5
28 = 2 × 2 × 7
Common = 2 × 2 = 4
👉 HCF = 4


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5. Find the HCF of 16 and 24 using the division method.
Solution:
Step 1: Divide 24 by 16 → remainder = 8
Step 2: Divide 16 by 8 → remainder = 0
👉 HCF = 8


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🔹 Medium Level

6. Find the HCF of 48, 60, and 72.
Solution:
48 = 2⁴ × 3
60 = 2² × 3 × 5
72 = 2³ × 3²
Common prime factors = 2² × 3 = 4 × 3 = 12
👉 HCF = 12


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7. Two numbers are 36 and 84. Find their HCF using Euclid’s Division Algorithm.
Solution:
Step 1: 84 ÷ 36 = 2, remainder = 12
Step 2: 36 ÷ 12 = 3, remainder = 0
👉 HCF = 12


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8. Find the HCF of 96 and 144 using prime factorization.
Solution:
96 = 2⁵ × 3
144 = 2⁴ × 3²
Common = 2⁴ × 3 = 16 × 3 = 48
👉 HCF = 48


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9. The HCF of two numbers is 14 and their LCM is 168. If one of the numbers is 56, find the other number.
Solution:
HCF × LCM = Product of two numbers
⇒ 14 × 168 = 56 × ?
⇒ 2352 = 56 × ?
⇒ ? = 2352 ÷ 56 = 42
👉 Other number = 42


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10. Three ropes of lengths 120 cm, 150 cm, and 180 cm need to be cut into equal pieces. What is the greatest possible length of each piece?
Solution:
HCF of 120, 150, and 180
120 = 2³ × 3 × 5
150 = 2 × 3 × 5²
180 = 2² × 3² × 5
Common = 2 × 3 × 5 = 30
👉 Greatest length = 30 cm

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✳️ What is Percentage?

Percentage is a way of expressing a number as a fraction of 100.

> 🔹 The word “percent” comes from Latin per centum, meaning “by the hundred.”

💡 Basic Definition:

> 1% = 1 out of 100

50% = 50 out of 100

100% = 100 out of 100 = Whole

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🧠 Understanding with Diagrams

🔘 Pie Chart (Pizza Example)

Imagine a pizza cut into 100 slices.

If you eat 25 slices, you ate 25% of the pizza.

Percentage Slices Visual Idea

25% 25 1/4 pizza

50% 50 Half pizza

100% 100 Full pizza

You can also use blocks (10x10 grid = 100 squares). Color 40 blocks = 40%

Conversion Between Forms

🔢 1. Fraction ↔️ Percentage

Multiply the fraction by 100

Example :- 

1]  2/3 ×100 = 200/3 = 66.6%

2]  1/4 ×100 = 100/4 = 25%

🔢 2. Decimal ↔️ Percentage

Multiply the decimal by 100 and add the % sign

1] 0.5 = 0.5 × 100 = 50%

      as we already know that 0.5 is half of 1 so that, 0.5 would be 50% of the number 1.

2]  0.08 = 0.08 × 100 = 8% 

similarly, when you multiply hundred with 0.08 you get 8% of the number 1. 

• 
  
• 1. Understanding Percentages (1% = 1 out of 100)

Basic Concept : What will we discuss in this section 

1. What is 25% of 100?

2. Convert 50% to a fraction.

3. What percent of 200 is 50?

4. Write 0.75 as a percent.

5. What is 100% of 50?

6. Convert 3/4 into percent.

7. Write 30% as a decimal.

8. What is 10% of 500?

9. If 20 out of 100 students are absent, what percentage is that?

10. Convert 0.2 to percent.

🟢 2. Fractions, Decimals & Percentages

11. Write 1/5 as a percent.

12. Convert 12.5% into a decimal.

13. Express 2.5 as a percentage.

14. What is 60% of 300?

15. What percent is 25 out of 50?

16. Write 0.6 as a percentage.

17. 1/2 is equal to how many percent?

18. What is 80% of 250?

19. Convert 45% into a decimal.

20. Find 15% of 240.

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🧠 Section B: Skill Practice (Medium Level)

🟡 3. Finding Percentages and Solving Word Problems

21. A boy scored 18 out of 20 in a test. What is his percentage?

22. What percent of ₹500 is ₹125?

23. Increase 200 by 25%.

24. Decrease 400 by 20%.

25. A jacket worth ₹800 is sold at 10% discount. Find the selling price.

26. A number is 30% more than 200. Find the number.

27. 60% of a class are boys. If there are 30 boys, find the total number of students.

28. A fruit basket has 40 apples and 10 are rotten. What percent are rotten?

29. Find 33⅓% of 180.

30. A man saves 20% of his income. If his income is ₹10,000, how much does he save?

🟡 4. Real-Life Applications

31. A shopkeeper gives 15% discount on a ₹1200 item. Find the discount and selling price.

32. A product marked at ₹2500 is sold for ₹2000. What is the percent discount?

33. A student got 360 marks out of 500. Find the percentage.

34. A quantity is increased by 10% and then again by 20%. What is the overall percentage increase?

35. If 75% of the class passed, and 30 students passed, how many total students were there?

36. Find 20% of ₹2,500.

37. Find 12% of 360.

38. What is 150% of 80?

39. If 5% of a number is 12, what is the number?

40. A number is 60% less than 200. Find the number.

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🧩 Section C: Mixed & Challenge Zone

🟠 5. Mixed Application and Puzzles

41. A bag has 80 marbles. 60% are red. How many red marbles are there?

42. What is the percent change if a value increases from 50 to 65?

43. A quantity decreases from 600 to 450. Find the percentage decrease.

44. A student gets 20% more marks than another who scored 75. What is the first student’s marks?

45. If 85% of a number is 170, find the number.

46. A TV was sold at a 25% discount for ₹9000. What was its original price?

47. A man spends 70% of his salary and saves ₹6000. What is his salary?

48. A book was marked ₹400, sold at ₹340. What percent discount was given?

49. If a student scores 95% in an exam of 800 marks, what are the marks obtained?

50. A student got 80% in Science and scored 120 marks. What was the full mark?

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Definition of Average

Average is a value that represents a set of numbers by a single number. It gives us an idea about the central value or typical value in a group of numbers.

In simple words: Average is the sum of all values divided by the number of values.

Average = 

Sum of all values / number of values 

https://youtu.be/uVbk4ZqEb4Y?feature=shared

Section A

1. Find the average of 4, 6, 8, 10.

2. Find the average of the first five natural numbers.

3. The marks obtained by Ravi in 3 subjects are 70, 80, and 90. Find his average marks.

4. A train covers distances of 100 km, 120 km, and 140 km on three days. What is the average distance?

5. Find the average of these numbers: 25, 35, 45, 55, and 65.

https://youtu.be/pYqaCFxy3vw?feature=shared

Section B 

6. The temperatures recorded on five days are 28°C, 30°C, 31°C, 27°C, and 29°C. Find the average temperature.

7. The heights (in cm) of 4 students are 150, 160, 145, and 155. Find the average height.

8. A batsman scored 48, 56, 72, and 64 runs in 4 innings. Find his average score.

9. The weights (in kg) of 6 children are 34, 36, 38, 40, 42, and 44. Find the average weight.

10. A student scored an average of 80 marks in 5 subjects. If his scores in 4 subjects are 75, 85, 78, and 82, find the marks in the 5th subject.

🔹 Question 1:

The average of 9 numbers is 45. If one number is removed, the average becomes 43. What is the number that was removed?

Solution:

Total of 9 numbers = 9 × 45 = 405

New total with 8 numbers = 8 × 43 = 344

Number removed = 405 − 344 = 61

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🔹 Question 2:

The average of 5 consecutive odd numbers is 49. What is the largest of these numbers?

Solution:

Let the numbers be: x−4, x−2, x, x+2, x+4

Their average = (5x)/5 = x

So, x = 49 (the middle number)

Largest number = x + 4 = 49 + 4 = 53

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🔹 Question 3:

A class has 30 students. The average weight of the class is 50 kg. If the teacher's weight is included, the average becomes 51 kg. What is the weight of the teacher?

Solution:

🧑‍🎓🧑‍🎓Total weight of 30 students = 30 × 50 = 1500 kg

New total with teacher = 31 × 51 = 1581 kg

🧑‍🏫Teacher’s weight = 1581 − 1500 = 81 kg

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🔹 Question 4:

The average score of 10 students is 72. If two scores, 64 and 58, are removed, what is the new average of the remaining students?

Solution:

Total of 10 students = 10 × 72 = 720

Scores removed = 64 + 58 = 122

Remaining total = 720 − 122 = 598

🎯New average = 598 ÷ 8 = 74.75

Answer= 74.75

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🔹 Question 5:

The average of three numbers is 60. The first number is twice the second and the third is 10 more than the second. Find the three numbers.

Solution:

Let the second number be x.

Then first = 2x, third = x + 10

Average = (2x + x + x + 10)/3 = 60

So, (4x + 10)/3 = 60

Multiply both sides: 4x + 10 = 180

4x = 170 → x = 42.5

So, numbers are:

Second = 42.5, First = 85, Third = 52.5

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🔹 Question 6:

A cricketer’s average score for 10 matches is 32 runs. His highest score exceeds his lowest score by 40. If these two scores are excluded, the average of the remaining 8 matches is 30. Find his highest score.

Solution:

Total score in 10 matches = 10 × 32 = 320

Total score of remaining 8 matches = 8 × 30 = 240

Sum of highest and lowest score = 320 − 240 = 80

Given difference = 40

Let lowest be x → highest = x + 40

x + (x + 40) = 80 → 2x + 40 = 80

2x = 40 → x = 20

Highest score = x + 40 = 60

Answer= 60

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