HCF and LCM: word problems

key notes :

Understanding HCF and LCM in Word Problems

HCF (Highest Common Factor): The largest number that divides two or more numbers exactly.

LCM (Lowest Common Multiple): The smallest number that is a multiple of two or more numbers.

Word Problems often involve real-life scenarios where you need to find the HCF or LCM to solve the problem.

Common Types of Word Problems

1. Finding Common Group Sizes (HCF)

Problem Type: You need to group items into the largest possible groups where each group has the same number of items.

Example Problem:

Sarah has 24 apples and 36 oranges. She wants to put them into baskets such that each basket has the same number of apples and the same number of oranges. What is the maximum number of baskets she can use?

Steps to Solve:

  1. Find the HCF of 24 and 36.
    • Prime Factorisation:
      • 24 = 2³ × 3
      • 36 = 2² × 3²
    • HCF: 2² × 3 = 12
  2. Answer: Sarah can use a maximum of 12 baskets.

2. Scheduling Events (LCM)

Problem Type: You need to find when two events will occur at the same time again.

Example Problem:

John’s favorite TV shows are on every 8 days and every 12 days. If both shows aired on the same day today, in how many days will they both air on the same day again?

Steps to Solve:

  1. Find the LCM of 8 and 12.
    • Prime Factorisation:
      • 8 = 2³
      • 12 = 2² × 3
    • LCM: 2³ × 3 = 24
  2. Answer: Both shows will air on the same day again in 24 days.

3. Arranging Items (HCF and LCM)

Problem Type: You need to find the best way to arrange items or containers.

Example Problem:

Jenny has two types of boxes: one can hold 9 books and the other can hold 15 books. She wants to arrange her books so that she uses the least number of boxes and all boxes are filled completely. What is the maximum number of books she can arrange in each box without leaving any book out?

Steps to Solve:

  1. Find the HCF of 9 and 15.
    • Prime Factorisation:
      • 9 = 3²
      • 15 = 3 × 5
    • HCF: 3
  2. Answer: Jenny can arrange 3 books in each box.

4. Combining Items (LCM)

Problem Type: You need to find the least number of items needed to complete a set or cycle.

Example Problem:

Alice and Bob are working on a project. Alice works every 5 days and Bob works every 7 days. If they worked together today, how many days from now will they both work on the project together again?

Steps to Solve:

  1. Find the LCM of 5 and 7.
    • Prime Factorisation:
      • 5 = 5¹
      • 7 = 7¹
    • LCM: 5 × 7 = 35
  2. Answer: Alice and Bob will work on the project together again in 35 days.

General Tips for Solving Word Problems

  1. Read the Problem Carefully: Identify what you are asked to find (HCF or LCM).
  2. Break Down the Problem: Determine which method to use based on the context of the problem.
  3. Solve Step-by-Step:
    • For HCF: Find the greatest common factor.
    • For LCM: Find the smallest common multiple.
  4. Check Your Answer: Verify if the solution makes sense in the context of the problem.

Practice Problems

  1. HCF Word Problem: Lisa wants to create identical gift bags for a party. She has 18 candies and 27 stickers. What is the maximum number of gift bags she can prepare so that each bag has the same number of candies and stickers, and all candies and stickers are used up?
    • Solution:
      • HCF of 18 and 27 is 9.
      • Answer: She can prepare 9 gift bags.
  2. LCM Word Problem:A school’s bell rings every 15 minutes for the first bell and every 20 minutes for the second bell. If both bells rang together at 8:00 AM, when will they ring together again?
    • Solution:
      • LCM of 15 and 20 is 60.
      • Answer: They will ring together again at 9:00 AM.
  3. HCF Word Problem: Two friends have different numbers of marbles: 42 and 56. They want to divide them into the largest possible groups with the same number of marbles per group. What is the largest number of marbles that can be in each group?
    • Solution:
      • HCF of 42 and 56 is 14.
      • Answer: The largest number of marbles in each group is 14.
  4. LCM Word Problem: Tom’s work schedule repeats every 10 days, and Jerry’s schedule repeats every 15 days. If they both worked on the same day today, how many days will it be before they work together again?
    • Solution:
      • LCM of 10 and 15 is 30.
      • Answer: They will work together again in 30 days.

let’s practice!