Apples come in boxes of 36 and Pears come in boxes of 20, a shop wants to purchase equal quantities of apples and pears. How many boxes of each should they buy to get equal quantities of each for the lowest possible price?


To solve this problem, we need to find the least number of boxes of apples and pears that result in equal quantities of fruit. Since the quantities per box are different (36 apples and 20 pears per box), the goal is to find the **Least Common Multiple (LCM)** of 36 and 20, as this will tell us the smallest total quantity of fruit that both apples and pears can match.

### Steps to find the LCM of 36 and 20:

1. **Prime factorization of 36**:
   \( 36 = 2^2 \times 3^2 \)

2. **Prime factorization of 20**:
   \( 20 = 2^2 \times 5 \)

3. **LCM Calculation**:
   To find the LCM, we take the highest powers of all the prime factors involved:
   \[
   LCM(36, 20) = 2^2 \times 3^2 \times 5 = 180
   \]
   Therefore, the least number of apples and pears that can be purchased in equal quantities is 180.

### Number of boxes:
- For apples: Each box contains 36 apples, so to get 180 apples, the shop needs:
  \[
  \frac{180}{36} = 5 \text{ boxes of apples}
  \]
- For pears: Each box contains 20 pears, so to get 180 pears, the shop needs:
  \[
  \frac{180}{20} = 9 \text{ boxes of pears}
  \]

Thus, the shop should buy **5 boxes of apples** and **9 boxes of pears** to have equal quantities of each at the lowest possible total cost.

Would you like further details on LCM or the prime factorization process? Here are 5 related questions:

1. How can the LCM help in solving real-world distribution problems?
2. What if the shop wanted a larger quantity of apples and pears? How would you adjust the calculations?
3. Can you always find the LCM by prime factorization, or are there faster methods?
4. How is the concept of LCM useful in scheduling or time management?
5. What are some applications of LCM in industrial processes?

**Tip**: When solving problems involving multiples, the LCM helps ensure you can match quantities across different units or products efficiently.

Apples come in boxes of 36 and Pears come in boxes of 20, a shop wants to purchase equal quantities of apples and pears. How many boxes of each should they buy to get equal quantities of each for the lowest possible price?

Discover how to determine the least number of boxes needed to purchase equal quantities of apples and pears by using the Least Common Multiple (LCM) method. This step-by-step solution covers key mathematical concepts like prime factorization, suitable for students in Grades 6-8.

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