Solved problems: Using repeated subtraction to divide

📚 Understanding Repeated Subtraction for Division

Repeated subtraction is a simple way to understand division. It involves subtracting the divisor (the number you're dividing by) from the dividend (the number you're dividing) repeatedly until you reach zero or a number less than the divisor. The number of times you subtract is the quotient (the answer).

📜 History and Background

Repeated subtraction is one of the earliest methods used to teach division. It provides a conceptual understanding of what division represents – breaking a larger quantity into equal smaller groups. Before the widespread use of calculators and long division algorithms, repeated subtraction was a practical method for performing division, particularly with smaller numbers.

🔑 Key Principles

• ➕ Dividend: The number being divided.
• ➖ Divisor: The number you are dividing by.
• 🔄 Subtraction: Repeatedly subtracting the divisor from the dividend.
• 🛑 Termination: Stop when you reach zero or a number less than the divisor.
• 🔢 Quotient: The number of times you subtracted the divisor.
• 📊 Remainder: The number left over (if any) when you can't subtract the divisor anymore.

✍️ How to Perform Repeated Subtraction

Here's a step-by-step guide:

1. Write down the division problem (e.g., 15 ÷ 3).
2. Subtract the divisor (3) from the dividend (15). 15 - 3 = 12
3. Subtract the divisor from the result again. 12 - 3 = 9
4. Continue subtracting until you reach zero or a number smaller than the divisor.
   • 9 - 3 = 6
   • 6 - 3 = 3
   • 3 - 3 = 0
5. Count how many times you subtracted. In this case, you subtracted 5 times. Therefore, 15 ÷ 3 = 5.

💡 Real-World Examples

Example 1: Sharing Cookies

Suppose you have 20 cookies and want to share them equally among 4 friends. How many cookies does each friend get?

Using repeated subtraction:

• 🍪 Start with 20 cookies.
• 🧑‍🤝‍🧑 Give 4 cookies to the first friend: 20 - 4 = 16
• 🧑‍🤝‍🧑 Give 4 cookies to the second friend: 16 - 4 = 12
• 🧑‍🤝‍🧑 Give 4 cookies to the third friend: 12 - 4 = 8
• 🧑‍🤝‍🧑 Give 4 cookies to the fourth friend: 8 - 4 = 4
• 🎁 Give 4 cookies to another (imaginary) friend: 4 - 4 = 0

You subtracted 5 times, so each friend gets 5 cookies. $20 \div 4 = 5$

Example 2: Dividing Apples

You have 28 apples and want to put them into bags, with each bag holding 7 apples. How many bags do you need?

• 🍎 Start with 28 apples.
• 🛍️ Fill the first bag: 28 - 7 = 21
• 🛍️ Fill the second bag: 21 - 7 = 14
• 🛍️ Fill the third bag: 14 - 7 = 7
• 🛍️ Fill the fourth bag: 7 - 7 = 0

You subtracted 4 times, so you need 4 bags. $28 \div 7 = 4$

✔️ Practice Quiz

Solve these using repeated subtraction:

• ❓ $24 \div 6 = ?$
• ❓ $36 \div 9 = ?$
• ❓ $42 \div 7 = ?$
• ❓ $16 \div 4 = ?$
• ❓ $25 \div 5 = ?$

✅ Conclusion

Repeated subtraction is a foundational method for understanding division. It visually demonstrates how division works and can be a helpful tool for learning basic division facts. While not as efficient as long division for larger numbers, it builds a strong conceptual base.