90 Divided By 18 With Remainder

90 Divided by 18 with Remainder: A Deep Dive into Division

The seemingly simple question, "What is 90 divided by 18 with remainder?" opens a door to a fascinating exploration of fundamental mathematical concepts. While a calculator can quickly provide the answer, understanding the process behind the calculation is crucial for building a strong foundation in arithmetic and related fields. This comprehensive guide will not only answer the question but also delve into the underlying principles of division, including divisors, dividends, quotients, and remainders. We’ll also explore practical applications and related mathematical concepts.

Understanding the Basics of Division

Division is one of the four basic arithmetic operations, alongside addition, subtraction, and multiplication. It's essentially the process of repeatedly subtracting a number (the divisor) from another number (the dividend) until you reach zero or a number smaller than the divisor. The result of this repeated subtraction is called the quotient, and any number left over is the remainder.

In the context of 90 divided by 18:

• Dividend: 90 (the number being divided)
• Divisor: 18 (the number by which we are dividing)
• Quotient: The result of the division (how many times 18 goes into 90).
• Remainder: The amount left over after the division.

Calculating 90 Divided by 18

Several methods can be used to calculate 90 divided by 18. Let's explore the most common approaches:

Method 1: Long Division

Long division is a traditional method that systematically breaks down the division process. Here’s how it works for 90 divided by 18:

1. Set up the problem: Write 90 inside the long division symbol (÷) and 18 outside.

2. Estimate: Ask yourself, "How many times does 18 go into 90?" You might estimate 5 (since 18 x 5 = 90).

3. Multiply: Multiply the divisor (18) by your estimate (5): 18 x 5 = 90

4. Subtract: Subtract the result (90) from the dividend (90): 90 - 90 = 0

5. Remainder: Since the remainder is 0, 18 divides 90 exactly.

Therefore, 90 divided by 18 is 5 with a remainder of 0.

Method 2: Using Multiplication Tables

A simpler method, particularly for smaller numbers like this, is to use multiplication tables. You simply look for the multiple of 18 that equals 90. Knowing your multiplication tables reveals that 18 x 5 = 90. This immediately gives us the quotient as 5 and a remainder of 0.

Method 3: Factoring

Factoring can be a useful technique, especially with larger numbers. We can find the prime factorization of both 90 and 18:

• 90 = 2 x 3 x 3 x 5
• 18 = 2 x 3 x 3

Notice that 18 is a factor of 90. Dividing 90 by 18 is essentially canceling out the common factors:

(2 x 3 x 3 x 5) / (2 x 3 x 3) = 5

Again, this confirms that the quotient is 5, and the remainder is 0.

Understanding Remainders in Division

While the example of 90 divided by 18 resulted in a remainder of 0 (meaning 18 is a factor of 90), many divisions produce a remainder. A remainder is the amount left over after dividing a number as completely as possible. It's always smaller than the divisor.

Example with a Remainder:

Let's consider 95 divided by 18:

1. Estimate: 18 goes into 95 approximately 5 times (18 x 5 = 90).

2. Multiply: 18 x 5 = 90

3. Subtract: 95 - 90 = 5

4. Remainder: The remainder is 5.

Therefore, 95 divided by 18 is 5 with a remainder of 5.

Real-World Applications of Division with Remainders

Division with remainders isn't just a theoretical exercise; it has numerous practical applications:

• Sharing Items: Imagine distributing 95 candies among 18 children equally. Each child would get 5 candies (the quotient), and there would be 5 candies left over (the remainder).

• Measurement: Converting units often involves division with remainders. For example, converting 95 inches into feet (12 inches per foot): 95 ÷ 12 = 7 feet with a remainder of 11 inches.

• Scheduling: If you have to complete 95 tasks in 18 days, you'd need to complete an average of 5 tasks per day (95 ÷ 18 ≈ 5), but you'd still have 5 tasks left over.

• Computer Science: Remainders are fundamental in computer science, particularly in algorithms and hashing.

• Finance: Calculating interest, splitting bills, and dividing profits can involve remainders.

Beyond the Basics: Related Mathematical Concepts

Understanding division with remainders lays the groundwork for more advanced mathematical concepts:

• Modular Arithmetic: Modular arithmetic uses remainders to perform arithmetic operations within a given range. For example, in modular arithmetic modulo 12 (clock arithmetic), 95 is equivalent to 11 (95 ÷ 12 = 7 remainder 11). This has applications in cryptography and computer science.

• Euclidean Algorithm: This algorithm uses the concept of remainders to find the greatest common divisor (GCD) of two numbers.

• Prime Numbers: Prime numbers are only divisible by 1 and themselves, meaning they always leave a remainder when divided by any other number.

• Divisibility Rules: Understanding divisibility rules helps quickly determine if a number is divisible by another without performing long division.

Conclusion

The seemingly simple calculation of 90 divided by 18, with its resulting quotient of 5 and remainder of 0, serves as a gateway to a deeper understanding of division, its applications, and its connection to more advanced mathematical concepts. By mastering the fundamentals of division with remainders, we build a solid foundation for success in various mathematical and real-world scenarios. From sharing candies to complex algorithms, the concept of the remainder plays a crucial role in numerous areas, highlighting the importance of a thorough understanding of this fundamental arithmetic operation. This exploration serves as a reminder that even seemingly basic math holds a world of complexity and practical application, making it a rewarding field to study and master.