90 Divided By 18 With Remainder

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

Dividing 90 by 18 might seem like a simple arithmetic problem, but it offers a fantastic opportunity to explore core mathematical concepts and delve into the intricacies of division with remainders. This comprehensive guide will not only solve 90 ÷ 18 but also unpack the underlying principles, explore various methods of solving the problem, and discuss the significance of remainders in mathematics and real-world applications.

Understanding Division and Remainders

Division is a fundamental arithmetic operation that involves splitting a quantity into equal parts. When we divide a number (the dividend) by another number (the divisor), we find out how many times the divisor goes into the dividend. Sometimes, the divisor doesn't divide the dividend perfectly, leaving a leftover amount called the remainder.

The general formula for division with a remainder is:

Dividend = (Divisor × Quotient) + Remainder

Where:

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

Solving 90 ÷ 18

Let's tackle the problem at hand: 90 divided by 18.

Method 1: Long Division

Long division is a traditional method for performing division, especially helpful with larger numbers.

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

2. Divide: Ask yourself, "How many times does 18 go into 90?" You might find it easier to think, "How many times does 18 go into 9 (tens place)?" It doesn't go at all. So you need to consider the number 90. 18 goes into 90 exactly 5 times (18 x 5 = 90).

3. Multiply: Multiply the quotient (5) by the divisor (18): 5 × 18 = 90

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

5. Remainder: The remainder is 0.

Therefore, 90 ÷ 18 = 5 with a remainder of 0.

Method 2: Repeated Subtraction

This method is conceptually simpler and helps visualize the division process. We repeatedly subtract the divisor (18) from the dividend (90) until we reach 0 or a number smaller than the divisor.

1. Start with the dividend: 90

2. Subtract the divisor: 90 - 18 = 72

3. Repeat subtraction:

   • 72 - 18 = 54
   • 54 - 18 = 36
   • 36 - 18 = 18
   • 18 - 18 = 0

4. Count subtractions: We subtracted 18 a total of 5 times.

5. Remainder: The remainder is 0.

Again, we find that 90 ÷ 18 = 5 with a remainder of 0.

Method 3: Factoring

This method uses the prime factorization of both numbers to simplify the division.

1. Find prime factors of 90: 90 = 2 × 3 × 3 × 5

2. Find prime factors of 18: 18 = 2 × 3 × 3

3. Cancel common factors: Both 90 and 18 share the factors 2, 3, and 3. Cancelling these gives us 5.

Therefore, 90 ÷ 18 = 5. The remainder is 0.

The Significance of the Remainder (even when it's zero)

While in this specific case, the remainder is 0, signifying a perfect division, understanding remainders is crucial in various mathematical contexts and real-world applications.

• Incomplete Divisions: In many scenarios, a perfect division isn't possible. For example, if you have 92 apples and want to divide them equally among 18 people, you'd have 5 apples per person with 2 apples remaining. The remainder represents the apples that can't be divided equally.

• Modular Arithmetic: Remainders are fundamental to modular arithmetic (often used in cryptography and computer science). The remainder when a number is divided by another number is its modulo. For instance, 92 mod 18 = 2.

• Real-world Applications: Imagine you need to transport 92 passengers in buses that hold 18 people each. You'd need 6 buses (5 full buses and one bus with only 2 passengers). The remainder dictates the need for an extra bus, despite having 5 full buses.

Expanding on Division Concepts

Let's consider variations and extensions related to the division of 90 by 18:

Decimal Division

While we've focused on whole numbers, it's worth considering decimal division. If the problem was slightly different, say 92 ÷ 18, we'd get a decimal answer. Long division would yield a quotient of approximately 5.111... indicating that 18 goes into 92 more than 5 times, but not a whole number of times.

Understanding Divisibility Rules

Divisibility rules provide quick ways to check if a number is divisible by another without performing long division. Since 90 is divisible by 18, it also meets divisibility rules for the factors of 18 (2, 3, 6, 9). This allows for faster calculations and mental estimations.

Applications in Geometry and Measurement

Division with remainders is frequently used in geometry and measurement. For example, if you have a rope of length 90 cm and need to cut it into pieces of 18 cm each, you'll get 5 pieces with no remainder. However, if the rope was 92 cm, you'd have 5 pieces of 18 cm and a remaining piece of 2 cm.

Conclusion

Dividing 90 by 18 results in a quotient of 5 and a remainder of 0, indicating a perfect division. However, this simple problem serves as a springboard for a deeper understanding of division, remainders, and their importance in various mathematical contexts and real-world situations. From long division to repeated subtraction and factorization, multiple methods exist to solve this type of problem. Understanding these concepts enhances your mathematical skills and provides the tools to approach more complex division problems confidently. Remember the significance of the remainder, even when it's zero; it provides valuable information and plays a crucial role in many mathematical and practical applications.