6 7 Divided By 3

Diving Deep into 67 Divided by 3: A Comprehensive Exploration of Division

This article provides a detailed explanation of how to divide 67 by 3, covering various methods, applications, and related mathematical concepts. Understanding division, particularly with numbers that don't divide evenly, is fundamental to many areas of mathematics and everyday life. We'll go beyond simply finding the answer and explore the underlying principles to solidify your understanding. Keywords: division, long division, remainders, fractions, decimals, mathematics, arithmetic.

Introduction: Understanding the Problem

The question "67 divided by 3" asks us to determine how many times 3 goes into 67. Here's the thing — this involves the fundamental arithmetic operation of division. Now, since 67 isn't a multiple of 3, we'll encounter a remainder, which will be crucial in understanding the complete solution. This exploration will cover several approaches to solve this problem, ranging from simple estimation to precise calculation using long division and decimal representation.

Method 1: Long Division - The Classic Approach

Long division is a systematic method for performing division with larger numbers. Let's break down the process step-by-step:

Set up the problem: Write the dividend (67) inside the long division symbol (⟌) and the divisor (3) outside.
```
3⟌67
```
Divide the tens digit: How many times does 3 go into 6? It goes in 2 times (2 x 3 = 6). Write the 2 above the 6.
```
  2
3⟌67
```
Multiply and subtract: Multiply the quotient (2) by the divisor (3) and write the result (6) below the 6 in the dividend. Subtract 6 from 6, resulting in 0.
```
  2
3⟌67
-6
--
 0
```
Bring down the next digit: Bring down the next digit from the dividend (7).
```
  2
3⟌67
-6
--
 07
```
Divide the ones digit: How many times does 3 go into 7? It goes in 2 times (2 x 3 = 6). Write the 2 above the 7.
```
  22
3⟌67
-6
--
 07
```
Multiply and subtract: Multiply the quotient (2) by the divisor (3) and write the result (6) below the 7. Subtract 6 from 7, resulting in 1. This is the remainder.
```
  22
3⟌67
-6
--
 07
 -6
 --
  1
```

Because of this, 67 divided by 3 is 22 with a remainder of 1. This can be written as 22 R 1.

Method 2: Repeated Subtraction

This method involves repeatedly subtracting the divisor (3) from the dividend (67) until the result is less than the divisor. Let's illustrate:

Start with the dividend: We begin with 67.
Repeated subtraction:
- 67 - 3 = 64
- 64 - 3 = 61
- 61 - 3 = 58
- 58 - 3 = 55
- 55 - 3 = 52
- 52 - 3 = 49
- 49 - 3 = 46
- 46 - 3 = 43
- 43 - 3 = 40
- 40 - 3 = 37
- 37 - 3 = 34
- 34 - 3 = 31
- 31 - 3 = 28
- 28 - 3 = 25
- 25 - 3 = 22
- 22 - 3 = 19
- 19 - 3 = 16
- 16 - 3 = 13
- 13 - 3 = 10
- 10 - 3 = 7
- 7 - 3 = 4
- 4 - 3 = 1
Count the subtractions: We subtracted 3 from 67 a total of 22 times before reaching a remainder of 1.

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This method confirms the result obtained through long division: 67 divided by 3 is 22 with a remainder of 1.

Method 3: Representing the Result as a Mixed Number

The remainder can be expressed as a fraction, converting the result into a mixed number. In practice, the remainder (1) becomes the numerator of a fraction, and the divisor (3) becomes the denominator. Thus, the result is 22 1/3. This representation is particularly useful in contexts where expressing the remainder as a fraction is more meaningful than a simple remainder.

Method 4: Converting to a Decimal

To obtain a decimal representation, we continue the division process beyond the remainder. After obtaining the remainder 1, we add a decimal point and a zero to the remainder:

Notice that the remainder 1 keeps reappearing, leading to a repeating decimal: 22.333... This is often represented as 22.3̅. The bar above the 3 indicates that the digit 3 repeats infinitely.

The Significance of Remainders

The remainder (1 in this case) signifies the portion of the dividend that is left over after the division is complete. It’s crucial because it indicates that 67 is not perfectly divisible by 3. Remainders have many practical applications:

Sharing: If you have 67 candies to share equally among 3 friends, each friend gets 22 candies, and you have 1 candy left over.
Measurement: If you need to cut a 67-inch rope into 3 equal pieces, each piece will be 22 and 1/3 inches long.
Programming: In computer programming, remainders are often used in modular arithmetic operations (modulo operator, %), useful for tasks like determining if a number is even or odd.

Real-World Applications of Division

Division is a fundamental arithmetic operation with broad applications across diverse fields:

Finance: Calculating per-share earnings, splitting bills, determining interest rates.
Engineering: Dividing materials equally, calculating ratios in construction, determining speeds.
Cooking: Dividing recipes for different serving sizes, adjusting ingredient amounts.
Everyday Life: Sharing items equally, calculating unit prices, figuring out travel time.

Frequently Asked Questions (FAQ)

Q1: What is the difference between a quotient and a remainder?

A1: The quotient is the result of the division (the whole number part). The remainder is the amount left over after the division. In 67 ÷ 3, the quotient is 22, and the remainder is 1.

Q2: Can a remainder be larger than the divisor?

A2: No. If the remainder is larger than the divisor, it means the division is incomplete. You can further divide the remainder by the divisor to obtain a more accurate result.

Q3: Why does dividing 67 by 3 result in a repeating decimal?

A3: This happens because 67 is not a multiple of 3. And the division process generates a remainder that keeps repeating, leading to an infinite repeating decimal. This is characteristic of rational numbers that cannot be expressed as a terminating decimal.

Q4: What are some alternative ways to check the answer?

A4: You can check your answer by multiplying the quotient by the divisor and adding the remainder: (22 x 3) + 1 = 67. This confirms that the calculation is correct. You can also use a calculator to verify your long division result.

Conclusion: Mastering Division and Beyond

Understanding how to divide 67 by 3 goes beyond simply finding the answer; it's about grasping the fundamental principles of division, handling remainders, and appreciating the different ways to represent the result. The exploration of this seemingly simple problem has revealed the rich tapestry of concepts underpinning even basic arithmetic operations. Mastering these concepts builds a strong foundation for more advanced mathematical studies and problem-solving. Whether using long division, repeated subtraction, or expressing the answer as a mixed number or decimal, the process demonstrates a crucial skill applicable across various mathematical and real-world scenarios. Remember, the key is not just to get the right answer, but to understand why it's the right answer and how it applies to a wider range of situations.