Solving Division Sum with Given Conditions and Its Multiple Solutions

Introduction

Division is a fundamental operation in mathematics, with its application ranging from basic arithmetic to more complex algorithms. One intriguing aspect of division sums is the calculation of the dividend when the remainder, quotient, and divisor are known. This article will explore a specific division sum where the divisor is 3 times the quotient and 6 times the remainder, with a given remainder of 2. We will also discuss the multiple solutions for this problem, providing a comprehensive guide for understanding and solving similar division sums.

Understanding the Problem

The problem involves the following conditions:

The divisor is 3 times the quotient. The divisor is 6 times the remainder. The remainder is 2.

Let's break down the problem step-by-step to find the dividend.

Step-by-Step Solution

Since the divisor is 3 times the quotient, we can represent the divisor as (3q), where (q) is the quotient.

Since the divisor is 6 times the remainder, and the remainder is 2, we can represent the divisor as (6 times 2 12).

Equating the two expressions for the divisor, we get:

[ 3q 12 ]

Solving for the quotient (q), we get:

[ q frac{12}{3} 4 ]

The dividend is the product of the divisor and the quotient. Therefore, the dividend is:

[ text{Dividend} 12 times 4 48 ]

Thus, the dividend is 48 when the remainder is 2.

Alternate Approach

We can also represent the divisor as (D), the quotient as (frac{D}{3}), and the remainder as (frac{D}{6}). Given the remainder is 2, we can write:

D? 6X

Quotient ? 2X

Remainder ? X

By multiplying 6 to each, given remainder (X 2), so divisor (6X 12). Quotient (2X 4).

Now, the dividend is the product of the divisor and the quotient, minus the remainder:

[ text{Dividend} 12 times 4 - 2 48 - 2 50 ]

Multiples of Solutions

In some cases, the divisor can be a multiple of 6. Let (D) be a divisor not divisible by 6. Then, we can take:

Let the divisor be (D).

The dividend (X DD/3 - DD/6 2D/3 - D/6).

Let (D 6):

[ X 6 times 2 - 6 div 6 12 - 1 11 ]

Let (D 7):

[ X 7 times 4 - 7 div 6 28 - 1 27 ]

Let (D 8):

[ X 8 times 2 - 8 div 6 16 - 1 15 ]

Let (D 21):

[ X 21 times 4 - 21 div 6 84 - 7 77 ]

There can be multiple answers based on the divisor chosen, all satisfying the conditions of the problem.

Conclusion

In conclusion, the dividend can be calculated based on the given conditions of the divisor, quotient, and remainder. Depending on the divisor chosen, the dividend will vary. Understanding these principles can help in solving related division sums and enhancing problem-solving skills in mathematics.