Solving Division Problems Involving Remainders and Quotients

Mathematics often presents us with intriguing problems that require us to apply various theorems and formulas to determine unknown values. One such problem involves a division scenario where the divisor, quotient, and remainder are related to each other. This article will walk you through the steps to solve such a problem using the given relationships.

Problem Statement

The divisor is 32 times the quotient and 12 times the remainder. If the remainder is 56, what is the dividend?

Understanding the Relationship

In a division expression, the relationship between the dividend, divisor, quotient, and remainder can be represented by the formula:

Dividend Divisor × Quotient Remainder

Solving the Problem Step by Step

Given that the remainder (R) is 56, we can find the divisor (D) using the relationship D 12 × R. This gives us:

D 12 × 56 672

Next, we use the relationship D 32 × Q (where Q is the quotient) to find the quotient. Substituting the value of D into this equation, we get:

672 32 × Q

Now, solve for Q:

Q 672 / 32 21

Finally, we use the formula Dividend D × Q R to find the dividend. Substituting the values of D, Q, and R, we get:

Dividend 672 × 21 56

Calculating 672 × 21:

672 × 21 14112

Adding the remainder:

Dividend 14112 56 14168

Conclusion

The dividend is 14168. This solution can be verified by checking the relationships between the divisor, quotient, and remainder.

Verification

To verify the solution, we can perform the division 14168 ÷ 672. The quotient should be 21 and the remainder should be 56, which confirms our solution.

Additional Examples

What if the remainder is 48? Then the divisor will be 5 times 48, which is 240. Using the relationships given, we can find the quotient and then the dividend as follows:

D 5 × 48 240

D 240, so 240 32 × Q

Solving for Q:

Q 240 / 32 7.5

Dividend 240 × 7.5 48 1800 48 1848

Checking the solution:

1848 ÷ 240 7 remainder 48

Thus, the verification confirms our solution.

Understanding and solving such problems using the relationships between divisor, dividend, quotient, and remainder is essential for many mathematical applications, including coding and algorithm design.