Solving for the Sum of Two Numbers Given Their Product

When dealing with algebraic equations, one common challenge is solving for specific values given a product or sum. In this article, we will explore a problem where you are given the product of two numbers and one of the numbers. We will walk through the process of finding the sum of these numbers step-by-step.

Problem Statement

Suppose that the product of two numbers (x) and (y) is 8, and one of the numbers, (x), is (frac{4}{3}). What is the sum of these two numbers?

Solution Steps

Step 1: Understand the Given Information

We know that the product of the two numbers (x) and (y) is 8, and one of the numbers, (x), is (frac{4}{3}).

Mathematically, this can be represented as:

[x cdot y 8]

Given that (x frac{4}{3}), we can substitute this value into the equation:

[frac{4}{3} cdot y 8]

Step 2: Solve for (y)

To solve for (y), we need to isolate (y). We do this by dividing both sides of the equation by (frac{4}{3}).

Recall that dividing by a fraction is equivalent to multiplying by its reciprocal:

[y 8 cdot left(frac{3}{4}right)]

Now, carry out the multiplication:

[y frac{8 cdot 3}{4} frac{24}{4} 6]

Step 3: Calculate the Sum of the Two Numbers

With both numbers identified, (x) and (y) are (frac{4}{3}) and 6, respectively. We need to find their sum:

[x y frac{4}{3} 6]

To add these, convert 6 to a fraction with a denominator of 3:

[6 frac{18}{3}]

Now add the fractions:

[x y frac{4}{3} frac{18}{3} frac{22}{3}]

Conclusion

The sum of the two numbers is (frac{22}{3}), which can also be expressed as 7 and (frac{1}{3}) if needed.

Additional Insights

This problem demonstrates how to solve for an unknown variable when given a product and one of the factors. The key steps involved are setting up the equation, isolating the unknown, and then performing arithmetic operations to find the desired result. Such skills are important in a wide range of real-world applications, including finance, engineering, and data analytics.