The Sum of Three Numbers and Their Ratios

In this article, we will discuss a common mathematical problem where the sum of three numbers and their given ratios are provided, and we need to find the value of the second number. We will solve a specific example using algebraic equations and ratios.

Understanding the Problem

Consider a scenario where the sum of three numbers is 80, denoted as x y z 80. Additionally, we know the ratios between these numbers: the ratio of the first number to the second is 2/6 or 1/3, and the ratio of the second number to the third is 6/8 or 3/4. Our goal is to find the exact value of the second number.

Solving the Problem Using Algebraic Equations

Let's denote the first number as x, the second number as y, and the third number as z. From the given ratios, we can express y and z in terms of x:

From the ratio x/y 2/6, we can write y 3x. From the ratio y/z 6/8, we can write z (4/3)y.

Substituting y 3x into the expression for z, we get:

z (4/3)(3x) 4x.

Setting Up the Equation

We now have the following system of equations:

x y z 80. y 3x. z 4x.

Substituting y and z in terms of x into the sum equation, we get:

x 3x 4x 80.

Solving for x

Combining like terms, we have:

8x 80.

Dividing both sides by 8:

x 10.

Finding y and z

Since y 3x and z 4x, we can substitute x 10 to find:

y 3(10) 30.

z 4(10) 40.

Verification

To verify our solution, we can substitute these values back into the original sum equation:

x y z 10 30 40 80.

This confirms that our solution is correct.

Summary

By using algebraic equations and the given ratios, we have successfully determined the values of the three numbers. Specifically, the second number is 30, and the first and third numbers are 10 and 40, respectively. This step-by-step approach provides a clear and systematic method for solving similar problems involving the sum of numbers and their ratios.

Additional Resources

For further practice and a deeper understanding of solving similar problems, consider exploring the following resources:

Solving linear equations Understanding and applying ratios in algebra Problem-solving techniques in mathematics

These resources will enhance your ability to tackle more complex problems involving ratios and sums in mathematics.