Finding the Numbers Given Their Ratio and Sum

The problem of finding numbers given their ratio and the sum is a common task in mathematical problem-solving. In this article, we will explore the solution to the problem where two numbers are in the ratio 3:7 and their sum is 410. Let's dive into the solution step by step.

Step-by-Step Solution

Let's denote the two numbers as x and y, where x is the smaller number and y is the larger number. According to the ratio, we have:

x : y 3 : 7

To find the numbers, we can use the following approach:

Define the relationship between the numbers using the ratio:

x : y 3 : 7

Thus, we can express y in terms of x:

y (7/3) * x

Sum of the numbers is given as 410:

x y 410

Substitute the expression for y from the ratio into the sum equation:

x (7/3) * x 410

Simplify the equation:

(3/3) * x (7/3) * x 410

(10/3) * x 410

Solve for x by multiplying both sides by 3/10:

x (410 * 3) / 10

x 123

Now, calculate y using the relationship y (7/3) * x:

y (7/3) * 123

y 287

Verification

Let's verify the obtained numbers:

The two numbers are 123 and 287.

Their sum is:

123 287 410

This matches the given sum.

Alternative Methods

There are several alternative methods to solve this problem:

Using the least common denominator (LCD) method:

3/10 * 410 123

7/10 * 410 287

Thus, the numbers are 123 and 287.

Using algebraic equations:

Express the two numbers as multiples of a common factor x:

3x 7x 410

Solve for x:

1 410

x 41

Thus, the numbers are:

3x 3 * 41 123

7x 7 * 41 287

Conclusion

We have solved the problem where two numbers are in the ratio 3:7 and their sum is 410. The two numbers are 123 and 287. This problem demonstrates the application of ratios and algebraic methods in solving mathematical problems.

If you need further assistance or more detailed explanations, please feel free to ask. Happy problem-solving!