Solving the Problem: Difference and Ratio of Two Numbers

In this article, we will explore the process of solving a problem involving the difference and ratio of two numbers. The problem is as follows:

The difference between two numbers is 8 and their ratio is 1:5. Find the two numbers and identify the larger one.

Problem Setup

Let’s denote the smaller number by x and the larger number by y. According to the problem, we have the following two equations:

The difference between the two numbers is 8: The ratio of the two numbers is 1:5.

Mathematically, this can be written as:

y - x 8 x/y 1/5

Solving the Equations

Let’s start by solving these equations step by step.

Expressing One Variable in Terms of the Other

From the second equation, we can express x in terms of y as:

x (1/5) * y

Substituting and Solving

Next, we substitute the expression for x into the first equation:

y - (1/5) * y 8

Combining the terms on the left side:

(4/5) * y 8

To solve for y, we multiply both sides by the reciprocal of (4/5):

y 8 * (5/4) 10

Finding the Other Variable

We now know that the larger number y is 10. To find the smaller number x, we substitute y 10 into the expression for x derived earlier:

x (1/5) * 10 2

Thus, the two numbers are 2 and 10, and the larger number is 10.

Verification

To verify our solution, let’s check the given conditions:

The difference between 10 and 2 is 8. The ratio of 10 to 2 is 1:5.

Our solution satisfies both conditions.

Alternative Approaches

There are multiple methods to solve this problem. Here are a few more approaches:

Approach 1

If the numbers 5 and 1 are given, their difference is 4. Doubling these numbers to get the required difference of 8 results in the numbers 10 and 2.

Approach 2

Let x and y be the two numbers.

The difference between the two numbers is 8: The ratio of the two numbers is 5:1.

Mathematically:

x - y 8 x/y 5

From the second equation, we can express x in terms of y:

x 5y

Substituting x 5y into the first equation:

5y - y 8

Combining the terms:

4y 8

Solving for y gives:

y 2

Substituting y 2 into x 5y gives:

x 5 * 2 10

Thus, the two numbers are 10 and 2, with 10 being the larger number.

Final Answer

The larger number in the problem is 10.