Solving Number Problems using Simultaneous Equations

In this article, we will delve into how to solve a specific type of number problem involving the difference and ratio of two whole numbers. We will explore different methods and provide step-by-step solutions to help you understand the concepts better.

Problem Statement

We are given two pieces of information:

The difference between two whole numbers is 66. The ratio of the two numbers is 2:5.

Let's denote these two numbers as x and y, with x y.

Solution Method 1: Using Simultaneous Equations

In this method, we will use the concept of simultaneous equations to solve for x and y.

Express the difference of the two numbers:

y - x 66

Express the ratio of the two numbers:

x/y 2/5

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

y 5x/2

Substitute the expression for y into the first equation:

5x/2 - x 66

Combine the terms and solve for x:

(5x - 2x) / 2 66

3x / 2 66

3x 132

x 44

Substitute x back into the expression for y:

y 5(44)/2 110

Therefore, the two numbers are 44 and 110.

Solution Method 2: Simplifying the Problem

The same problem can also be solved by directly simplifying the ratio and using the difference given.

Simplify the ratio 2:5 to a common factor:

2x and 5x

Express the difference equation:

5x - 2x 66

3x 66

Solve for x:

x 66/3 22

Find y:

y 5x 5(22) 110

Therefore, the two numbers are 22 and 110.

Solution Method 3: Using Ratio and Substitution

Express the two numbers in terms of the ratio 2:5:

x 2A

y 5A

Note that the difference is 66:

5A - 2A 66

3A 66

Solve for A:

A 66/3 22

Find x and y:

x 2(22) 44

y 5(22) 110

Therefore, the two numbers are 44 and 110.

Solution Method 4: Negative Numbers

Consider the possibility of negative numbers:

x -110/3, y -44/3

Note that the difference is positive, but the actual numbers are negative:

110/3 - 44/3 66

-44/3 / -110/3 2/5 (This ratio is still valid with negative numbers)

Therefore, the two numbers can also be -110/3 and -44/3.

Verification

Verify the difference:

110 - 44 66

Verify the ratio:

44/110 2/5

Both conditions are satisfied.

Conclusion

There are multiple methods to solve this class of number problems. The key is to express the given information in mathematical terms and use algebraic techniques to find the solution. Understanding the steps involved in each method will enhance your problem-solving skills in mathematics.

Keywords: number problems, simultaneous equations, ratio and difference