How Can We Divide 100 into Two Numbers So That Their Difference is 1?

Looking for a way to divide 100 into two numbers with a difference of 1? This article explores the mathematical method to solve this problem using algebraic equations. We'll delve into the steps, analyze the solutions, and discuss the implications of using fractions.

Understanding the Problem

The problem at hand is to divide the number 100 into two parts such that the difference between these two parts is exactly 1. This can be written as:

1. Sum of the two numbers: x y 100

2. Difference between the two numbers: x - y 1

Solving the Equations

To find the values of x and y, we can solve these equations simultaneously. Let's start by expressing one variable in terms of the other.

Step 1: Express x in terms of y

From the second equation, we have:

x - y 1

x y 1

Step 2: Substitute this expression into the first equation

Now, substitute x y 1 into the first equation:

(y 1) y 100

2y 1 100

Solving for y:

2y 99

y 49.5

Step 3: Find the value of x

Substitute y 49.5 back into the equation x y 1:

x 49.5 1

x 50.5

Thus, the two numbers are:

x 50.5

y 49.5

The sum of these two numbers is indeed 100, and their difference is 1 as required.

Alternative Methods

Let's explore some alternative methods to solve this problem.

Method 1: Direct Subtraction

A direct subtraction method can also provide the solution:

To divide 100 into two numbers such that one is 0.5 bigger than the other:

100 - 0.5 49.5

49.5 0.5 50.5

So the two numbers are 49.5 and 50.5.

Method 2: Quadratic Equation Approach

Another way to solve the problem is using a quadratic equation. Let's consider:

xy 100

x - y 1

Expressing x in terms of y:

x y 1

Substitute x y 1 into the first equation:

(y 1)y 100

y^2 y - 100 0

Solving the quadratic equation using the quadratic formula a 1, b 1, c -100:

y frac{{-1 pm sqrt{{1^2 - 4 cdot 1 cdot (-100)}}}}{2 cdot 1}

y frac{{-1 pm sqrt{{1 400}}}}{2}

y frac{{-1 pm sqrt{{401}}}}{2}

y frac{{-1 pm 20.025}}{2}

This gives us two values:

y_1 frac{{19.025}}{2} approx 9.5125

y_2 frac{{-21.025}}{2} approx -10.5125 (not applicable as y must be positive)

Thus, y 9.5125 and x y 1 10.5125.

Implications of Using Fractions

It is important to note that in this context, using fractions is necessary. When dealing with the sum and difference of two numbers being integers, one of the numbers must be odd and the other even for their difference to be 1. Since 100 is an even number, the numbers must include a fractional part. Thus, 49.5 and 50.5 are the correct solution.

Partitioning 100 into two numbers with a difference of 1 using an even number like 00 and 0 would not be mathematically correct, as it would not satisfy the conditions given.

Conclusion

Dividing 100 into two numbers with a difference of 1 requires the use of fractions. The solution 49.5 and 50.5 satisfies both conditions: their sum is 100, and their difference is 1.

Further Reading

Explore related topics such as number partitioning, arithmetic division, and the use of quadratic equations in solving mathematical problems.