Number Problem Solution

This problem asks us to find a specific number that has a unique relationship with two other numbers, 75 and 117. The key phrase here is "as much greater than 75 as it is smaller than 117." This tells us that the distance between the unknown number and 75 is exactly the same as the distance between 117 and the unknown number. In essence, the unknown number is precisely in the middle of 75 and 117.

Understanding the Unknown Number Relationship

Let's represent the unknown number we need to find with a variable, say x. The problem statement gives us two conditions:

• The number x is greater than 75. The difference is expressed as \(x - 75\).
• The number x is smaller than 117. The difference is expressed as \(117 - x\).

The problem states that these two differences are equal. This allows us to set up an algebraic equation to find the value of x.

Setting Up the Number Equation

Based on our understanding, we can form the following equation:

\[x - 75 = 117 - x\]

Solving for the Unknown Number

Now, let's solve this linear equation for x step-by-step:

1. Isolate the variable terms: Add \(x\) to both sides of the equation.

   \[x - 75 + x = 117 - x + x\]

   \[2x - 75 = 117\]

2. Isolate the constant terms: Add 75 to both sides of the equation.

   \[2x - 75 + 75 = 117 + 75\]

   \[2x = 192\]

3. Solve for x: Divide both sides by 2.

   \[\frac{2x}{2} = \frac{192}{2}\]

   \[x = 96\]

So, the unknown number is 96.

Alternative Method: Finding the Midpoint Number

Since the unknown number is equidistant from 75 and 117, it is essentially the midpoint or the average of these two numbers. We can calculate the average by summing the two numbers and dividing by 2.

\[\text{Unknown Number} = \frac{\text{First Number} + \text{Second Number}}{2}\]

Substituting the given values:

\[\text{Unknown Number} = \frac{75 + 117}{2}\]

\[\text{Unknown Number} = \frac{192}{2}\]

\[\text{Unknown Number} = 96\]

Both methods yield the same result, confirming that the number is 96.

Verifying the Number

Let's verify our answer to ensure it satisfies the original condition:

• Is 96 greater than 75? Yes, by \(96 - 75 = 21\).
• Is 96 smaller than 117? Yes, by \(117 - 96 = 21\).

Since the difference (21) is the same in both cases, the number 96 correctly fits the description given in the problem.

The number that is as much greater than 75 as it is smaller than 117 is 96.