Finding the Number Equally Distant from 50 and 84

The problem asks us to find a number that is the same distance away from 50 as it is from 84. Let's call the unknown number \(x\).

According to the question:

• The number is as much greater than 50: This means the difference between the number and 50, which can be written as \(x - 50\).
• The number is as much lesser than 84: This means the difference between 84 and the number, which can be written as \(84 - x\).

The phrase "as much greater than 50 as it is lesser than 84" tells us that these two differences are equal. So, we can set up an equation:

\[x - 50 = 84 - x\]

Now, we need to solve this equation for \(x\). To do this, we can gather the \(x\) terms on one side and the constant terms on the other side.

Add \(x\) to both sides of the equation:

\[x - 50 + x = 84 - x + x\] \[2x - 50 = 84\]

Add 50 to both sides of the equation:

\[2x - 50 + 50 = 84 + 50\] \[2x = 134\]

Finally, divide both sides by 2 to find the value of \(x\):

\[\frac{2x}{2} = \frac{134}{2}\] \[x = 67\]

So, the number is 67.

We can check this by verifying if 67 is the same distance from 50 and 84:

• Distance from 50: \(67 - 50 = 17\)
• Distance from 84: \(84 - 67 = 17\)

Since \(17 = 17\), our answer is correct. The number is indeed 67.

Revision Table: Key Concepts

Additional Information: Finding the Midpoint

Another way to think about a number that is "as much greater than A as it is lesser than B" is that the number is exactly in the middle of A and B. In other words, the number is the average (or midpoint) of A and B.

The formula for the average of two numbers, A and B, is:

\[\text{Average} = \frac{A + B}{2}\]

In this problem, A = 50 and B = 84. Let's use the average formula to find the number:

\[\text{Number} = \frac{50 + 84}{2}\] \[\text{Number} = \frac{134}{2}\] \[\text{Number} = 67\]

This method gives the same result and provides a quicker way to solve this specific type of problem.