Calculating (0.1² - 0.025²) ÷ (0.1 - 0.025)

The problem asks us to evaluate the expression: \((0.1^2 - 0.025^2) \div (0.1 - 0.025)\). This expression involves squares and division of decimal numbers.

Let's analyze the structure of the expression. The numerator is in the form of a difference of squares, \(a^2 - b^2\), where \(a = 0.1\) and \(b = 0.025\). The denominator is the difference of the same two numbers, \(a - b\).

We can use the algebraic identity for the difference of squares, which states:

\(a^2 - b^2 = (a - b)(a + b)\)

Using this identity, we can rewrite the numerator of the given expression:

\(0.1^2 - 0.025^2 = (0.1 - 0.025)(0.1 + 0.025)\)

Now substitute this back into the original expression:

\(\frac{0.1^2 - 0.025^2}{0.1 - 0.025} = \frac{(0.1 - 0.025)(0.1 + 0.025)}{0.1 - 0.025}\)

Assuming that the denominator is not zero (which is true since \(0.1 - 0.025 = 0.075 \neq 0\)), we can cancel the term \((0.1 - 0.025)\) from both the numerator and the denominator:

\(\frac{\cancel{(0.1 - 0.025)}(0.1 + 0.025)}{\cancel{(0.1 - 0.025)}} = 0.1 + 0.025\)

Now, we just need to perform the addition:

\(0.1 + 0.025\)

To add these decimal numbers, we can align the decimal points:

\(\begin{array}{c} \phantom{+} 0.100 \\ + 0.025 \\ \hline \phantom{+} 0.125 \end{array}\)

So, the result of the expression is \(0.125\).

Let's compare this result with the given options:

• Option 1: 0.325
• Option 2: 0.125
• Option 3: 0.625
• Option 4: 0.25

The calculated value \(0.125\) matches Option 2.

Step-by-Step Calculation Summary

1. Identify the structure of the expression as a difference of squares divided by the difference of the base numbers.
2. Recall or apply the difference of squares formula: \(a^2 - b^2 = (a - b)(a + b)\).
3. Substitute the numbers into the formula: \(0.1^2 - 0.025^2 = (0.1 - 0.025)(0.1 + 0.025)\).
4. Rewrite the original expression using the factored numerator: \(\frac{(0.1 - 0.025)(0.1 + 0.025)}{0.1 - 0.025}\).
5. Cancel out the common term \((0.1 - 0.025)\) from the numerator and the denominator.
6. The expression simplifies to \(0.1 + 0.025\).
7. Perform the addition: \(0.1 + 0.025 = 0.125\).
8. Match the result with the provided options.

Revision Table: Key Concepts

Additional Information: Understanding Difference of Squares

The difference of squares identity, \(a^2 - b^2 = (a - b)(a + b)\), is a fundamental concept in algebra. It is useful for factoring expressions, simplifying fractions like in this problem, and solving equations.

Let's see why the formula \(a^2 - b^2 = (a - b)(a + b)\) works:

Consider the product \((a - b)(a + b)\). Using the distributive property (or FOIL method):

• First terms: \(a \times a = a^2\)
• Outer terms: \(a \times b = ab\)
• Inner terms: \(-b \times a = -ab\)
• Last terms: \(-b \times b = -b^2\)

Adding these terms together:

\((a - b)(a + b) = a^2 + ab - ab - b^2\)

The terms \(+ab\) and \(-ab\) cancel each other out:

\((a - b)(a + b) = a^2 - b^2\)

This confirms the identity. In our problem, recognizing this pattern allowed us to simplify a complex-looking division into a simple addition.