Solving the Cubic Expression Problem

The problem asks us to evaluate a given mathematical expression involving powers of decimal numbers and find which option matches the result. The expression is: \(\frac{{{{\left( {0.2} \right)}^3} - {{\left( {0.1} \right)}^3}}}{{{{\left( {0.2\; + \;0.1} \right)}^2}}}\)

We need to simplify this expression step by step.

Step-by-Step Calculation

Analyzing the Expression

Let's look at the structure of the expression. The numerator is the difference of cubes of 0.2 and 0.1. The denominator is the square of the sum of 0.2 and 0.1.

We can identify this pattern by letting \(a = 0.2\) and \(b = 0.1\). The expression becomes:

\[\frac{{a^3 - b^3}}{{(a + b)^2}}\]

We know the algebraic identity for the difference of cubes: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).

Calculating the Numerator

The numerator is \({{\left( {0.2} \right)}^3} - {{\left( {0.1} \right)}^3}\). Using the identity with \(a = 0.2\) and \(b = 0.1\):

• \(a - b = 0.2 - 0.1 = 0.1\)
• \(a^2 = (0.2)^2 = 0.2 \times 0.2 = 0.04\)
• \(ab = (0.2)(0.1) = 0.02\)
• \(b^2 = (0.1)^2 = 0.1 \times 0.1 = 0.01\)

So, the numerator is:

\(a^3 - b^3 = (a - b)(a^2 + ab + b^2) = (0.1)(0.04 + 0.02 + 0.01)\)

\(= (0.1)(0.07)\)

\(= 0.007\)

Calculating the Denominator

The denominator is \({{\left( {0.2\; + \;0.1} \right)}^2}\). First, calculate the sum inside the parenthesis:

• \(0.2 + 0.1 = 0.3\)

Now, square the sum:

\((0.3)^2 = 0.3 \times 0.3 = 0.09\)

So, the denominator is \(0.09\).

Final Calculation

Now we divide the numerator by the denominator:

\[\frac{\text{Numerator}}{\text{Denominator}} = \frac{0.007}{0.09}\]

To simplify the division of decimals, we can multiply both the numerator and the denominator by 1000 to remove the decimal places. This is equivalent to moving the decimal point 3 places to the right in both numbers (to make the numerator a whole number).

\[\frac{0.007 \times 1000}{0.09 \times 1000} = \frac{7}{90}\]

The value of the expression is \(\frac{7}{90}\).

Comparing with Options

Let's compare our result with the given options:

1. \(\frac{3}{{40}}\)
2. \(\frac{1}{{18}}\)
3. \(-\frac{7}{{90}}\)
4. \(\frac{7}{{90}}\)

Our calculated value, \(\frac{7}{90}\), matches option 4.

Revision Table: Key Concepts

Additional Information: Related Math Concepts

• Other Algebraic Identities: Besides \(a^3 - b^3\), other useful identities include \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\), \((a+b)^2 = a^2 + 2ab + b^2\), and \((a-b)^2 = a^2 - 2ab + b^2\). Recognizing these patterns can simplify complex expressions.
• Working with Decimals and Fractions: Often, converting decimals to fractions can make calculations clearer, especially when dealing with exact values. For example, \(0.2 = \frac{2}{10} = \frac{1}{5}\) and \(0.1 = \frac{1}{10}\). Our final division \(\frac{0.007}{0.09}\) is equivalent to \(\frac{7/1000}{9/100} = \frac{7}{1000} \times \frac{100}{9} = \frac{7 \times 100}{1000 \times 9} = \frac{7 \times 1}{10 \times 9} = \frac{7}{90}\).
• Quantitative Aptitude: Questions like this are common in quantitative aptitude tests, requiring both algebraic understanding and arithmetic skills with decimals and fractions. Practicing such problems helps in improving speed and accuracy.