The value of $\frac{{0.0203 \times 2.92}}{{0.7 \times 0.0365 \times 2.9}} \div \frac{{{{\left( {12.12} \right)}^2} - {{\left( {8.12} \right)}^2}}}{{{{\left( {0.25} \right)}^2} + \left( {0.25} \right)\left( {19.99} \right)}}$ is:

Question

The value of $\frac{{0.0203 \times 2.92}}{{0.7 \times 0.0365 \times 2.9}} \div \frac{{{{\left( {12.12} \right)}^2} - {{\left( {8.12} \right)}^2}}}{{{{\left( {0.25} \right)}^2} + \left( {0.25} \right)\left( {19.99} \right)}}$  is:

Prepp Team · Accepted Answer

Evaluate Complex Mathematical Expression Step by Step

The problem asks us to evaluate a complex mathematical expression involving fractions, decimals, powers, multiplication, and division. The expression is given as:

$\frac{{0.0203 \times 2.92}}{{0.7 \times 0.0365 \times 2.9}} \div \frac{{{{\left( {12.12} \right)}^2} - {{\left( {8.12} \right)}^2}}}{{{{\left( {0.25} \right)}^2} + \left( {0.25} \right)\left( {19.99} \right)}}$

To solve this, we will evaluate the two fractions separately and then perform the division.

Calculating the First Fraction Value

The first fraction is $\frac{{0.0203 \times 2.92}}{{0.7 \times 0.0365 \times 2.9}}$.

Let's analyze the numbers in the numerator and the denominator:

Numerator: $0.0203$ and $2.92$.
Denominator: $0.7$, $0.0365$, and $2.9$.

We can rewrite these decimals as fractions or look for common factors. Let's try to simplify by converting to integers multiplied by powers of 10:

$0.0203 = 203 \times 10^{-4}$
$2.92 = 292 \times 10^{-2}$
$0.7 = 7 \times 10^{-1}$
$0.0365 = 365 \times 10^{-4}$
$2.9 = 29 \times 10^{-1}$

Substitute these into the first fraction:

$\frac{{(203 \times 10^{-4}) \times (292 \times 10^{-2})}}{{(7 \times 10^{-1}) \times (365 \times 10^{-4}) \times (29 \times 10^{-1})}}$

Combine powers of 10 in numerator and denominator:

$\frac{{203 \times 292 \times 10^{-6}}}{{7 \times 365 \times 29 \times 10^{-6}}}$

Cancel the $10^{-6}$ terms:

$\frac{{203 \times 292}}{{7 \times 365 \times 29}}$

Now, let's find common factors among the integers:

$203 = 7 \times 29$
$292 = 4 \times 73$
$365 = 5 \times 73$

Substitute these factors into the fraction:

$\frac{{(7 \times 29) \times (4 \times 73)}}{{7 \times (5 \times 73) \times 29}}$

Cancel the common factors $7$, $29$, and $73$:

$\frac{{\cancel{7} \times \cancel{29} \times 4 \times \cancel{73}}}{{\cancel{7} \times 5 \times \cancel{73} \times \cancel{29}}}$

The first fraction simplifies to:

$\frac{4}{5}$

Calculating the Second Fraction Value Using Algebraic Identity

The second fraction is $\frac{{{{\left( {12.12} \right)}^2} - {{\left( {8.12} \right)}^2}}}{{{{\left( {0.25} \right)}^2} + \left( {0.25} \right)\left( {19.99} \right)}}$.

Let's evaluate the numerator using the algebraic identity $a^2 - b^2 = (a-b)(a+b)$.

Here, $a = 12.12$ and $b = 8.12$.

Numerator $= (12.12 - 8.12)(12.12 + 8.12)$

$12.12 - 8.12 = 4$
$12.12 + 8.12 = 20.24$

Numerator $= 4 \times 20.24 = 80.96$.

Now, let's evaluate the denominator. We can factor out $0.25$.

Denominator $= (0.25)^2 + (0.25)(19.99) = 0.25 \times (0.25 + 19.99)$

$0.25 + 19.99 = 20.24$

Denominator $= 0.25 \times 20.24$.

Now substitute the simplified numerator and denominator back into the second fraction:

$\frac{80.96}{0.25 \times 20.24}$

Notice that $80.96 = 4 \times 20.24$. Substitute this into the fraction:

$\frac{4 \times 20.24}{0.25 \times 20.24}$

Cancel the common term $20.24$:

$\frac{4}{0.25}$

Since $0.25 = \frac{1}{4}$, we have:

$\frac{4}{1/4} = 4 \times 4 = 16$

The second fraction simplifies to $16$.

Performing the Final Division of Fractions

The original expression is the first fraction divided by the second fraction:

$\text{First Fraction} \div \text{Second Fraction}$

Substitute the simplified values:

$\frac{4}{5} \div 16$

Dividing by a number is the same as multiplying by its reciprocal. The reciprocal of $16$ is $\frac{1}{16}$.

$\frac{4}{5} \times \frac{1}{16}$

Multiply the numerators and the denominators:

$\frac{4 \times 1}{5 \times 16} = \frac{4}{80}$

Simplify the resulting fraction by dividing the numerator and denominator by their greatest common divisor, which is 4:

$\frac{4 \div 4}{80 \div 4} = \frac{1}{20}$

Convert the fraction $\frac{1}{20}$ to a decimal:

$\frac{1}{20} = \frac{1 \times 5}{20 \times 5} = \frac{5}{100} = 0.05$

Thus, the value of the given expression is $0.05$.

Revision Table: Key Mathematical Concepts

Concept	Description	Application in Problem
Decimal Operations	Performing arithmetic operations (multiplication, division) with decimal numbers.	Used throughout the calculation of both fractions.
Scientific Notation	Writing numbers as a product of a number between 1 and 10 and a power of 10.	Used to simplify the first fraction by managing powers of 10.
Algebraic Identity: $a^2 - b^2$	The difference of squares identity: $a^2 - b^2 = (a-b)(a+b)$.	Used to simplify the numerator of the second fraction efficiently.
Factoring Expressions	Taking out a common factor from terms in an expression.	Used to simplify the denominator of the second fraction by factoring out $0.25$.
Fraction Division	Dividing by a fraction (or a whole number) is equivalent to multiplying by its reciprocal. $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$.	Used for the final step to combine the results of the two fractions.

Additional Information on Expression Simplification

When faced with complex mathematical expressions, breaking them down into smaller, manageable parts is a crucial strategy. In this problem, evaluating the two main fractions separately made the calculation much simpler.

Recognizing patterns and applying mathematical identities, like the difference of squares $a^2 - b^2 = (a-b)(a+b)$, can significantly reduce computation time and effort. Similarly, factoring common terms, as done in the denominator of the second fraction, helps simplify expressions before performing calculations.

Working with decimals can sometimes be tricky. Converting them to fractions or using scientific notation can help in managing the magnitude of numbers and identifying common factors for simplification, especially in fractions involving many decimal numbers.

Always double-check calculations, particularly when dealing with multiple steps and different types of operations. Simplifying fractions to their lowest terms at intermediate steps can also prevent large number calculations.