Problem 26 - Algebra and Trigonometry

Question

\(21-26=\) Perform the indicated operations. \(\begin{array}{ll}{\text { (a) } \frac{2-\frac{3}{4}}{\frac{1}{2}-\frac{1}{3}}} & {\text { (b) } \frac{\frac{2}{5}+\frac{1}{2}}{\frac{1}{10}+\frac{3}{15}}}\end{array}\)

Step-by-Step Solution

Answer

(a) \(\frac{15}{2}\), (b) 3.

Step 1: Simplifying the Numerator of Expression (a)

For the numerator of expression (a), we have \(2 - \frac{3}{4}\). To subtract these, convert 2 to a fraction with a denominator of 4. Thus, \(2 = \frac{8}{4}\). Now, subtract: \(\frac{8}{4} - \frac{3}{4} = \frac{5}{4}\).

Step 2: Simplifying the Denominator of Expression (a)

For the denominator of expression (a), we have \(\frac{1}{2} - \frac{1}{3}\). Find a common denominator, which is 6. Convert the fractions: \(\frac{1}{2} = \frac{3}{6}\) and \(\frac{1}{3} = \frac{2}{6}\). Now, subtract: \(\frac{3}{6} - \frac{2}{6} = \frac{1}{6}\).

Step 3: Divide the Simplified Fractions for Expression (a)

Now, divide the results from Step 1 and Step 2: \(\frac{\frac{5}{4}}{\frac{1}{6}}\). Dividing by a fraction is the same as multiplying by its reciprocal, so multiply: \(\frac{5}{4} \times \frac{6}{1} = \frac{30}{4}\). Simplify the result: \(\frac{30}{4} = \frac{15}{2}\).

Step 4: Simplifying the Numerator of Expression (b)

For the numerator of expression (b), add \(\frac{2}{5} + \frac{1}{2}\). Find a common denominator, which is 10. Convert the fractions: \(\frac{2}{5} = \frac{4}{10}\) and \(\frac{1}{2} = \frac{5}{10}\). Now, add: \(\frac{4}{10} + \frac{5}{10} = \frac{9}{10}\).

Step 5: Simplifying the Denominator of Expression (b)

For the denominator of expression (b), add \(\frac{1}{10} + \frac{3}{15}\). Find a common denominator, which is 30. Convert the fractions: \(\frac{1}{10} = \frac{3}{30}\) and \(\frac{3}{15} = \frac{6}{30}\). Now, add: \(\frac{3}{30} + \frac{6}{30} = \frac{9}{30}\). Simplify the result: \(\frac{9}{30} = \frac{3}{10}\).

Step 6: Divide the Simplified Fractions for Expression (b)

Now, divide the results from Step 4 and Step 5: \(\frac{\frac{9}{10}}{\frac{3}{10}}\). Since both fractions have the same denominator, this is \(\frac{9}{3} = 3\).

Key Concepts

Fraction Addition

Adding fractions involves combining two or more fractions together to get a single fraction. To do this straightforwardly, it's crucial that the fractions share the same denominator. This matching denominator is often referred to as a common denominator. For example:

• Let's say you want to add \(\frac{2}{5} + \frac{1}{2}\).
• First, find a common denominator. Here, the lowest common denominator for 5 and 2 is 10.
• Convert the fractions to equivalent fractions with denominator 10: \(\frac{2}{5} = \frac{4}{10}\) and \(\frac{1}{2} = \frac{5}{10}\).
• Then, just add the numerators: \(\frac{4}{10} + \frac{5}{10} = \frac{9}{10}\).

This process makes adding fractions easy once you have a common footing with the denominators.

Fraction Subtraction

Subtracting fractions is like adding them, only you're taking one fraction away from another. The critical part is ensuring fractions have a common denominator before subtracting the numerators. Consider this example:

• To subtract \(2 - \frac{3}{4}\), we first convert 2 into a fraction by representing it as \(\frac{8}{4}\) to have a common denominator of 4.
• Now, subtract the numerators: \(\frac{8}{4} - \frac{3}{4} = \frac{5}{4}\).

Rearanging a whole number to a fraction, while seeming complex, simplifies subtraction when you take fractions away from whole numbers. Always simplify the result if possible.

Common Denominators

Finding a common denominator is essential in both adding and subtracting fractions. A common denominator is a shared multiple of the denominators of the two fractions. Let's revisit finding one:

• For the fractions \(\frac{1}{2}\) and \(\frac{1}{3}\), the least common denominator is 6.
• Convert each to an equivalent fraction: \(\frac{1}{2} = \frac{3}{6}\) and \(\frac{1}{3} = \frac{2}{6}\).
• Now, operations like subtraction or addition become straightforward: \(\frac{3}{6} - \frac{2}{6} = \frac{1}{6}\).

By using a common denominator, fractions become compatible, and easy arithmetic can be performed.

Fraction Division

Dividing fractions might initially sound complicated, but it's a breeze when you know one simple trick: multiplying by the reciprocal. The reciprocal of a fraction reverses the numerator and the denominator. Here’s how it works:

• For the operation \(\frac{\frac{5}{4}}{\frac{1}{6}}\), take the reciprocal of \(\frac{1}{6}\) to get \(\frac{6}{1}\).
• Then simply multiply: \(\frac{5}{4} \times \frac{6}{1} = \frac{30}{4}\).
• Simplify if possible: \(\frac{30}{4} = \frac{15}{2}\).

Remember, dividing is just flipping the fraction around and multiplying. This method makes fraction division a simple maneuver of multiplication.