Problem 22 - Algebra and Trigonometry

Question

\(21-26=\) Perform the indicated operations. \(\begin{array}{ll}{\text { (a) } \frac{2}{3}-\frac{3}{5}} & {\text { (b) } 1+\frac{5}{8}-\frac{1}{6}}\end{array}\)

Step-by-Step Solution

Answer

(a) \(\frac{1}{15}\); (b) \(1\frac{11}{24}\).

Step 1: Find a Common Denominator for (a)

For \( \frac{2}{3} - \frac{3}{5} \), we need a common denominator. The least common multiple of 3 and 5 is 15.

Step 2: Convert to Equivalent Fractions for (a)

Convert \( \frac{2}{3} \) to \( \frac{10}{15} \) by multiplying the numerator and denominator by 5. Convert \( \frac{3}{5} \) to \( \frac{9}{15} \) by multiplying the numerator and denominator by 3.

Step 3: Perform the Subtraction for (a)

Subtract the equivalent fractions: \( \frac{10}{15} - \frac{9}{15} = \frac{1}{15} \).

Step 4: Simplify (b) by Adding Whole Number

For \( 1 + \frac{5}{8} - \frac{1}{6} \), convert 1 to \( \frac{8}{8} \) so it can be added to \( \frac{5}{8} \). This results in \( \frac{8}{8} + \frac{5}{8} = \frac{13}{8} \).

Step 5: Find a Common Denominator for (b)

Determine the least common multiple of 8 and 6, which is 24.

Step 6: Convert to Equivalent Fractions for (b)

Convert \( \frac{13}{8} \) to \( \frac{39}{24} \) by multiplying the numerator and denominator by 3. Convert \( \frac{1}{6} \) to \( \frac{4}{24} \) by multiplying the numerator and denominator by 4.

Step 7: Perform the Subtraction for (b)

Subtract the equivalent fractions: \( \frac{39}{24} - \frac{4}{24} = \frac{35}{24} \).

Step 8: Simplify the Result for (b)

Since \( \frac{35}{24} \) is an improper fraction, convert it to a mixed number: \( 1 \frac{11}{24} \).

Key Concepts

Fractions

Fractions are numbers representing parts of a whole. They consist of a numerator—the number of parts we have—and a denominator—the total number of equal parts the whole is divided into. Understanding fractions allows us to comprehend quantities that fall between whole numbers.

• Example: In the fraction \( \frac{3}{4} \), 3 is the numerator and 4 is the denominator.
• Fractions can either be proper (numerator is less than the denominator) or improper (numerator is greater than the denominator).

Recognizing fractions is crucial as they are used extensively across different areas of mathematics.

Common Denominator

A common denominator is a shared multiple of the denominators of two or more fractions. Having the same denominator makes it easier to perform arithmetic operations like addition or subtraction on fractions.

• To find a common denominator, identify the least common multiple (LCM) of the denominators.
• Example: For fractions with denominators of 3 and 5, the LCM is 15.

Once a common denominator is determined, you can convert each fraction to an equivalent fraction with the same denominator, setting the stage for straightforward addition or subtraction.

Improper Fractions

Improper fractions have numerators larger than or equal to their denominators. This type of fraction represents a quantity greater than or equal to one whole.

• Example: \( \frac{35}{24} \) is an improper fraction because 35 is greater than 24.

Improper fractions are often converted to mixed numbers for easier interpretation or comparison. However, in calculations, they can be left as they are if that simplifies the process.

Mixed Numbers

Mixed numbers combine a whole number with a proper fraction, providing an intuitive sense of a number greater than a whole. They offer an easy way to grasp more complex amounts that do not fit neatly into fractions.

• Example: \( 1 \frac{11}{24} \) is a mixed number where 1 is the whole number and \( \frac{11}{24} \) is the fraction.
• To convert an improper fraction into a mixed number, divide the numerator by the denominator.

Understanding both improper fractions and mixed numbers, and how to transition between them, improves fluency in fraction calculations.