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Chapter 0: Problem 22

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

Short Answer

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

Step by step solution

01

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.
02

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.
03

Perform the Subtraction for (a)

Subtract the equivalent fractions: \( \frac{10}{15} - \frac{9}{15} = \frac{1}{15} \).
04

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 \( \frac{8}{8} + \frac{5}{8} = \frac{13}{8} \).
05

Find a Common Denominator for (b)

Determine the least common multiple of 8 and 6, which is 24.
06

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.
07

Perform the Subtraction for (b)

Subtract the equivalent fractions: \( \frac{39}{24} - \frac{4}{24} = \frac{35}{24} \).
08

Simplify the Result for (b)

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fractions
Fractions are numbers representing parts of a whole. They consist of a numerator鈥攖he number of parts we have鈥攁nd a denominator鈥攖he 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.

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Most popular questions from this chapter

Easy Powers That Look Hard Calculate these expressions in your head. Use the Laws of Exponents to help you. $$ \begin{array}{ll}{\text { (a) } \frac{18^{5}}{9^{5}}} & {\text { (b) } 20^{6} \cdot(0.5)^{6}}\end{array} $$

\(65-70\) m Simplify the fractional expression. (Expressions like these arise in calculus.) $$ \sqrt{1+\left(x^{3}-\frac{1}{4 x^{3}}\right)^{2}} $$

Use scientific notation, the Laws of Exponents, and a calculator to perform the indicated operations. State your answer correct to the number of significant digits indicated by the given data. $$ \left(7.2 \times 10^{-9}\right)\left(1.806 \times 10^{-12}\right) $$

Simplify the expression and eliminate any negative exponent(s). $$ \frac{\left(x y^{2} z^{3}\right)^{4}}{\left(x^{3} y^{2} z\right)^{3}} $$

Complete the following tables. What happens to the \(n\) th root of 2 as \(n\) gets large? What about the \(n\) th root of \(\frac{1}{2} ?\) \(\begin{array}{|c|c|}\hline n & {2^{1 / n}} \\ \hline 1 & {} \\ {2} & {} \\\ {5} \\ {10} \\ {100} & {} \\ \hline\end{array}\) \(\begin{array}{|c|c|}\hline n & {\left(\frac{1}{2}\right)^{1 / n}} \\ \hline 1 & {} \\ {2} & {} \\ {5} & {} \\ {10} \\ {100} & {} \\ \hline\end{array}\) Construct a similar table for \(n^{1 / n} .\) What happens to the \(n\) th root of \(n\) as \(n\) gets large?
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