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Chapter 5: Problem 17

Do each calculation by hand, and then check your results with a calculator. Express your answers as fractions. a. \(3-\frac{5}{6}\) b. \(\frac{1}{4}+\frac{5}{12}\) c. \(\frac{3}{4} \cdot \frac{2}{9}\) d. \(\frac{1}{5}+\frac{2}{3}+\frac{3}{4}\)

Short Answer

Expert verified
a. \(\frac{13}{6}\); b. \(\frac{2}{3}\); c. \(\frac{1}{6}\); d. \(\frac{97}{60}\).

Step by step solution

01

Convert Whole Number to Fraction for Part a

First, convert the whole number 3 into a fraction with a denominator of 6. This gives us \( \frac{18}{6} \), because 3 is equivalent to \( \frac{18}{6} \).
02

Subtract the Fractions for Part a

Subtract \( \frac{5}{6} \) from \( \frac{18}{6} \). \( \frac{18}{6} - \frac{5}{6} = \frac{13}{6} \). The result of part a is \( \frac{13}{6} \).
03

Find a Common Denominator for Part b

For \( \frac{1}{4} + \frac{5}{12} \), we need a common denominator. The least common denominator of 4 and 12 is 12.
04

Convert and Add Fractions for Part b

Convert \( \frac{1}{4} \) to \( \frac{3}{12} \) and add \( \frac{5}{12} \): \( \frac{3}{12} + \frac{5}{12} = \frac{8}{12} \). Simplify \( \frac{8}{12} \) to \( \frac{2}{3} \). The result of part b is \( \frac{2}{3} \).
05

Multiply the Fractions for Part c

Multiply \( \frac{3}{4} \) by \( \frac{2}{9} \): \( \frac{3}{4} \times \frac{2}{9} = \frac{6}{36} \). Simplify \( \frac{6}{36} \) to \( \frac{1}{6} \). The result of part c is \( \frac{1}{6} \).
06

Find a Common Denominator for Part d

For \( \frac{1}{5} + \frac{2}{3} + \frac{3}{4} \), find the least common denominator of 5, 3, and 4, which is 60.
07

Convert and Add Fractions for Part d

Convert each fraction: \( \frac{1}{5} = \frac{12}{60}, \frac{2}{3} = \frac{40}{60}, \frac{3}{4} = \frac{45}{60} \), and add them: \( \frac{12}{60} + \frac{40}{60} + \frac{45}{60} = \frac{97}{60} \). The result of part d is \( \frac{97}{60} \).

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Key Concepts

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

Fraction Addition
When you're adding fractions, the first step is to make sure they have the same denominator. This is because only fractions with the same bottom number (denominator) can be directly added together. If they don't match, you'll need to adjust the fractions to achieve a common denominator.
Let's consider the example of adding \(\frac{1}{4}\) and \(\frac{5}{12}\). Here, the least common denominator (LCD) of 4 and 12 is 12.
  • Convert \(\frac{1}{4}\) into \(\frac{3}{12}\), since \(1 \times 3 = 3\) and \(4 \times 3 = 12\).
  • Then, you can add the fractions: \(\frac{3}{12} + \frac{5}{12} = \frac{8}{12}\).
  • Finally, simplify the result by dividing both the numerator and the denominator by their greatest common divisor, which in this case is 4, giving you \(\frac{2}{3}\).

Always check if your answer can be simplified further. Simplification helps in achieving the cleanest form of a fraction.
Fraction Subtraction
Subtracting fractions follows a process similar to addition. The key is ensuring the fractions being subtracted have the same denominator.
For instance, when you face a problem like subtracting \( \frac{5}{6} \) from 3, first convert the whole number into a fraction, such as \(3 = \frac{18}{6} \). This way both fractions share a common denominator.
  • Subtract: \( \frac{18}{6} - \frac{5}{6} = \frac{13}{6} \).

You don't need to simplify \(\frac{13}{6}\) further because it's already in its simplest form. Don't forget that when working with mixed numbers or whole numbers, converting them simplifies the subtraction process.
Fraction Multiplication
Multiplying fractions is straightforward and doesn't require a common denominator. You simply multiply the numerators and then the denominators.
Consider multiplying \( \frac{3}{4} \) by \( \frac{2}{9} \):
  • Multiply the numerators: \(3 \times 2 = 6\).
  • Multiply the denominators: \(4 \times 9 = 36\).
  • Combine them into a new fraction: \( \frac{6}{36} \).
  • Simplify the fraction by dividing the top and bottom by their greatest common divisor, which is 6 here, giving \( \frac{1}{6} \).

This is how you achieve the simplest form of a fraction after multiplication. Always check for simplification as it reduces the fraction to its lowest terms.
Least Common Denominator
Finding a common denominator is crucial for adding or subtracting fractions with different denominators. The least common denominator (LCD) is the smallest number that all denominators can divide into evenly.
Taking the example from the problem \(\frac{1}{5} + \frac{2}{3} + \frac{3}{4}\), we need to find the LCD of 5, 3, and 4.
  • Identify the multiples of each number.
  • The smallest multiple common to 5, 3, and 4 is 60.
  • Convert each fraction: \(\frac{1}{5}=\frac{12}{60}, \frac{2}{3}=\frac{40}{60}, \frac{3}{4}=\frac{45}{60}\).
  • Add them together: \(\frac{12}{60} + \frac{40}{60} + \frac{45}{60} = \frac{97}{60}\).

Use the LCD to handle fractions with diverse denominators, simplifying calculations both for addition and subtraction.

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Most popular questions from this chapter

Consider the inequality \(y<2-0.5 x\). a. Graph the boundary line for the inequality on axes scaled from \(-6\) to 6 on each axis. (a) b. Determine whether each given point satisfies \(y<2-0.5 x\). Plot the point on the graph you drew in \(4 a\). Label the point \(T\) (true) if it is part of the solution or F (false) if it is not part of the solution region. (a) i. \((1,2)\) ii. \((4,0)\) iii. \((2,-3)\) iv. \((-2,-1)\) c. Use your results from \(4 b\) to shade the half-plane that represents the inequality. (a)

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Find three values of the variable that satisfy each inequality. a. \(5+2 a>21\) b. \(7-3 b<28\) c. \(-11.6+2.5 c<8.2\) d. \(4.7-3.25 d>-25.3\)
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