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Chapter 1: Problem 121

Write a numerical expression for each phrase, and simplify the expression. The sum of \(\frac{1}{2}\) and \(\frac{5}{8},\) times the difference of \(\frac{3}{5}\) and \(\frac{1}{3}\)

Short Answer

Expert verified
\(\frac{3}{10}\)

Step by step solution

01

Write the numerical expression

Start by translating the given phrase into a numerical expression. The phrase is 'The sum of \(\frac{1}{2}\) and \(\frac{5}{8}\), times the difference of \(\frac{3}{5}\) and \(\frac{1}{3}\).' This gives us the expression: \((\frac{1}{2} + \frac{5}{8}) \times (\frac{3}{5} - \frac{1}{3})\).
02

Simplify the sum

Simplify \(\frac{1}{2} + \frac{5}{8}\). To do this, find a common denominator, which in this case is 8. Convert \(\frac{1}{2}\) to \(\frac{4}{8}\). Now, add the fractions: \(\frac{4}{8} + \frac{5}{8} = \frac{9}{8}\).
03

Simplify the difference

Simplify \(\frac{3}{5} - \frac{1}{3}\). Find a common denominator, which is 15. Convert \(\frac{3}{5}\) to \(\frac{9}{15}\) and \(\frac{1}{3}\) to \(\frac{5}{15}\). Now, subtract the fractions: \(\frac{9}{15} - \frac{5}{15} = \frac{4}{15}\).
04

Multiply the results

Multiply the results from Step 2 and Step 3. \((\frac{9}{8}) \times (\frac{4}{15})\). Multiply the numerators and the denominators: \(\frac{9 \times 4}{8 \times 15} = \frac{36}{120}\).
05

Simplify the final fraction

Simplify \(\frac{36}{120}\). Find the greatest common divisor of 36 and 120, which is 12. Divide both the numerator and the denominator by 12: \(\frac{36 \div 12}{120 \div 12} = \frac{3}{10}\).

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Key Concepts

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

Fractions
Fractions represent parts of a whole. They consist of a numerator (top number) and a denominator (bottom number). For example, \( \frac{1}{2}\) means 1 part out of 2 total parts. Knowing how to handle fractions is crucial for solving equations involving them.
When working with fractions, it is important to understand a few key principles:
  • Finding a Common Denominator: This is essential when adding or subtracting fractions. For instance, when adding \( \frac{1}{2}\) and \( \frac{5}{8}\), the common denominator is 8.
  • Conversion: You often need to convert fractions to have the same denominator. In our example, \( \frac{1}{2}\) is converted to \( \frac{4}{8}\).
  • Adding/Subtracting: Once the denominators are the same, add or subtract the numerators. For example, \( \frac{4}{8} + \frac{5}{8} = \frac{9}{8}\).
Arithmetic Operations
Arithmetic operations include addition, subtraction, multiplication, and division. In our problem, we need to perform both addition and subtraction before multiplying:
  • Addition: To add fractions, ensure they have a common denominator. As we did with \( \frac{1}{2}\) and \( \frac{5}{8}\), we first converted \( \frac{1}{2}\) to \( \frac{4}{8}\).
  • Subtraction: Similar to addition, we need a common denominator. For \( \frac{3}{5}\) and \( \frac{1}{3}\), the common denominator is 15. We converted \( \frac{3}{5}\) to \( \frac{9}{15}\) and \( \frac{1}{3}\) to \( \frac{5}{15}\), then subtracted to get \( \frac{4}{15}\).
  • Multiplication: To multiply fractions, multiply the numerators and the denominators. For instance, \( \frac{9}{8} \times \frac{4}{15} = \frac{36}{120}\).
These basic arithmetic operations are fundamental for simplifying and solving expressions.
Simplification
Simplification makes expressions easier to understand and work with. Here, we simplified the final fraction:
Simplification involves several steps:
  • Identify Common Factors: Look for the greatest common divisor (GCD). For \( \frac{36}{120}\), the GCD is 12.
  • Reduce the Fraction: Divide both the numerator and denominator by the GCD. For \( \frac{36}{120}\), dividing by 12 gives \( \frac{3}{10}\).
This step ensures the fraction is in its simplest form, making it easier to understand and use in further calculations.

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Most popular questions from this chapter

Solve each problem. The surface, or rim, of a canyon is at altitude \(0 .\) On a hike down into the canyon, a party of hikers stops for a rest at \(130 \mathrm{~m}\) below the surface. The hikers then descend another \(54 \mathrm{~m}\). Write the new altitude as a signed number.

Solve each problem. The lowest temperature ever recorded in Arkansas was \(-29^{\circ} \mathrm{F}\). The highest temperature ever recorded there was \(149^{\circ} \mathrm{F}\) more than the lowest. What was this highest temperature? (Data from National Climatic Data Center.)

Write each expression without parentheses. $$ -(-z+5 w-9 y) $$

Write each expression without parentheses. $$ -(-5 c-4 d) $$

Write a numerical expression for each phrase, and simplify the expression. The product of -9 and \(2,\) added to 9
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