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Chapter 3: Problem 21

What is the quotient of \(7 \frac{2}{3} \div 2 \frac{1}{2}\) ? Leave the answer as a reduced improper fraction. Write the correct numerator in the box.

Short Answer

Expert verified
The quotient is \(\frac{46}{15}\). The numerator is 46.

Step by step solution

01

Convert Mixed Numbers to Improper Fractions

First, convert the mixed numbers to improper fractions. For \(7 \frac{2}{3}\), multiply the whole number 7 by the denominator 3, and then add the numerator 2. The result is the new numerator: \(7 \times 3 + 2 = 21 + 2 = 23\). So, \(7 \frac{2}{3} = \frac{23}{3}\).For \(2 \frac{1}{2}\), multiply the whole number 2 by the denominator 2, and then add the numerator 1. The result is the new numerator: \(2 \times 2 + 1 = 4 + 1 = 5\). So, \(2 \frac{1}{2} = \frac{5}{2}\).
02

Write the Division as a Multiplication

Rewrite the division of fractions as the multiplication of the first fraction by the reciprocal of the second fraction. This means we change \(\frac{23}{3} \div \frac{5}{2}\) to \(\frac{23}{3} \times \frac{2}{5}\).
03

Multiply the Fractions

To multiply the fractions, multiply the numerators together and the denominators together:\(\frac{23}{3} \times \frac{2}{5} = \frac{23 \times 2}{3 \times 5} = \frac{46}{15}\).
04

Simplify the Fraction

Check if the fraction can be simplified. The greatest common divisor (GCD) of 46 and 15 is 1, so the fraction \(\frac{46}{15}\) is already in its simplest form. The final answer remains \(\frac{46}{15}\).

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Key Concepts

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

mixed numbers
Mixed numbers are numbers that contain both a whole number and a fraction part. For example, in the number \(7 \frac{2}{3}\), 7 is the whole number and \(\frac{2}{3}\) is the fraction part. Mixed numbers are often used in day-to-day life.

To perform arithmetic operations with mixed numbers, it's easier if we convert them into improper fractions. This simplifies calculations. An improper fraction has a numerator larger than its denominator. To convert a mixed number to an improper fraction, follow these steps:
  • Multiply the whole number by the denominator of the fraction
  • Add the numerator of the fraction to the result
  • This gives you the new numerator, which you place over the original denominator

For example, converting \(7 \frac{2}{3}\) to an improper fraction:

  • Multiply 7 (whole number) by 3 (denominator): 7 脳 3 = 21
  • Add the numerator: 21 + 2 = 23
  • So, \(7 \frac{2}{3}\) is \(\frac{23}{3}\)
improper fractions
Improper fractions have a numerator that is greater than or equal to the denominator. These fractions represent values greater than or equal to 1. In mathematical operations, improper fractions are highly versatile.

Once a mixed number is converted to an improper fraction, arithmetic calculations become straightforward. For example: \(7 \frac{2}{3}\) is converted to \(\frac{23}{3}\), making it easier to handle in addition, subtraction, multiplication, and division.

Here鈥檚 a quick recap:
  • Improper fractions simplify complex mixed numbers
  • They aid in straightforward arithmetic operations
  • They are useful in converting between different types of fractions
reciprocal
A reciprocal of a fraction is simply flipping its numerator and denominator. For instance, the reciprocal of \(\frac{5}{2}\) is \(\frac{2}{5}\).

Reciprocals are especially useful when dividing fractions. When we divide by a fraction, we multiply by its reciprocal. For example:
  • The reciprocal of \(\frac{5}{2}\) is \(\frac{2}{5}\)
  • So, \(\frac{23}{3} \div \frac{5}{2}\) becomes \(\frac{23}{3} \times \frac{2}{5}\)
This transformation simplifies the division process, turning it into a multiplication problem, which is easier to solve.
simplifying fractions
Simplifying fractions means reducing them to their simplest form. A fraction is in its simplest form if the Greatest Common Divisor (GCD) of its numerator and denominator is 1. For example, if we have \(\frac{46}{15}\), we need to check:
  • Find the GCD of 46 and 15, which is 1
  • Since the GCD is 1, \(\frac{46}{15}\) is already in its simplest form
Simplification makes fractions easier to understand and work with. Always ensure your final answer is in simplified form to make sure it鈥檚 fully correct.

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Most popular questions from this chapter

Eli's daughter had a birthday party at Molly Mouse's Pizza Planet. There were two tables of girls. The girls left to play the games and Eli saw that one table had left \(\frac{1}{2}\) of a pie and the other had left \(\frac{2}{3}\) of a pie. Eli added the two and came up with one full pie and one piece left over. What is the smallest number of pieces that each pie was cut into? Write your answer in the box.

$$ \left(-\frac{1}{3}\right)^2 $$ A. \(-\frac{2}{9}\) B. \(-\frac{1}{9}\) C. \(\frac{1}{9}\) D. \(\frac{2}{9}\)

$$ 16^{-\frac{1}{2}} $$ A. 8 B. \(\frac{1}{4}\) C. \(-\frac{1}{4}\) D. -4

Reduce \(\frac{144}{216}\) to its lowest terms. A. \(\frac{18}{27}\) B. \(\frac{2}{3}\) C. \(\frac{36}{54}\) D. \(\frac{72}{108}\)

Tomasz multiplied \(\frac{3}{4}\) by \(1 \frac{1}{5}\) and got \(\frac{3}{4}\) for an answer. Did he do something wrong? A. No, that is the correct answer. B. Yes, he did not change the mixed number to an improper fraction. C. Yes, he forgot to invert the second fraction. D. Yes, he did not add 1 when he converted the mixed number to an improper fraction (he multiplied \(\frac{3}{4}\) by \(\frac{5}{5}\) rather than by \(\frac{6}{5}\) ).
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