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Chapter 3: Problem 22

Find the following quotients. $$12 \div 3 \frac{1}{6}$$

Short Answer

Expert verified
The quotient of \( 12 \div 3 \frac{1}{6} \) is \( \frac{72}{19} \).

Step by step solution

01

Convert Mixed Number to Improper Fraction

The divisor in the expression is a mixed number: \[ 3 \frac{1}{6} \]First, convert it into an improper fraction. Multiply the whole number (3) by the denominator (6) and add the numerator (1):\[ 3 \times 6 + 1 = 19 \]So, \( 3 \frac{1}{6} \) becomes \( \frac{19}{6} \).
02

Divide by a Fraction

When dividing by a fraction, instead multiply by its reciprocal. The reciprocal of \( \frac{19}{6} \) is \( \frac{6}{19} \). Therefore, \( 12 \div \frac{19}{6} \) becomes \( 12 \times \frac{6}{19} \).
03

Multiply and Simplify

Multiply the whole number by the fraction:\[ 12 \times \frac{6}{19} = \frac{12 \times 6}{19} = \frac{72}{19} \]This fraction cannot be simplified further as 72 and 19 have no common factors other than 1.

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Key Concepts

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

Mixed Numbers
A mixed number is a combination of a whole number and a proper fraction. Proper fractions have numerators smaller than their denominators. In the example of \(3 \frac{1}{6}\), 3 is the whole number, and \(\frac{1}{6}\) is the fraction. Mixed numbers are often used in everyday math to express quantities like 3 and a fraction more, such as in recipes or measurements. To perform operations like addition, subtraction, or division, it鈥檚 typically easier to work with improper fractions. This is why, when dividing by a mixed number, the first step is converting it into an improper fraction.
  • Whole numbers are those without fractions or decimals.
  • A proper fraction has a numerator smaller than its denominator.
  • Mixed numbers can be easily converted for calculations.
Improper Fractions
Improper fractions have numerators that are equal to or larger than their denominators. This is helpful in calculations because they represent a number greater than or equal to 1, which simplifies mathematical operations. Converting a mixed number to an improper fraction involves multiplying the whole number by the denominator, adding the numerator, and putting the result over the original denominator. This conversion provides a straightforward form for performing arithmetic operations such as division or multiplication.
馃搹 Example: \(3 \frac{1}{6}\) converts to \(\frac{19}{6}\).
  • This method assists in performing calculations without misunderstandings.
  • Improper fractions efficiently express larger quantities.
  • They are easy to manipulate in equations.
Reciprocal
The reciprocal of a fraction is obtained by swapping its numerator and denominator. When you need to divide by a fraction, you multiply by its reciprocal. This method leverages the concept that dividing by a number is the same as multiplying by its reciprocal, effectively transforming a division problem into a multiplication one. It turns a potentially complex operation into a simpler one, making it easier to solve. In the exercise given, the reciprocal of \(\frac{19}{6}\) is \(\frac{6}{19}\), which changes the division \(12 \div \frac{19}{6}\) into the multiplication \(12 \times \frac{6}{19}\).
  • Reciprocal operation simplifies division into multiplication.
  • Turns complex division into straightforward multiplication.
  • Essential in solving fractions-involved equations.

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Most popular questions from this chapter

A column on horse racing in the Daily News in Los Angeles reported that the horse Action This Day ran 3 furlongs in 35 \(\frac{1}{5}\) seconds and another horse, Halfbridled, went two-fifths of a second faster. How many seconds did it take Halfbridled to run 3 furlongs?

Arithmetic Sequences Recall that an arithmetic sequence is a sequence in which each term comes from the previous term by adding the same number each time. For example, the sequence \(1, \frac{3}{2}, 2, \frac{5}{2}, \ldots\) is an arithmetic sequence that starts with the number 1. Then each term after that is found by adding \(\frac{1}{2}\) to the previous term. By observing this fact, we know that the next term in the sequence will be \(\frac{5}{2}+\frac{1}{2}=\frac{6}{2}=3\) Find the next number in each arithmetic sequence below. $$1, \frac{4}{3}, \frac{5}{3}, 2, \dots$$

These problems review the four basic operations with fractions from this chapter. Perform the indicated operations. $$\frac{3}{7}-\frac{2}{7}$$

The following problems all involve the concept of borrowing. Subtract in case. \(9 \frac{4}{7}-7 \frac{2}{3}\)

The following problems all involve the concept of borrowing. Subtract in case. \(18 \frac{5}{12}-9 \frac{3}{4}\)
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