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Chapter 5: Problem 33

Find each quotient. Use an area model if necessary. $$12 \div \frac{4}{9}$$

Short Answer

Expert verified
The quotient of \(12 \div \frac{4}{9}\) is 27.

Step by step solution

01

Understand the Problem

The problem requires dividing a whole number, 12, by a fraction, \( \frac{4}{9} \). This is equivalent to determining how many times \( \frac{4}{9} \) fits into 12.
02

Reciprocate the Fraction

To divide by a fraction, we multiply by its reciprocal. The reciprocal of \( \frac{4}{9} \) is \( \frac{9}{4} \). Therefore, \( 12 \div \frac{4}{9} = 12 \times \frac{9}{4} \).
03

Convert Whole Number to a Fraction

Rewrite the whole number 12 as a fraction to simplify multiplication: \( \frac{12}{1} \).
04

Multiply the Fractions

Multiply the fractions: \( \frac{12}{1} \times \frac{9}{4} = \frac{12 \times 9}{1 \times 4} = \frac{108}{4} \).
05

Simplify the Fraction

Divide the numerator and the denominator of \( \frac{108}{4} \) by their greatest common divisor, which is 4: \( \frac{108 \div 4}{4 \div 4} = \frac{27}{1} \).
06

Convert the Fraction to a Whole Number

Since \( \frac{27}{1} \) is equivalent to 27, the quotient is 27.

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Key Concepts

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

Reciprocal
Understanding what a reciprocal is can greatly simplify division issues involving fractions. A reciprocal of a number is essentially what you multiply that number by to get the value of 1.
For a fraction, this means flipping the fraction upside down. Thus, the reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \). With this understanding, we can transform division into a much easier multiplication problem.
  • Take the fraction you want to find the reciprocal for. For example, \( \frac{4}{9} \).
  • Flip it, so the numerator becomes the denominator and vice versa, making it \( \frac{9}{4} \).
Remember, the reciprocal only applies when you're dividing by a fraction. It's a handy trick that changes a complicated division problem into a simple multiplication task.
Multiplication of Fractions
Once you've found the reciprocal of a fraction during a division problem, the next step is to multiply. Multiplying fractions is straightforward compared to other operations. Here's how you do it:
  • First, express the whole number as a fraction. In the case of 12, this becomes \( \frac{12}{1} \).
  • Then, multiply the numerators together and the denominators together. This means for multiplying \( \frac{12}{1} \times \frac{9}{4} \), you calculate \( 12 \times 9 \) for the top of the fraction and \( 1 \times 4 \) for the bottom.
  • This results in \( \frac{108}{4} \).
Multiplying fractions might seem tricky initially, but it relies simply on multiplying straight across, which can become a quick and easy process once practiced a few times.
Simplifying Fractions
After multiplying fractions, the answer may not always be in its simplest form. Simplifying fractions ensures the answer is clean and concise. Here's how you do it:
  • Identify the greatest common divisor (GCD) of the fraction's numerator and denominator. In our example, the fraction we got was \( \frac{108}{4} \).
  • The GCD of 108 and 4 is 4. Divide both the numerator and the denominator by the GCD: \( \frac{108 \div 4}{4 \div 4} \).
  • This simplification reduces the fraction to \( \frac{27}{1} \) which is simply 27.
By simplifying the fraction, not only does it make it easier to understand, but it also makes calculations involving that fraction much simpler. Always check if your fractions can be further simplified, making answers more elegant and comprehensible.

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Most popular questions from this chapter

Write each fraction or mixed number as a decimal. Use a bar to show a repeating decimal. $$-7 \frac{4}{5}$$

The denominator of a fraction is 4 more than the numerator. If both the numerator and denominator are increased by \(1,\) the resulting fraction equals \(\frac{1}{2} .\) Find the original fraction.

Find each product. Write in simplest form. $$-\frac{5}{12} \cdot 1 \frac{1}{7}$$

Solve each equation. Check your solution. $$\frac{1}{3} n=\frac{2}{9}$$

Find each sum or difference. Write in simplest form. $$\frac{3}{5}+\frac{1}{3}$$
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