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Chapter 4: Problem 61

Divide the fractions, and simplify your result. $$\frac{8}{21} \div 2$$

Short Answer

Expert verified
The simplified result is \( \frac{4}{21} \).

Step by step solution

01

Understand Division of a Fraction by a Whole Number

When you divide a fraction by a whole number, you can rewrite the division as multiplication by the reciprocal of the whole number. For instance, dividing by 2 is the same as multiplying by its reciprocal, which is \( \frac{1}{2} \).
02

Rewrite the Problem Using Multiplication

Replace the division operation with multiplication by the reciprocal of the whole number. This changes \( \frac{8}{21} \div 2 \) into \( \frac{8}{21} \times \frac{1}{2} \).
03

Multiply the Fractions

To multiply two fractions, multiply the numerators together and the denominators together. Hence, \( \frac{8}{21} \times \frac{1}{2} = \frac{8 \times 1}{21 \times 2} = \frac{8}{42} \).
04

Simplify the Resulting Fraction

Simplify \( \frac{8}{42} \) by finding the greatest common divisor (GCD) of 8 and 42, which is 2. Divide both the numerator and the denominator by their GCD: \( \frac{8 \div 2}{42 \div 2} = \frac{4}{21} \).

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Key Concepts

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

Reciprocal
A reciprocal is a fraction that is flipped upside down. For example, the reciprocal of a whole number like 2 is \( \frac{1}{2} \). When dividing fractions, finding the reciprocal is key because it allows you to turn division into multiplication, making the calculation easier.

  • To find the reciprocal of a number, place 1 over the number if it is a whole number.
  • If it's a fraction, simply swap the numerator and the denominator.
By using the reciprocal in our exercise, \( \frac{8}{21} \div 2 \) becomes \( \frac{8}{21} \times \frac{1}{2} \). This transformation is what allows us to multiply instead of dividing.
Simplifying Fractions
Simplifying fractions means reducing them to their simplest form. A fraction is in its simplest form when the numerator and the denominator have no common factors other than 1.

  • First, find any common factors between the numerator and denominator.
  • Then, divide both the numerator and the denominator by the greatest common divisor (GCD).
Through simplification, a fraction is easier to understand and work with. In the solution, after multiplying, we found \( \frac{8}{42} \), which simplifies to \( \frac{4}{21} \) by dividing both by their GCD.
Greatest Common Divisor
The greatest common divisor (GCD) is the largest number that divides both the numerator and the denominator without leaving a remainder. Finding it is crucial for simplifying fractions.

  • You can find the GCD through listing factors or using the Euclidean algorithm.
  • Divide both numbers by their GCD to simplify the fraction.
In the exercise, we found the GCD of 8 and 42 to be 2, allowing us to simplify \( \frac{8}{42} \) to \( \frac{4}{21} \). This not only makes the fraction smaller but also easier to interpret and use.

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Most popular questions from this chapter

Simplify the complex rational expression. $$\frac{\frac{7}{8}+\frac{1}{9}}{\frac{8}{9}-\frac{1}{6}}$$

Solve the equation and simplify your answer. $$-\frac{9}{7} x+\frac{9}{2}=-\frac{5}{2}$$

Solve the equation and simplify your answer. $$-\frac{3}{2} x+\frac{8}{3}=\frac{7}{9} x-\frac{1}{2}$$

Is \(3 / 8\) a solution of the equation \(x-\frac{5}{9}=-\frac{13}{72}\) ?

A trapezoid has bases measuring \(3 \frac{1}{8}\) and \(6 \frac{1}{2}\) feet, respectively. The height of the trapezoid is 3 feet. Find the area of the trapezoid.
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