Problem 54
Question
Perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\frac{7}{8} \div \frac{2}{3}$$
Step-by-Step Solution
Verified Answer
The answer is \(\frac{21}{16}\).
1Step 1: Write the Reciprocal of the Second Fraction
To divide two fractions, you must multiply the first one by the reciprocal of the second one. The reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\). So, the current operation becomes: \( \frac{7}{8} \times \frac{3}{2}\).
2Step 2: Multiply the Fractions
Multiply the numerators to find the new numerator and multiply the denominators to find the new denominator: \(\frac{7 \times 3}{8 \times 2} = \frac{21}{16}\).
3Step 3: Simplify the Fraction
The fraction \(\frac{21}{16}\) is already in its simplest form, as 21 and 16 have no common factors other than 1. So, the final solution is \(\frac{21}{16}\).
Key Concepts
Reciprocal of a FractionMultiplying FractionsSimplifying Fractions
Reciprocal of a Fraction
The reciprocal of a fraction is created by swapping the numerator (top number) and the denominator (bottom number) of the original fraction. For instance, the reciprocal of a fraction like \( \frac{a}{b} \) is \( \frac{b}{a} \). This concept is crucial when it comes to dividing fractions because division can be thought of as multiplication by the reciprocal.
In our exercise, the reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \). When you divide \( \frac{7}{8} \) by \( \frac{2}{3} \), you actually multiply \( \frac{7}{8} \) by \( \frac{3}{2} \), the reciprocal of \( \frac{2}{3} \). Understanding this principle helps resolve what might first seem like a complex division problem into a simpler multiplication problem, which is more straightforward to solve.
In our exercise, the reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \). When you divide \( \frac{7}{8} \) by \( \frac{2}{3} \), you actually multiply \( \frac{7}{8} \) by \( \frac{3}{2} \), the reciprocal of \( \frac{2}{3} \). Understanding this principle helps resolve what might first seem like a complex division problem into a simpler multiplication problem, which is more straightforward to solve.
Multiplying Fractions
Multiplying fractions is a simpler operation compared to addition, subtraction, or division of fractions. To multiply two fractions, you multiply the numerators with each other to get the new numerator, and multiply the denominators with each other to get the new denominator. The resulting fraction represents the product of the initial fractions.
Following this rule, when you multiply the fractions \( \frac{7}{8} \) and \( \frac{3}{2} \), you'll end up with \( \frac{7 \times 3}{8 \times 2} \), which equals \( \frac{21}{16} \). It is vital to multiply straight across; do not cross-multiply, as that's a technique used for solving proportions rather than multiplying fractions.
Following this rule, when you multiply the fractions \( \frac{7}{8} \) and \( \frac{3}{2} \), you'll end up with \( \frac{7 \times 3}{8 \times 2} \), which equals \( \frac{21}{16} \). It is vital to multiply straight across; do not cross-multiply, as that's a technique used for solving proportions rather than multiplying fractions.
Simplifying Fractions
Simplifying a fraction means reducing it to its smallest, or simplest, form where the numerator and denominator are as small as possible. This is done by dividing both the numerator and the denominator by their greatest common factor, if one exists beyond 1.
For our example, the fraction \( \frac{21}{16} \) is already in its simplest form because 21 and 16 do not have any common divisors other than 1. If there had been a common factor, say 'c', you would divide both the numerator and the denominator by 'c' to simplify the fraction. Constantly looking for ways to simplify fractions is essential, as it makes further arithmetic operations and comparisons much easier to manage. Simplified fractions are also easier to understand and interpret when applying the results in practical situations.
For our example, the fraction \( \frac{21}{16} \) is already in its simplest form because 21 and 16 do not have any common divisors other than 1. If there had been a common factor, say 'c', you would divide both the numerator and the denominator by 'c' to simplify the fraction. Constantly looking for ways to simplify fractions is essential, as it makes further arithmetic operations and comparisons much easier to manage. Simplified fractions are also easier to understand and interpret when applying the results in practical situations.
Other exercises in this chapter
Problem 54
Simplify each series of additions and subtractions. $$8-2+5-13$$
View solution Problem 54
Determine whether the given number is a solution of the equation. $$5 a-3=2 a+6 ; 3$$
View solution Problem 55
In Exercises \(47-76\), perform the indicated division or state that the expression is undefined. $$\frac{7}{0}$$
View solution Problem 55
Simplify each algebraic expression. $$-7 b+10+(-b)+(-6)$$
View solution