Problem 22
Question
Simplify. $$-\frac{2}{3}\left(\frac{5}{8}\right) \div \frac{2}{7}$$
Step-by-Step Solution
Verified Answer
The simplified expression is \(-1\frac{11}{24}\).
1Step 1: Multiplication of Fractions
Multiply \(-\frac{2}{3}\) by \(\frac{5}{8}\). Multiplication of fractions is done by multiplying the numerators together to get the new numerator and the denominators together to get the new denominator. Thus, \(-\frac{2}{3} \times \frac{5}{8} = -\frac{10}{24}\), which can be simplified to \(-\frac{5}{12}\).
2Step 2: Division of Fractions
Next, divide the result by \(\frac{2}{7}\). Recall that dividing fractions is the same as multiplying by the reciprocal. Thus, \(-\frac{5}{12} \div \frac{2}{7} = -\frac{5}{12} \times \frac{7}{2}\).
3Step 3: Simplify the Result
Multiply \(-\frac{5}{12} \times \frac{7}{2}\) to get \(-\frac{35}{24}\). This fraction can further be simplified. Convert it into a mixed fraction to get \(-1\frac{11}{24}\).
Key Concepts
Multiplication of FractionsDivision of FractionsReciprocal of a Fraction
Multiplication of Fractions
When multiplying fractions, the process is straightforward. You multiply across the numerators to get the numerator of the product, and you multiply across the denominators to get the denominator of the product. For instance, if we have two fractions, \(\frac{a}{b}\) and \(\frac{c}{d}\), the multiplication of these fractions is \(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\).
In our exercise, we multiply \( -\frac{2}{3}\) by \(\frac{5}{8}\) to obtain \( -\frac{2 \times 5}{3 \times 8}\), which simplifies to \( -\frac{10}{24}\), and upon simplification, we can reduce it to \( -\frac{5}{12}\). Remember always to simplify the product by looking for common factors between the numerator and denominator and dividing them to achieve the simplest form of the fraction.
In our exercise, we multiply \( -\frac{2}{3}\) by \(\frac{5}{8}\) to obtain \( -\frac{2 \times 5}{3 \times 8}\), which simplifies to \( -\frac{10}{24}\), and upon simplification, we can reduce it to \( -\frac{5}{12}\). Remember always to simplify the product by looking for common factors between the numerator and denominator and dividing them to achieve the simplest form of the fraction.
Division of Fractions
Dividing fractions might sound complicated, but it's actually quite simple once you know the rule: to divide by a fraction, you multiply by its reciprocal. What is a reciprocal, you ask? For a given fraction \(\frac{x}{y}\), the reciprocal is \(\frac{y}{x}\) where you swap the numerator and the denominator.
This means that to divide one fraction by another, such as \(\frac{a}{b} \div \frac{c}{d}\), you would actually perform the operation \(\frac{a}{b} \times \frac{d}{c}\). Applying this to our example, \( -\frac{5}{12} \div \frac{2}{7}\) becomes \( -\frac{5}{12} \times \frac{7}{2}\). The next step is to multiply these two fractions, as you would normally do.
This means that to divide one fraction by another, such as \(\frac{a}{b} \div \frac{c}{d}\), you would actually perform the operation \(\frac{a}{b} \times \frac{d}{c}\). Applying this to our example, \( -\frac{5}{12} \div \frac{2}{7}\) becomes \( -\frac{5}{12} \times \frac{7}{2}\). The next step is to multiply these two fractions, as you would normally do.
Reciprocal of a Fraction
Understanding the reciprocal of a fraction is key to dividing fractions effectively. The reciprocal, also known as the multiplicative inverse, is what you get when you flip a fraction. If you have a fraction \(\frac{n}{m}\), its reciprocal is \(\frac{m}{n}\).
For nonzero numbers, the product of a number and its reciprocal is always 1 because \(\frac{n}{m} \times \frac{m}{n} = \frac{n \times m}{m \times n} = 1\). This property makes the reciprocal special and useful, especially in division. When dividing fractions such as \(\frac{a}{b} \div \frac{c}{d}\), we use the reciprocal of \(\frac{c}{d}\), which is \(\frac{d}{c}\), and proceed with multiplication.
For nonzero numbers, the product of a number and its reciprocal is always 1 because \(\frac{n}{m} \times \frac{m}{n} = \frac{n \times m}{m \times n} = 1\). This property makes the reciprocal special and useful, especially in division. When dividing fractions such as \(\frac{a}{b} \div \frac{c}{d}\), we use the reciprocal of \(\frac{c}{d}\), which is \(\frac{d}{c}\), and proceed with multiplication.