Answer :
- Multiply the first two fractions: $\frac{2}{5} \times \frac{3}{4} = \frac{3}{10}$.
- Divide the result by the third fraction, which is equivalent to multiplying by its reciprocal: $\frac{3}{10} \div \frac{8}{5} = \frac{3}{10} \times \frac{5}{8}$.
- Multiply the fractions: $\frac{3}{10} \times \frac{5}{8} = \frac{15}{80}$.
- Simplify the fraction: $\frac{15}{80} = \boxed{\frac{3}{16}}$.
### Explanation
1. Understanding the Problem
We are asked to evaluate the expression $\frac{2}{5} \times \frac{3}{4} \div \frac{8}{5}$. This involves multiplication and division of fractions. Remember that when we have a series of multiplication and division operations, we perform them from left to right. Also, dividing by a fraction is the same as multiplying by its reciprocal.
2. Multiplying the First Two Fractions
First, let's perform the multiplication: $\frac{2}{5} \times \frac{3}{4}$. To multiply fractions, we multiply the numerators and the denominators:$$\frac{2}{5} \times \frac{3}{4} = \frac{2 \times 3}{5 \times 4} = \frac{6}{20}$$We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:$$\frac{6}{20} = \frac{6 \div 2}{20 \div 2} = \frac{3}{10}$$
3. Dividing by the Third Fraction
Next, we need to divide the result by $\frac{8}{5}$. Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of $\frac{8}{5}$ is $\frac{5}{8}$. So, we have:$$\frac{3}{10} \div \frac{8}{5} = \frac{3}{10} \times \frac{5}{8}$$Now, we multiply these fractions:$$\frac{3}{10} \times \frac{5}{8} = \frac{3 \times 5}{10 \times 8} = \frac{15}{80}$$
4. Simplifying the Result
Finally, we simplify the fraction $\frac{15}{80}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 5:$$\frac{15}{80} = \frac{15 \div 5}{80 \div 5} = \frac{3}{16}$$So, the value of the expression is $\frac{3}{16}$.
5. Final Answer
Therefore, the value of $\frac{2}{5} \times \frac{3}{4} \div \frac{8}{5}$ is $\boxed{\frac{3}{16}}$.
### Examples
Understanding how to multiply and divide fractions is essential in many real-life situations. For example, when you're cooking and need to adjust a recipe, you often have to multiply or divide fractions to get the right amount of ingredients. Another example is when you're calculating distances on a map or determining proportions in construction or design. Knowing how to work with fractions accurately helps ensure your calculations are correct and your results are precise.
- Divide the result by the third fraction, which is equivalent to multiplying by its reciprocal: $\frac{3}{10} \div \frac{8}{5} = \frac{3}{10} \times \frac{5}{8}$.
- Multiply the fractions: $\frac{3}{10} \times \frac{5}{8} = \frac{15}{80}$.
- Simplify the fraction: $\frac{15}{80} = \boxed{\frac{3}{16}}$.
### Explanation
1. Understanding the Problem
We are asked to evaluate the expression $\frac{2}{5} \times \frac{3}{4} \div \frac{8}{5}$. This involves multiplication and division of fractions. Remember that when we have a series of multiplication and division operations, we perform them from left to right. Also, dividing by a fraction is the same as multiplying by its reciprocal.
2. Multiplying the First Two Fractions
First, let's perform the multiplication: $\frac{2}{5} \times \frac{3}{4}$. To multiply fractions, we multiply the numerators and the denominators:$$\frac{2}{5} \times \frac{3}{4} = \frac{2 \times 3}{5 \times 4} = \frac{6}{20}$$We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:$$\frac{6}{20} = \frac{6 \div 2}{20 \div 2} = \frac{3}{10}$$
3. Dividing by the Third Fraction
Next, we need to divide the result by $\frac{8}{5}$. Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of $\frac{8}{5}$ is $\frac{5}{8}$. So, we have:$$\frac{3}{10} \div \frac{8}{5} = \frac{3}{10} \times \frac{5}{8}$$Now, we multiply these fractions:$$\frac{3}{10} \times \frac{5}{8} = \frac{3 \times 5}{10 \times 8} = \frac{15}{80}$$
4. Simplifying the Result
Finally, we simplify the fraction $\frac{15}{80}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 5:$$\frac{15}{80} = \frac{15 \div 5}{80 \div 5} = \frac{3}{16}$$So, the value of the expression is $\frac{3}{16}$.
5. Final Answer
Therefore, the value of $\frac{2}{5} \times \frac{3}{4} \div \frac{8}{5}$ is $\boxed{\frac{3}{16}}$.
### Examples
Understanding how to multiply and divide fractions is essential in many real-life situations. For example, when you're cooking and need to adjust a recipe, you often have to multiply or divide fractions to get the right amount of ingredients. Another example is when you're calculating distances on a map or determining proportions in construction or design. Knowing how to work with fractions accurately helps ensure your calculations are correct and your results are precise.