Problem 107
Question
For exercises 97-114, evaluate. $$ -\frac{3}{8} \div \frac{1}{2} \cdot \frac{1}{5}+\frac{1}{4} $$
Step-by-Step Solution
Verified Answer
\frac{1}{10}\
1Step 1: Simplify the Division
First, simplify the division. Evaluate \( -\frac{3}{8} \div \frac{1}{2} \) by multiplying by the reciprocal of \ \frac{1}{2} \: \( -\frac{3}{8} \times 2 = -\frac{3}{8} \times \frac{2}{1} = -\frac{6}{8} = -\frac{3}{4} \).
2Step 2: Multiply the Result by \ \frac{1}{5} \
Next, take the result from the previous step \( -\frac{3}{4} \) and multiply it by \ \frac{1}{5} \: \( -\frac{3}{4} \times \frac{1}{5} = -\frac{3}{20} \).
3Step 3: Add \ \frac{1}{4} \
Finally, add \ \frac{1}{4} \ to the result from Step 2: \( -\frac{3}{20} + \frac{1}{4}\). Obtain a common denominator (20) to perform the addition: \( -\frac{3}{20} + \frac{5}{20} = \frac{2}{20} \). Simplify the fraction: \( \frac{2}{20} = \frac{1}{10} \).
Key Concepts
Division of FractionsMultiplication of FractionsAddition of FractionsSimplifying Fractions
Division of Fractions
To divide fractions, you need to multiply by the reciprocal. The reciprocal of a fraction is created by swapping its numerator and denominator. For example, the reciprocal of \(\frac{1}{2}\) is \(\frac{2}{1}\).
Consider the original problem where we had to divide \(-\frac{3}{8} \) by \(\frac{1}{2}\). Instead of dividing, we multiply \(-\frac{3}{8} \) by \(\frac{2}{1}\). This changes the operation from division to multiplication and makes it easier to manage.
Therefore, \(-\frac{3}{8} \div \frac{1}{2} = -\frac{3}{8} \times \frac{2}{1}\), simplifying to \(-\frac{6}{8}\), which reduces to \(-\frac{3}{4}\).
Consider the original problem where we had to divide \(-\frac{3}{8} \) by \(\frac{1}{2}\). Instead of dividing, we multiply \(-\frac{3}{8} \) by \(\frac{2}{1}\). This changes the operation from division to multiplication and makes it easier to manage.
Therefore, \(-\frac{3}{8} \div \frac{1}{2} = -\frac{3}{8} \times \frac{2}{1}\), simplifying to \(-\frac{6}{8}\), which reduces to \(-\frac{3}{4}\).
Multiplication of Fractions
Multiplying fractions is straightforward. Simply multiply the numerators together and the denominators together.
In the example, we multiply \(-\frac{3}{4} \times \frac{1}{5}\). To do this, multiply the numerators: -3 \times 1 = -3.
Then, multiply the denominators: 4 \times 5 = 20.
The result is: \(-\frac{3}{20}\).
Remember to always check if you can simplify fractions after multiplication, though in this specific case, \(-\frac{3}{20}\) cannot be simplified further.
In the example, we multiply \(-\frac{3}{4} \times \frac{1}{5}\). To do this, multiply the numerators: -3 \times 1 = -3.
Then, multiply the denominators: 4 \times 5 = 20.
The result is: \(-\frac{3}{20}\).
Remember to always check if you can simplify fractions after multiplication, though in this specific case, \(-\frac{3}{20}\) cannot be simplified further.
Addition of Fractions
To add fractions, they need a common denominator. This means both fractions must have the same bottom number.
For our example, we need to add \(-\frac{3}{20} \) and \(\frac{1}{4}\). Since the denominators are different, find the least common multiple (LCM).
The LCM of 20 and 4 is 20.
Convert \(\frac{1}{4}\) to \(\frac{5}{20}\) by multiplying both the numerator and denominator by 5.
Now, add the fractions: \(-\frac{3}{20} \) + \(\frac{5}{20}\) = \(\frac{2}{20}\).
For our example, we need to add \(-\frac{3}{20} \) and \(\frac{1}{4}\). Since the denominators are different, find the least common multiple (LCM).
The LCM of 20 and 4 is 20.
Convert \(\frac{1}{4}\) to \(\frac{5}{20}\) by multiplying both the numerator and denominator by 5.
Now, add the fractions: \(-\frac{3}{20} \) + \(\frac{5}{20}\) = \(\frac{2}{20}\).
Simplifying Fractions
After performing operations with fractions, always simplify your results. Simplifying means reducing the fraction to its smallest form.
To do this, divide both the numerator and the denominator by their greatest common divisor (GCD).
In our case, we have \(\frac{2}{20}\).
The GCD of 2 and 20 is 2.
Divide both the numerator and the denominator by 2 to get: \(\frac{2 \div 2}{20 \div 2} = \frac{1}{10}\).
Hence, \(\frac{1}{10}\) is the simplified form of \(\frac{2}{20}\).
Always check if there's a common factor for further simplification.
To do this, divide both the numerator and the denominator by their greatest common divisor (GCD).
In our case, we have \(\frac{2}{20}\).
The GCD of 2 and 20 is 2.
Divide both the numerator and the denominator by 2 to get: \(\frac{2 \div 2}{20 \div 2} = \frac{1}{10}\).
Hence, \(\frac{1}{10}\) is the simplified form of \(\frac{2}{20}\).
Always check if there's a common factor for further simplification.
Other exercises in this chapter
Problem 104
For exercises 101-104, evaluate. Use a scientific or a graphing calculator. $$ -8^{2}-4-15 \div(-3) $$
View solution Problem 106
For exercises 97-114, evaluate. $$ \left[\frac{1}{6}+2\left(\frac{1}{3}\right)\right]^{2} $$
View solution Problem 108
For exercises \(85-108\), write \(>\) or \(
View solution Problem 108
For exercises 97-114, evaluate. $$ -\frac{2}{9} \div \frac{1}{6} \cdot \frac{1}{4}+\frac{1}{5} $$
View solution