Problem 95
Question
For exercises 95-97, evaluate. $$ \frac{5}{21}+\frac{2}{21} $$
Step-by-Step Solution
Verified Answer
\frac{1}{3}
1Step 1: Identify the fractions
Both fractions \(\frac{5}{21}\) and \(\frac{2}{21}\) share the same denominator, 21.
2Step 2: Add the numerators
Since the denominators are the same, add the numerators together: \(5 + 2 = 7.\)
3Step 3: Write the result as a single fraction
Combine the sum of the numerators over the common denominator: \(\frac{7}{21}\).
4Step 4: Simplify the fraction
Simplify \(\frac{7}{21}\) by dividing the numerator and the denominator by their greatest common divisor, which is 7: \(\frac{7 \div 7}{21 \div 7} = \frac{1}{3}\).
Key Concepts
common denominatorsum of numeratorssimplifying fractions
common denominator
When adding fractions, it's crucial that the fractions you are adding have the same denominator. A common denominator is simply the shared bottom number of the fractions involved.
If the denominators are already the same, like in our example \(\frac{5}{21} + \frac{2}{21}\), you can easily proceed to the next step.
A shared denominator allows the fractions to be directly comparable and combinable.
If you encounter fractions with different denominators, you will first need to find a common denominator. This involves finding the least common multiple (LCM) of the denominators.
If the denominators are already the same, like in our example \(\frac{5}{21} + \frac{2}{21}\), you can easily proceed to the next step.
A shared denominator allows the fractions to be directly comparable and combinable.
If you encounter fractions with different denominators, you will first need to find a common denominator. This involves finding the least common multiple (LCM) of the denominators.
sum of numerators
Once you have a common denominator, adding the fractions becomes straightforward. You focus on the numerators, the numbers above the fraction line.
In our example, both fractions \(\frac{5}{21}\) and \(\frac{2}{21}\) share the same denominator, 21.
To add these fractions:
In our example, both fractions \(\frac{5}{21}\) and \(\frac{2}{21}\) share the same denominator, 21.
To add these fractions:
- Add only the numerators: \(5 + 2 = 7\)
- Keep the common denominator the same: 21
simplifying fractions
The final step in working with fractions often involves simplifying the result. A simplified fraction is one where the numerator and the denominator are as small as possible, with no common factors other than 1.
In our example, \(\frac{7}{21}\) can be simplified. To do this, you need to find the greatest common divisor (GCD) of the numerator and the denominator. In this case, the GCD is 7.
Simplifying fractions allows for more straightforward results and often makes it easier to understand and use the fraction in further calculations.
In our example, \(\frac{7}{21}\) can be simplified. To do this, you need to find the greatest common divisor (GCD) of the numerator and the denominator. In this case, the GCD is 7.
- Divide the numerator and the denominator by their GCD: \(\frac{7 \div 7}{21 \div 7} = \frac{1}{3}\)
Simplifying fractions allows for more straightforward results and often makes it easier to understand and use the fraction in further calculations.
Other exercises in this chapter
Problem 95
For exercises \(95-98\), evaluate. $$ \frac{\frac{10}{3}}{\frac{2}{3}} $$
View solution Problem 95
For exercises \(95-98\), evaluate. \(\frac{3}{4}+\frac{5}{6}\)
View solution Problem 95
\text { Describe how to divide two fractions. }
View solution Problem 96
For exercises \(95-98\), evaluate. $$ \frac{5}{7}-\frac{2}{9} $$
View solution