Chapter 2: Problem 4
Calculate (a) \(\frac{3}{4}+\frac{1}{3}\) (b) \(\frac{1}{2}+\frac{3}{5}\) (c) \(\frac{5}{6}-\frac{1}{3}\) (d) \(\frac{10}{11}-\frac{1}{2}\) (e) \(\frac{4}{7}+\frac{1}{2}-\frac{2}{3}\)
Short Answer
Expert verified
Question: Perform the following additions and subtractions with fractions and provide the answers in their simplest form:
(a) \(\frac{3}{4}+\frac{1}{3}\)
(b) \(\frac{1}{2}+\frac{3}{5}\)
(c) \(\frac{5}{6}-\frac{1}{3}\)
(d) \(\frac{10}{11}-\frac{1}{2}\)
(e) \(\frac{4}{7}+\frac{1}{2}-\frac{2}{3}\)
Answers:
(a) \(\frac{13}{12}\)
(b) \(\frac{11}{10}\)
(c) \(\frac{1}{2}\)
(d) \(\frac{9}{22}\)
(e) \(\frac{17}{42}\)
Step by step solution
01
Find the common denominator
The lowest common denominator (LCD) between 4 and 3 is 12. The LCD is the smallest number that both denominators can divide into, and in this case, 12 is the smallest number that both 4 and 3 can divide into.
02
Convert the fractions to their equivalent forms with the LCD
We want fractions with a denominator of 12, so we need to change \(\frac{3}{4}\) and \(\frac{1}{3}\) accordingly:
\(\frac{3}{4} \times \frac{3}{3} = \frac{9}{12}\)
\(\frac{1}{3} \times \frac{4}{4} = \frac{4}{12}\)
03
Add the fractions
Now that the fractions have the same denominator, we can add them together:
\(\frac{9}{12}+\frac{4}{12}=\frac{9+4}{12}=\frac{13}{12}\)
04
Simplify (if possible)
In this case, the result is already in its simplest form, so the final answer is \(\frac{13}{12}\).
(b) \(\frac{1}{2}+\frac{3}{5}\)
05
Find the common denominator
The lowest common denominator (LCD) between 2 and 5 is 10.
06
Convert the fractions to their equivalent forms with the LCD
We want fractions with a denominator of 10, so we need to change \(\frac{1}{2}\) and \(\frac{3}{5}\) accordingly:
\(\frac{1}{2} \times \frac{5}{5} = \frac{5}{10}\)
\(\frac{3}{5} \times \frac{2}{2} = \frac{6}{10}\)
07
Add the fractions
Now that the fractions have the same denominator, we can add them together:
\(\frac{5}{10}+\frac{6}{10}=\frac{5+6}{10}=\frac{11}{10}\)
08
Simplify (if possible)
In this case, the result is already in its simplest form, so the final answer is \(\frac{11}{10}\).
(c) \(\frac{5}{6}-\frac{1}{3}\)
09
Find the common denominator
The lowest common denominator (LCD) between 6 and 3 is 6.
10
Convert the fractions to their equivalent forms with the LCD
We can leave \(\frac{5}{6}\) as it is since the denominator is already 6.
\(\frac{1}{3} \times \frac{2}{2} = \frac{2}{6}\)
11
Subtract the fractions
Now that the fractions have the same denominator, we can subtract them:
\(\frac{5}{6}-\frac{2}{6}=\frac{5-2}{6}=\frac{3}{6}\)
12
Simplify (if possible)
We can simplify the result by dividing both the numerator and denominator by 3, resulting in \(\frac{1}{2}\).
(d) \(\frac{10}{11}-\frac{1}{2}\)
13
Find the common denominator
The lowest common denominator (LCD) between 11 and 2 is 22.
14
Convert the fractions to their equivalent forms with the LCD
We want fractions with a denominator of 22, so we need to change \(\frac{10}{11}\) and \(\frac{1}{2}\) accordingly:
\(\frac{10}{11} \times \frac{2}{2} = \frac{20}{22}\)
\(\frac{1}{2} \times \frac{11}{11} = \frac{11}{22}\)
15
Subtract the fractions
Now that the fractions have the same denominator, we can subtract them:
\(\frac{20}{22}-\frac{11}{22}=\frac{20-11}{22}=\frac{9}{22}\)
16
Simplify (if possible)
In this case, the result is already in its simplest form, so the final answer is \(\frac{9}{22}\).
(e) \(\frac{4}{7}+\frac{1}{2}-\frac{2}{3}\)
17
Finding the common denominator and converting the fractions
The common denominator for all three fractions is 42, and we need to change each fraction accordingly:
\(\frac{4}{7} \times \frac{6}{6} = \frac{24}{42}\)
\(\frac{1}{2} \times \frac{21}{21} = \frac{21}{42}\)
\(\frac{2}{3} \times \frac{14}{14} = \frac{28}{42}\)
18
Adding and subtracting the fractions
Now that the fractions have the same denominator, we can add and subtract them:
\(\frac{24}{42}+\frac{21}{42}-\frac{28}{42}=\frac{24+21-28}{42}=\frac{17}{42}\)
The final answer for (e) is \(\frac{17}{42}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Common Denominator
When dealing with fractions, a common denominator is crucial for performing addition or subtraction. The common denominator is essentially the smallest number that both denominators can evenly divide into. To find this number, look for the least common multiple (LCM) of the denominators involved.
Let's break it down:
For instance, if you want to add \( \frac{3}{4} \) and \( \frac{1}{3} \), the denominators are 4 and 3. The LCM or the smallest common denominator for 4 and 3 is 12. This means you need to express both fractions with a denominator of 12 to perform addition or subtraction effectively. Finding the common denominator is the first step in managing fraction arithmetic.
Let's break it down:
- Identify the denominators of the fractions.
- Determine the smallest number that is a multiple of both denominators.
For instance, if you want to add \( \frac{3}{4} \) and \( \frac{1}{3} \), the denominators are 4 and 3. The LCM or the smallest common denominator for 4 and 3 is 12. This means you need to express both fractions with a denominator of 12 to perform addition or subtraction effectively. Finding the common denominator is the first step in managing fraction arithmetic.
Equivalent Fractions
Equivalent fractions are different fractions that represent the same value or proportion. To create equivalent fractions, you simply multiply or divide the numerator and the denominator of a fraction by the same non-zero number. The value of the fraction remains unchanged because you're essentially multiplying by 1.
Consider this example:
This conversion makes \( \frac{3}{4} \) equivalent to \( \frac{9}{12} \), allowing it to be directly compared or combined with other fractions also expressed with a denominator of 12. Understanding equivalent fractions is vital for modifying and combining fractions during calculations.
Consider this example:
- We have \( \frac{3}{4} \) and we need to convert it to have a denominator of 12.
- Multiply both the numerator and denominator by 3: \( \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \).
This conversion makes \( \frac{3}{4} \) equivalent to \( \frac{9}{12} \), allowing it to be directly compared or combined with other fractions also expressed with a denominator of 12. Understanding equivalent fractions is vital for modifying and combining fractions during calculations.
Fraction Simplification
Fraction simplification is the process of reducing fractions to their simplest form, where the numerator and denominator have no common factors other than 1. Simplifying is often the final step after performing operations like addition or subtraction to ensure the result is as simple as possible.
How do you simplify fractions?
For example, if you have \( \frac{6}{12} \), you can simplify it by dividing both numbers by their GCF, which is 6. Thus, \( \frac{6}{12} = \frac{1}{2} \). Simplification helps in achieving a cleaner and more comprehensible fraction.
How do you simplify fractions?
- Identify the greatest common factor (GCF) of the numerator and denominator.
- Divide both the numerator and the denominator by this GCF.
For example, if you have \( \frac{6}{12} \), you can simplify it by dividing both numbers by their GCF, which is 6. Thus, \( \frac{6}{12} = \frac{1}{2} \). Simplification helps in achieving a cleaner and more comprehensible fraction.
Addition and Subtraction of Fractions
To perform addition or subtraction with fractions, they must first have a common denominator. Only then can you combine or subtract the numerators directly while keeping the common denominator the same.
Let's look at an example:
For subtraction, the same principle applies. Once the fractions share a common denominator, subtract the numerators and simplify the result if necessary. Practicing these steps ensures accuracy when dealing with fraction arithmetic in both simple and complex problems.
Let's look at an example:
- Add \( \frac{5}{6} \) and \( \frac{1}{3} \). First, find a common denominator, which is 6 in this case.
- Convert \( \frac{1}{3} \) to an equivalent fraction: \( \frac{1 \times 2}{3 \times 2} = \frac{2}{6} \) so it matches the denominator of \( \frac{5}{6} \).
- Add them: \( \frac{5}{6} + \frac{2}{6} = \frac{7}{6} \).
For subtraction, the same principle applies. Once the fractions share a common denominator, subtract the numerators and simplify the result if necessary. Practicing these steps ensures accuracy when dealing with fraction arithmetic in both simple and complex problems.