Chapter 3: Problem 21
Find each sum or difference. Give your answers in lowest terms. If an answer is greater than \(1,\) write it as a mixed number. $$\frac{25}{32}+\frac{7}{24}$$
Short Answer
Expert verified
1 \(\frac{7}{96}\)
Step by step solution
01
- Determine the Least Common Denominator (LCD)
Find the least common multiple (LCM) of the denominators 32 and 24. The LCM of 32 and 24 is 96.
02
- Convert Fractions to Common Denominator
Rewrite \(\frac{25}{32}\) and \(\frac{7}{24}\) with a denominator of 96. \[ \frac{25}{32} = \frac{25 \times 3}{32 \times 3} = \frac{75}{96} \] and \[ \frac{7}{24} = \frac{7 \times 4}{24 \times 4} = \frac{28}{96} \]
03
- Add the Fractions
Add the numerators and keep the denominator the same: \[ \frac{75}{96} + \frac{28}{96} = \frac{75 + 28}{96} = \frac{103}{96} \]
04
- Simplify the Fraction
Since \(\frac{103}{96}\) is an improper fraction (numerator is greater than the denominator), convert it to a mixed number: \[ \frac{103}{96} = 1 \frac{7}{96} \]
05
- Verify the Answer
Verify the sum \(\frac{103}{96}\) is in lowest terms. The greatest common divisor (GCD) of 7 and 96 is 1, so \(\frac{7}{96}\) is already in lowest terms.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Least Common Denominator
To add or subtract fractions with different denominators, we need a common denominator. The least common denominator (LCD) is the smallest number that both denominators divide into evenly. For example, to add \(\frac{25}{32}\) and \(\frac{7}{24}\), we need to find the LCD of 32 and 24. We do this by finding the least common multiple (LCM) of these numbers.
The LCM of 32 and 24 is 96, so we use 96 as our common denominator. This allows us to rewrite each fraction with the same denominator, making it possible to add the fractions together.
The LCM of 32 and 24 is 96, so we use 96 as our common denominator. This allows us to rewrite each fraction with the same denominator, making it possible to add the fractions together.
Mixed Numbers
Mixed numbers consist of an integer and a fraction. They are used to express improper fractions in a more readable form. An improper fraction has a numerator larger than the denominator. Converting it to a mixed number involves dividing the numerator by the denominator.
For instance, after adding our fractions \(\frac{75}{96} + \frac{28}{96} = \frac{103}{96}\), we get an improper fraction. We convert \(\frac{103}{96}\) to a mixed number by dividing 103 by 96, which gives 1 with a remainder of 7. So, \(\frac{103}{96} = 1 \frac{7}{96}\).
For instance, after adding our fractions \(\frac{75}{96} + \frac{28}{96} = \frac{103}{96}\), we get an improper fraction. We convert \(\frac{103}{96}\) to a mixed number by dividing 103 by 96, which gives 1 with a remainder of 7. So, \(\frac{103}{96} = 1 \frac{7}{96}\).
Simplifying Fractions
Simplifying fractions means reducing them to their lowest terms. This happens when the numerator and denominator are divided by their greatest common divisor (GCD).
For example, if after adding or subtracting fractions, we get a fraction like \(\frac{6}{8}\), we simplify it by dividing both the numerator and denominator by 2 (the GCD of 6 and 8). Simplified, this fraction becomes \(\frac{3}{4}\).
In our solution, the fraction \(\frac{7}{96}\) is already in its simplest form because 7 and 96 have no common divisors other than 1. So, no further simplification is needed.
For example, if after adding or subtracting fractions, we get a fraction like \(\frac{6}{8}\), we simplify it by dividing both the numerator and denominator by 2 (the GCD of 6 and 8). Simplified, this fraction becomes \(\frac{3}{4}\).
In our solution, the fraction \(\frac{7}{96}\) is already in its simplest form because 7 and 96 have no common divisors other than 1. So, no further simplification is needed.
Improper Fractions
Improper fractions have numerators larger than their denominators. They represent values greater than or equal to 1. Converting improper fractions to mixed numbers helps make them easier to understand.
For example, \(\frac{103}{96}\) is an improper fraction. When we divide 103 by 96, we get 1 as the quotient and 7 as the remainder. Hence, \(\frac{103}{96}\) converts to the mixed number \(1 \frac{7}{96}\).
Understanding how to work with improper fractions and convert them to mixed numbers is crucial for many math problems.
For example, \(\frac{103}{96}\) is an improper fraction. When we divide 103 by 96, we get 1 as the quotient and 7 as the remainder. Hence, \(\frac{103}{96}\) converts to the mixed number \(1 \frac{7}{96}\).
Understanding how to work with improper fractions and convert them to mixed numbers is crucial for many math problems.