Chapter 3: Problem 25
Find each sum or difference, showing each step of your work. Give your answers in lowest terms. If an answer is greater than 1 , write it as a mixed number. $$2 \frac{1}{2}-\frac{7}{9}$$
Short Answer
Expert verified
1 \frac{13}{18}
Step by step solution
01
Convert Mixed Number to Improper Fraction
First, convert the mixed number to an improper fraction. For the mixed number 2 \frac{1}{2}, the improper fraction is calculated as follows:\[2 \frac{1}{2} = \frac{2 \times 2 + 1}{2} = \frac{5}{2}\]
02
Find a Common Denominator
To subtract the fractions, ensure they have a common denominator. The denominators are 2 and 9. The least common multiple (LCM) of 2 and 9 is 18.
03
Convert Fractions to Equivalent Fractions
Convert \( \frac{5}{2} \) and \(\frac{7}{9}\) to fractions with a denominator of 18:\( \frac{5}{2} = \frac{5 \times 9}{2 \times 9} = \frac{45}{18}\)\( \frac{7}{9} = \frac{7 \times 2}{9 \times 2} = \frac{14}{18}\)
04
Subtract the Fractions
Subtract the numerators of the equivalent fractions and keep the common denominator:\( \frac{45}{18} - \frac{14}{18} = \frac{45 - 14}{18} = \frac{31}{18}\)
05
Convert Improper Fraction to Mixed Number
Convert the improper fraction \(\frac{31}{18}\) to a mixed number by dividing the numerator by the denominator:\( \frac{31}{18} = 1 \frac{13}{18}\). Thus, the improper fraction \(\frac{31}{18}\) is equivalent to the mixed number \(1 \frac{13}{18}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
mixed numbers
A mixed number is a whole number combined with a fraction. For example, in the expression 2 \(\frac{1}{2}\), 2 is the whole number and \(\frac{1}{2}\) is the fraction. Mixed numbers are often used in daily life, such as in cooking or measuring lengths. To work with mixed numbers in mathematical operations, it's often necessary to convert them to improper fractions first. This makes calculations easier.
To convert a mixed number to an improper fraction, multiply the whole number by the denominator of the fraction, then add the numerator. Place this sum over the original denominator. For instance:
\(2 \frac{1}{2} = \frac{2 \times 2 + 1}{2} = \frac{5}{2}\)
This improper fraction is now easier to work with in operations like addition, subtraction, multiplication, and division.
To convert a mixed number to an improper fraction, multiply the whole number by the denominator of the fraction, then add the numerator. Place this sum over the original denominator. For instance:
\(2 \frac{1}{2} = \frac{2 \times 2 + 1}{2} = \frac{5}{2}\)
This improper fraction is now easier to work with in operations like addition, subtraction, multiplication, and division.
improper fractions
An improper fraction has a numerator larger than or equal to its denominator. For example, \(\frac{5}{2}\) and \(\frac{9}{9}\) are improper fractions. They can be converted back to mixed numbers if needed. Improper fractions are essential in fraction arithmetic because they make operations straightforward. To convert an improper fraction to a mixed number, divide the numerator by the denominator. The quotient is the whole number part, and the remainder is the numerator of the fractional part, over the original denominator. For example:
\(\frac{31}{18} = 1 \frac{13}{18}\)
\(\frac{31}{18} = 1 \frac{13}{18}\)
least common multiple (LCM)
Finding the least common multiple (LCM) of two numbers is crucial when adding or subtracting fractions with different denominators. The LCM of two numbers is the smallest number that both original numbers can divide into without leaving a remainder. To find the LCM of 2 and 9:
- List the multiples of each number:
- Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, ...
- Multiples of 9: 9, 18, 27, 36, ...
- Identify the smallest common multiple, which is 18.
equivalent fractions
Equivalent fractions represent the same value, even though they have different numerators and denominators. For example, \(\frac{1}{2}\) is equivalent to \(\frac{2}{4}\) and \(\frac{3}{6}\). To convert fractions to equivalent forms with a common denominator (such as the LCM), multiply both the numerator and the denominator by the same number.
For example, to convert \(\frac{5}{2}\) and \(\frac{7}{9}\) to have a common denominator of 18:
For example, to convert \(\frac{5}{2}\) and \(\frac{7}{9}\) to have a common denominator of 18:
- Multiply the numerator and denominator of \(\frac{5}{2}\) by 9:
\(\frac{5}{2} = \frac{5 \times 9}{2 \times 9} = \frac{45}{18}\) - Multiply the numerator and denominator of \(\frac{7}{9}\) by 2:
\(\frac{7}{9} = \frac{7 \times 2}{9 \times 2} = \frac{14}{18}\)
simplification
Simplification in fractions involves reducing a fraction to its lowest terms. This means making the numerator and denominator as small as possible while keeping the same value. For instance, \(\frac{4}{8}\) simplifies to \(\frac{1}{2}\) because both 4 and 8 are divisible by 4. To simplify a fraction:
- Find the greatest common divisor (GCD) of the numerator and the denominator.
- Divide both the numerator and the denominator by the GCD.