Working with Fractions
Working with Fractions: Mastering the Parts of a Whole
This chapter dives into the world of fractions, a fundamental concept used to represent parts of a whole. You’ll explore identifying and simplifying fractions, performing operations with them, converting between mixed numbers and improper fractions, and utilizing the least common multiple (LCM) and greatest common factor (GCF).
Identifying and Simplifying Fractions
A fraction represents a part of a whole divided into equal parts. It consists of two parts:
- Numerator (Top Number): Represents the number of parts being considered.
- Denominator (Bottom Number): Represents the total number of equal parts the whole is divided into.
Examples:
- 1/2: One out of two equal parts of a whole pie.
- 3/4: Three out of four equal slices of a cake.
Simplifying Fractions:
A fraction is in its simplest form when the numerator and denominator have no common factors (numbers that divide both) other than 1. We can simplify fractions by dividing both the numerator and denominator by their greatest common factor (GCD).
Example: Simplify 6/12
- Find the GCF of 6 and 12: GCF(6, 12) = 6
- Divide both numerator and denominator by 6: 6/12 = (6/6) / (12/6) = 1/2 (Simplest form)
Practice Problems:
- Identify the numerator and denominator in the fraction 5/8. (Numerator: 5, Denominator: 8)
- Simplify the fraction 10/25. (10/25 = 2/5)
Adding, Subtracting, Multiplying, and Dividing Fractions
Performing operations with fractions requires considering the relationship between the numerators and denominators.
Adding and Subtracting Fractions:
- Fractions must have the same denominator (like units) to be added or subtracted directly.
- If the fractions have different denominators, find the least common multiple (LCM) of the denominators and rewrite the fractions with the LCM as the new denominator.
- Add or subtract the numerators, keeping the denominator the same.
Example: Add 1/3 + 1/4
- LCM(3, 4) = 12 (Least common multiple)
- Rewrite fractions: 1/3 = (4/4) x (1/3) = 4/12 ; 1/4 = (3/3) x (1/4) = 3/12
- Add: 4/12 + 3/12 = 7/12
Multiplying Fractions:
- Multiply the numerators and the denominators separately.
Example: Multiply 2/5 x 3/4
- 2/5 x 3/4 = (2 x 3) / (5 x 4) = 6/20 (Simplify: 6/20 = 3/10)
Dividing Fractions:
- Dividing by a fraction is the same as multiplying by the reciprocal of the fraction (flip the numerator and denominator).
Example: Divide 1/2 ÷ 3/4
- Reciprocal of 3/4 is 4/3
- 1/2 ÷ 3/4 = 1/2 x 4/3 = (1 x 4) / (2 x 3) = 4/6 (Simplify: 4/6 = 2/3)
Practice Problems:
- Solve: 2/3 + 1/4 = ( ) (Find LCM first)
- Solve: 3/5 x 2/7 = ( )
- Solve: 1/4 ÷ 2/3 = ( ) (Flip the divisor)
Converting Between Mixed Numbers and Improper Fractions
Mixed Numbers:
A mixed number combines a whole number and a fraction. It represents a whole divided into parts, where some parts are whole and some are fractional.
Example: 1 ¾ represents 1 whole and ¾ of another whole.
Improper Fractions:
An improper fraction has a numerator that is greater than or equal to the denominator. It represents a quantity larger than a whole.
Example: 5/3 represents a quantity larger than dividing something into 3 parts.
Conversions:
- Mixed Number to Improper Fraction: Multiply the whole number part by the denominator, add the product to the numerator, and use the denominator as the new denominator.
Example: Convert 1 ¾ to an improper fraction
1 ¾ = (1 x 4) + 3 / 4 = 7/4
Improper Fraction to Mixed Number: Divide the numerator by the denominator. The quotient becomes the whole number part of the mixed number (ignore any remainder for now). The remainder becomes the new numerator, and the original denominator remains the same. If the division results in a zero remainder, it’s already a whole number, not a mixed number.
Example: Convert 7/4 to a mixed number
- 7 ÷ 4 = 1 R 3 (Quotient 1, Remainder 3)
- The improper fraction becomes 1 ¾ (1 whole and ¾ remaining)
Practice Problems:
1. Convert the following mixed number to an improper fraction: 2 ½ (2 x 2 + ½ = 5/2)
2. Convert the following improper fraction to a mixed number: 11/3 (11 ÷ 3 = 3 R 2, so 3 ⅔)
Least Common Multiple (LCM) and Greatest Common Factor (GCF)
The LCM and GCF are essential concepts for working with fractions.
- Least Common Multiple (LCM): The smallest number that is a multiple of two or more integers. It’s like finding the least common denominator for fractions.
- Greatest Common Factor (GCF): The largest number that is a factor of two or more integers. It helps simplify fractions by dividing both numerator and denominator by the GCF.
We can find the LCM and GCF using various methods, including prime factorization (breaking down numbers into their prime factors) or listing multiples.
Example:
Find the LCM and GCF of 6 and 12:
-
- Prime factorization: 6 = 2 x 3; 12 = 2 x 2 x 3
- LCM: Find the highest power of each prime factor present in either number. LCM = 2 x 2 x 3 = 12
- GCF: Find the factors common to both numbers raised to the lowest power. GCF = 2 x 3 = 6
Understanding LCM and GCF is crucial for adding, subtracting, and comparing fractions effectively.
Practice Problems:
- Find the LCM of 8 and 10: (LCM = 40)
- Find the GCF of 15 and 25: (GCF = 5)
By mastering these concepts and practicing the various operations, you’ll be well-equipped to handle fractions with confidence, a valuable skill for the HESI A2 exam and various applications. This chapter has equipped you with a strong foundation in working with fractions. Remember, consistent practice and a clear understanding of these concepts will lead to success.