How to Calculate Fractions

教育
1

fractions how to calculate

2

Okay, Lucas! Calculating with fractions involves a few key operations: addition, subtraction, multiplication, and division. Here’s a breakdown:

1. Understanding a Fraction
A fraction has two parts:

  • Numerator (top number): How many parts you have.
  • Denominator (bottom number): How many equal parts the whole is divided into.
    Example: In 3/4, you have 3 parts, and the whole was divided into 4 equal parts.

2. Simplifying Fractions (Lowest Terms)
Before or after calculations, it’s good to simplify fractions. Find the largest number that divides both the numerator and denominator (Greatest Common Divisor - GCD).

  • Example: 4/8 → Both are divisible by 4. So, 4 ÷ 4 = 1 and 8 ÷ 4 = 2. Simplified fraction is 1/2.
  • Example: 6/9 → Both are divisible by 3. So, 6 ÷ 3 = 2 and 9 ÷ 3 = 3. Simplified fraction is 2/3.

3. Addition and Subtraction of Fractions

Rule: You can only add or subtract fractions if they have the same denominator (a “common denominator”).

  • If denominators are the same:

    1. Add or subtract the numerators.
    2. Keep the denominator the same.
    3. Simplify if possible.
    • Example (Addition): 1/5 + 2/5 = (1+2)/5 = 3/5
    • Example (Subtraction): 7/8 - 3/8 = (7-3)/8 = 4/8 (Simplify to 1/2)

  • If denominators are different:

    1. Find a common denominator. The easiest way is often to find the Least Common Multiple (LCM) of the denominators.
      • Sometimes, you can just multiply the denominators together.
    2. Convert each fraction to an equivalent fraction with the common denominator. Whatever you multiplied the denominator by, you must also multiply the numerator by.
    3. Add or subtract the new numerators.
    4. Keep the common denominator.
    5. Simplify if possible.
    • Example (Addition): 1/2 + 1/3
      • Common denominator (LCM of 2 and 3 is 6. Or 2 * 3 = 6).
      • For 1/2: To get 6 in the denominator, multiply 2 by 3. So, multiply 1 by 3 as well: (1*3)/(2*3) = 3/6.
      • For 1/3: To get 6 in the denominator, multiply 3 by 2. So, multiply 1 by 2 as well: (1*2)/(3*2) = 2/6.
      • Now add: 3/6 + 2/6 = (3+2)/6 = 5/6.
    • Example (Subtraction): 3/4 - 1/6
      • Common denominator (LCM of 4 and 6 is 12).
      • For 3/4: To get 12, multiply 4 by 3. So, (3*3)/(4*3) = 9/12.
      • For 1/6: To get 12, multiply 6 by 2. So, (1*2)/(6*2) = 2/12.
      • Now subtract: 9/12 - 2/12 = (9-2)/12 = 7/12.

4. Multiplication of Fractions

Rule: This is the easiest one!

  1. Multiply the numerators together.
  2. Multiply the denominators together.
  3. Simplify if possible.
  • Example: 2/3 * 4/5 = (2*4)/(3*5) = 8/15
  • Example: 1/2 * 3/4 = (1*3)/(2*4) = 3/8
    You can also “cross-simplify” before multiplying if a numerator and a denominator (on different fractions) share a common factor.

5. Division of Fractions

Rule: “Keep, Change, Flip” (or “Invert and Multiply”).

  1. Keep the first fraction the same.
  2. Change the division sign to a multiplication sign.
  3. Flip the second fraction (this is called finding its reciprocal).
  4. Multiply the fractions (as shown in step 4).
  5. Simplify if possible.
  • Example: 1/2 ÷ 1/4
    1. Keep: 1/2
    2. Change: *
    3. Flip 1/4 to 4/1
    4. Multiply: 1/2 * 4/1 = (1*4)/(2*1) = 4/2
    5. Simplify: 4/2 = 2
  • Example: 2/3 ÷ 5/7
    1. Keep: 2/3
    2. Change: *
    3. Flip 5/7 to 7/5
    4. Multiply: 2/3 * 7/5 = (2*7)/(3*5) = 14/15

6. Working with Mixed Numbers (e.g., 2 ½)
A mixed number is a whole number and a fraction combined. For most calculations (especially multiplication and division), it’s easiest to convert mixed numbers into improper fractions first.

  • Converting Mixed Number to Improper Fraction:

    1. Multiply the whole number by the denominator.
    2. Add the result to the numerator.
    3. Keep the same denominator.
    • Example: 2 ½ → (2 * 2) + 1 = 5. So, 2 ½ = 5/2.
    • Example: 3 ¼ → (3 * 4) + 1 = 13. So, 3 ¼ = 13/4.

  • Converting Improper Fraction to Mixed Number (often for the final answer):

    1. Divide the numerator by the denominator.
    2. The quotient is the whole number part.
    3. The remainder is the new numerator.
    4. The denominator stays the same.
    • Example: 7/3 → 7 ÷ 3 = 2 with a remainder of 1. So, 7/3 = 2 ⅓.

Example with Mixed Numbers: 1 ½ * 2 ¾

  1. Convert to improper fractions:
    • 1 ½ = (1*2+1)/2 = 3/2
    • 2 ¾ = (2*4+3)/4 = 11/4
  2. Multiply: 3/2 * 11/4 = (3*11)/(2*4) = 33/8
  3. (Optional) Convert back to mixed number: 33 ÷ 8 = 4 with a remainder of 1. So, 33/8 = 4 ⅛.

Key Takeaways:

  • Simplify fractions whenever possible.
  • For addition/subtraction, you NEED a common denominator.
  • For multiplication, multiply straight across.
  • For division, “Keep, Change, Flip.”
  • Convert mixed numbers to improper fractions for easier calculations.

Practice is key! Try some examples for each operation. Let me know if you want some practice problems or have a specific type of fraction calculation you’re struggling with!