Simplifying Fractions and Finding Common Denominators

Definition of Simplification

  • In mathematics, the instruction to simplify means to carry out the indicated operation or operations and leave the final answer in its simplest form.

Addition and Subtraction of Fractions with Like Denominators

  • When two or more fractions share the exact same denominator:
    • Combine the numerators into a single fraction over the common denominator.
    • Perform the addition or subtraction operation in the numerator.
    • Simplify the numerator to write the final result.

Addition and Subtraction of Fractions with Unlike Denominators

  • When fractions do not share the same denominator:
    • Determine whether one denominator can be multiplied by an integer to match the other denominator.
    • If no single integer multiplier works, determine the common denominator by multiplying the two original denominators together.
    • Convert each fraction into an equivalent fraction with the new common denominator.
    • Perform the indicated addition or subtraction across the numerators over the common denominator.
    • Simplify the numerator to get the final answer.

Fundamental Rule of Equivalent Fractions

  • Any operation performed on the denominator of a fraction must also be performed on the numerator.
  • Altering only the denominator without making the proportional change to the numerator changes the overall value of the fraction, resulting in errors.
  • Analogy: Changing only the denominator and leaving the numerator untouched is like treating one child in a family while ignoring another—both the numerator and denominator must be altered together so that the value remains equivalent.

Worked Examples

  • Example 1: Fraction Addition with Like Denominators

    • Expression:     617+217\frac{6}{17} + \frac{2}{17}
    • Step 1: Identify that both fractions have the same denominator of 1717.
    • Step 2: Combine the fractions into a single fraction over the shared denominator 1717:     6+217\frac{6 + 2}{17}
    • Step 3: Simplify the numerator by adding 6+2=86 + 2 = 8:     817\frac{8}{17}

  • Example 2: Fraction Subtraction with Like Denominators

    • Expression:     411−111\frac{4}{11} - \frac{1}{11}
    • Step 1: Identify that both fractions have the same denominator of 1111.
    • Step 2: Combine the fractions into a single fraction over the shared denominator 1111:     4−111\frac{4 - 1}{11}
    • Step 3: Simplify the numerator by subtracting 4−1=34 - 1 = 3:     311\frac{3}{11}

  • Example 3: Fraction Subtraction with Unlike Denominators

    • Expression:     59−14\frac{5}{9} - \frac{1}{4}
    • Step 1: Compare denominators 99 and 44. Since 4×integer=94 \times \text{integer} = 9 is not possible, find the common denominator by multiplying 9×4=369 \times 4 = 36.
    • Step 2: Convert the first fraction 59\frac{5}{9} into an equivalent fraction with a denominator of 3636 by multiplying both numerator and denominator by 44:     5×49×4=2036\frac{5 \times 4}{9 \times 4} = \frac{20}{36}
    • Step 3: Convert the second fraction 14\frac{1}{4} into an equivalent fraction with a denominator of 3636 by multiplying both numerator and denominator by 99:     1×94×9=936\frac{1 \times 9}{4 \times 9} = \frac{9}{36}
    • Step 4: Combine into a single fraction over the common denominator 3636:     20−936\frac{20 - 9}{36}
    • Step 5: Simplify the numerator (20−9=1120 - 9 = 11):     1136\frac{11}{36}

  • Example 4: Fraction Addition with Unlike Denominators

    • Expression:     37+25\frac{3}{7} + \frac{2}{5}
    • Step 1: Check if denominator 55 can be multiplied by an integer to obtain 77:     5×1=55 \times 1 = 55×2=105 \times 2 = 105×3=155 \times 3 = 15     Because 77 is skipped, a common denominator is found by multiplying the two denominators together: 7×5=357 \times 5 = 35.
    • Step 2: Convert the first fraction 37\frac{3}{7} by multiplying both numerator and denominator by 55:     3×57×5=1535\frac{3 \times 5}{7 \times 5} = \frac{15}{35}
    • Step 3: Convert the second fraction 25\frac{2}{5} by multiplying both numerator and denominator by 77:     2×75×7=1435\frac{2 \times 7}{5 \times 7} = \frac{14}{35}
    • Step 4: Write as a sum of equivalent fractions:     1535+1435\frac{15}{35} + \frac{14}{35}
    • Step 5: Combine into a single fraction over the denominator 3535:     15+1435\frac{15 + 14}{35}
    • Step 6: Simplify the numerator (15+14=2915 + 14 = 29):     2935\frac{29}{35}

Questions and Discussion

  • Strategy for Finding Common Denominators:

    • Question: Is it best to pick a random number or try out different numbers to determine a common denominator?
    • Response: Test multiples sequentially starting with smaller factors (e.g., 5×1=55 \times 1 = 5, 5×2=105 \times 2 = 10, 5×3=155 \times 3 = 15). If the target denominator cannot be obtained through integer multiplication, multiply both denominators together (7×5=357 \times 5 = 35) and alter both fractions accordingly.

  • Grade Inquiries:

    • Question: Can individual marks be disclosed privately after class?
    • Response: Yes, individual marks can be reviewed privately after class, though official marks are also posted on Meskenaz.