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Fraction and Mixed Number Operations

Fraction and Mixed Number Operations

When solving math problems that include a mix of addition, subtraction, multiplication, and fractions, you need to combine your fraction calculation skills with the order of operations.

The Golden Rules for Mixed Numbers

Before you multiply or divide, always convert mixed numbers into improper fractions.

For example, to convert 1231\frac{2}{3}, multiply the whole number by the denominator (1ร—3=31 \times 3 = 3), then add the numerator (3+2=53 + 2 = 5). Keep the denominator the same: 123=531\frac{2}{3} = \frac{5}{3}.

Applying the Order of Operations

Remember PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction). Multiplication and division always happen before addition and subtraction unless parentheses tell you otherwise.

Example 1: Fractions with Multiple Operations

Evaluate: 23ร—45+16\frac{2}{3} \times \frac{4}{5} + \frac{1}{6}

  1. Multiply first: 23ร—45=815\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}
  2. Add next: Now solve 815+16\frac{8}{15} + \frac{1}{6}. You need a common denominator. The least common multiple (LCM) of 15 and 6 is 30. 8ร—215ร—2=1630\frac{8 \times 2}{15 \times 2} = \frac{16}{30} 1ร—56ร—5=530\frac{1 \times 5}{6 \times 5} = \frac{5}{30}
  3. Find the sum: 1630+530=2130\frac{16}{30} + \frac{5}{30} = \frac{21}{30}
  4. Simplify: Divide the top and bottom by 3 to get 710\frac{7}{10}.

Example 2: Working with Mixed Numbers

Evaluate: 314โˆ’123ร—23\frac{1}{4} - 1\frac{2}{3} \times 2

  1. Convert to improper fractions: 314=1343\frac{1}{4} = \frac{13}{4} 123=531\frac{2}{3} = \frac{5}{3} 2=212 = \frac{2}{1} The problem is now: 134โˆ’53ร—21\frac{13}{4} - \frac{5}{3} \times \frac{2}{1}
  2. Multiply first: 53ร—21=103\frac{5}{3} \times \frac{2}{1} = \frac{10}{3}
  3. Subtract: Now solve 134โˆ’103\frac{13}{4} - \frac{10}{3}. The common denominator for 4 and 3 is 12. 13ร—34ร—3=3912\frac{13 \times 3}{4 \times 3} = \frac{39}{12} 10ร—43ร—4=4012\frac{10 \times 4}{3 \times 4} = \frac{40}{12} 3912โˆ’4012=โˆ’112\frac{39}{12} - \frac{40}{12} = -\frac{1}{12}

Example 3: Finding a Fraction "Of" a Number

Find 34\frac{3}{4} of 2252\frac{2}{5}

In math, the word "of" usually means multiply.

  1. Set up the multiplication and convert the mixed number: 34ร—125\frac{3}{4} \times \frac{12}{5}
  2. Multiply (and cross-simplify if you can): Notice that 12 and 4 can both be divided by 4. 31ร—35=95\frac{3}{1} \times \frac{3}{5} = \frac{9}{5}
  3. Convert back to a mixed number: 95=145\frac{9}{5} = 1\frac{4}{5}