Fractions, Decimals, and Mixed Numbers

1.2: Fractions, Decimals, and Mixed Numbers. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 2 of 53 parts Practice Flashcards

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Equivalent fractions and decimals

Objective: Convert fractions to decimals by division.

Prerequisite skills: integer operations and equivalent ratios. A fraction a/b is the quotient a ÷ b. Multiplying numerator and denominator by the same nonzero number preserves the value. Terminating decimals can be written over a power of ten and reduced. Common error: dividing denominator by numerator. Restrictions and interpretation: A denominator must be nonzero; convert a mixed number to an improper fraction before multiplying.

Formulas

$a/b = (ka)/(kb), b ≠ 0, k ≠ 0$

Worked examples

Try itWhat decimal equals 22/20?Show answer

Divide 22 by 20, or multiply numerator and denominator by 5: 110/100 = 1.1. Therefore the requested answer is 1.1.

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Equivalent fractions and decimals

Objective: Convert fractions to decimals by division.

Formulas

$a/b = (ka)/(kb), b ≠ 0, k ≠ 0$

Worked examples

Try itWhat decimal equals 22/20?Show answer

Divide 22 by 20, or multiply numerator and denominator by 5: 110/100 = 1.1. Therefore the requested answer is 1.1.

Adding fractions

Objective: Find a common denominator before adding.

Addition combines equal-sized parts. Rewrite each fraction using a common denominator, add numerators, and keep that denominator. Simplify afterward. Adding denominators invents a different unit. Common error: adding denominators as well as numerators. Restrictions and interpretation: A denominator must be nonzero; convert a mixed number to an improper fraction before multiplying.

Formulas

$a/b + c/d = (ad + bc)/(bd)$

Worked examples

Try itEvaluate 22/3 + 24/6.Show answer

Rewrite 22/3 as 44/6. Add 44/6 + 24/6 = 68/6 = 34/3. Therefore the requested answer is 34/3.

Mixed-number products and reciprocal division

Objective: Use improper fractions and reciprocals in a multistep calculation.

A mixed number q r/s equals (qs + r)/s. Division by a nonzero fraction is multiplication by its reciprocal. Cancel factors only within products; do not cancel terms across addition. Common error: multiplying by the divisor instead of its reciprocal. Restrictions and interpretation: A denominator must be nonzero; convert a mixed number to an improper fraction before multiplying. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.

Formulas

$(a/b) ÷ (c/d) = ad/(bc), b,c,d ≠ 0$

Worked examples

Try itEvaluate (24 1/2) ÷ (2/3).Show answer

Convert 24 1/2 to 49/2. Multiply by 3/2: (49 × 3)/(2 × 2) = 147/4 = 36.75. Therefore the requested answer is 36.75.

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