One-Step Equations with Fractions & Decimals - Sixth Grade Mathematics

Solving one-step equations with rational numbers applies properties of equality to fractional and decimal values, using reciprocal multiplication and decimal alignment.

Solving Equations Involving Decimals

Equations with decimals follow the exact same operational rules as whole number equations. Perform the inverse operation using standard decimal arithmetic.

Case A: Decimal Addition (x + 3.4 = 8.1)
        x + 3.4 =  8.1
          - 3.4   -3.4    <-- Subtract 3.4 from both sides
        --------------
        x       =  4.7

Case B: Decimal Multiplication (0.6y = 4.2)
        Divide both sides by 0.6:
        y = 4.2 / 0.6
        Shift decimal 1 place: y = 42 / 6 = 7.

Solving Equations Involving Fractions

When equations contain fractions, finding common denominators or multiplying by reciprocals isolates the variable efficiently.

Addition and Subtraction with Fractions

When solving an equation such as x - 3/8 = 1/2: Add 3/8 to both sides: x = 1/2 + 3/8. Find the common denominator (LCD = 8): 1/2 = 4/8. Add fractions: x = 4/8 + 3/8 = 7/8.

Multiplication by a Fraction: The Reciprocal Shortcut

When a variable is multiplied by a fraction (such as 2/3 * x = 8), dividing both sides by 2/3 is equivalent to multiplying both sides by its reciprocal (3/2).

The Reciprocal Multiplication Strategy:
                (2 / 3) * x  =  8

Step 1: Multiply BOTH sides by the reciprocal (3 / 2):
        (3 / 2) * (2 / 3) * x  =  (8 / 1) * (3 / 2)

Step 2: The left side simplifies to 1x:
        x  =  (8 * 3) / (1 * 2)  =  24 / 2

Step 3: Simplify:
        x  =  12.
Check: 2/3 of 12 = 8. (Confirmed!)

Clearing Fractions by Multiplying by the LCD

When an equation contains multiple fractions, you can eliminate all denominators in a single step by multiplying every term on both sides by the Least Common Denominator (LCD).

Solve: x / 4 + 1 / 2 = 5 / 4
Step 1: The LCD of 4, 2, and 4 is 4.
Step 2: Multiply every term by 4:
        4 * (x / 4) + 4 * (1 / 2) = 4 * (5 / 4)
Step 3: Denominators cancel completely:
        x + 2 = 5
Step 4: Subtract 2:
        x = 3.

Exercises

Exercise 1. Solve each decimal equation: (a) x + 4.85 = 9.2; (b) y - 3.6 = 11.45; (c) 0.8m = 5.6.

Exercise 2. Solve each fraction equation: (a) w + 2/5 = 7/10; (b) k - 1/3 = 5/6; (c) (3/4)x = 15.

Exercise 3. A container of juice is 2/3 full, containing 18 ounces. Write a multiplication equation with a fraction to find the total container capacity c, and solve it.

Solutions

Solution to Exercise 1. For (a) x + 4.85 = 9.2: subtract 4.85 from both sides: x = 9.20 - 4.85 = 4.35. For (b) y - 3.6 = 11.45: add 3.6 to both sides: y = 11.45 + 3.60 = 15.05. For (c) 0.8m = 5.6: divide both sides by 0.8: m = 5.6 / 0.8 = 56 / 8 = 7.

Solution to Exercise 2. For (a) w + 2/5 = 7/10: subtract 2/5 from both sides: w = 7/10 - 4/10 = 3/10. For (b) k - 1/3 = 5/6: add 1/3 to both sides: k = 5/6 + 2/6 = 7/6 = 1 and 1/6. For (c) (3/4)x = 15: multiply both sides by reciprocal 4/3: x = 15 * (4/3) = (15 / 3) * 4 = 5 * 4 = 20.

Solution to Exercise 3. The problem states that 2/3 of the capacity c equals 18: (2/3)c = 18. Multiply both sides by the reciprocal 3/2: c = 18 * (3/2). Divide 18 by 2 to get 9, then multiply by 3: c = 9 * 3 = 27. The total capacity of the container is 27 ounces.