Writing Equations from Word Problems - Sixth Grade Mathematics

Translating real-world situations into algebraic equations defines variables, models physical constraints, and enables systematic problem solving across science, commerce, and daily life.

The Four-Phase Problem Solving Architecture

Setting up and solving an equation from a word problem requires structured mathematical modeling.

+-------------------------------------------------------------------------+
|                  EQUATION WORD PROBLEM ARCHITECTURE                     |
|                                                                         |
|   Phase 1: Define the Variable. Explicitly state what quantity the      |
|            letter represents: "Let x = ...", including proper units.    |
|   Phase 2: Construct the Equation. Translate the narrative relationships|
|            into an algebraic equation with an equal sign.               |
|   Phase 3: Solve the Equation. Apply properties of equality to isolate  |
|            the variable.                                                |
|   Phase 4: Interpret and Check. State the final answer in a complete    |
|            sentence with units, and verify reasonableness.              |
+-------------------------------------------------------------------------+

Phase 1: Explicitly Defining the Variable

Never begin writing an equation without first declaring your variable. Poor Practice: Starting calculations without stating what x means. Professional Mathematical Practice: "Let d = the distance traveled in miles." Defining the variable clarifies exactly what your final calculated number represents.

Identifying the Governing Mathematical Relationship

Look for the underlying structural pattern described in the narrative.

+-------------------------------------------------------------------------+
|                  COMMON REAL-WORLD EQUATION PATTERNS                    |
+-------------------+-----------------------+-----------------------------+
| Scenario Pattern  | Narrative Description | Equation Form               |
+-------------------+-----------------------+-----------------------------+
| Combining Parts   | A starting amount is  | x + a = Total               |
| to a Total        | increased by another  |                             |
+-------------------+-----------------------+-----------------------------+
| Remaining Balance | An amount is removed  | Total - x = Remaining       |
| / Difference      | or spent              | or  x - a = Balance         |
+-------------------+-----------------------+-----------------------------+
| Equal Groups /    | Items are split or    | r * x = Total  (e.g., rate) |
| Rates             | multiplied in batches | Total / x = Size per group  |
+-------------------+-----------------------+-----------------------------+

Worked Case Study:
Sophia earned $75 babysitting this weekend. This was $18 more than she
earned last weekend. How much did Sophia earn last weekend?

Step 1 (Define Variable):
Let m = the amount of money Sophia earned last weekend (in dollars).

Step 2 (Construct Equation):
Sophia's earnings last weekend plus $18 equals this weekend's total:
m + 18 = 75

Step 3 (Solve):
Subtract 18 from both sides:
m = 75 - 18
m = 57

Step 4 (Interpret & Verify):
Sophia earned $57 last weekend.
Check: $57 + $18 = $75. (Verified!)

Exercises

Exercise 1. A bakery sold 148 cupcakes on Saturday. This was 4 times the number of cupcakes sold on Monday. Define a variable, write a multiplication equation, and determine how many cupcakes were sold on Monday.

Exercise 2. Ethan had a wooden board. After cutting off a piece measuring 2.4 meters, the remaining piece of board was 3.8 meters long. Write an equation with a variable for the original length of the board, and solve it.

Exercise 3. A school bus transported 45 students on a field trip. If 1/3 of the students on the bus were sixth graders, write a division or fraction equation to find the total number of sixth graders s, and solve it.

Solutions

Solution to Exercise 1. Phase 1: Let c = the number of cupcakes sold on Monday. Phase 2: Saturday's sales (148) was 4 times Monday's sales: 4c = 148. Phase 3: Solve by dividing both sides by 4: c = 148 / 4 = 37. Phase 4: Exactly 37 cupcakes were sold on Monday. (Check: 4 * 37 = 148).

Solution to Exercise 2. Phase 1: Let L = original length of the board in meters. Phase 2: Cutting off 2.4 meters leaves 3.8 meters: L - 2.4 = 3.8. Phase 3: Solve by adding 2.4 to both sides: L = 3.8 + 2.4 = 6.2 meters. Phase 4: The original length of the board was 6.2 meters. (Check: 6.2 - 2.4 = 3.8).

Solution to Exercise 3. Phase 1: Let s = the number of sixth-grade students. Phase 2: Since 1/3 of the 45 students are sixth graders, write: s = (1/3) * 45 (or 3s = 45). Phase 3: Solve: s = 45 / 3 = 15. Phase 4: There were 15 sixth-grade students on the bus. (Check: 15 * 3 = 45).