Turn a word problem into an equation

• Define an unknown variable with a precise contextual meaning and unit.
• Translate quantities and comparison language into one coherent equation.
• Check the mathematical solution against the original wording and realistic constraints.

Build the idea

Worked example

1. Step 1

   Let s be the number of students. The count must be a nonnegative whole number because partial students are not meaningful.

   $$s=\text{number of students}$$

2. Step 2

   The booking fee contributes twelve dollars once, and the student charge contributes seven s dollars. Their sum equals the paid total.

   $$12+7s=110$$

3. Step 3

   Subtract twelve to get ninety-eight, then divide by seven. The result fourteen is a valid whole-number count.

   $$7s=98\Rightarrow s=14$$

• Multiplying the fixed booking fee by the student count.

  A fixed fee occurs once. Only the phrase per student indicates a quantity that scales with the count.

• Reporting fourteen dollars instead of fourteen students.

  Return to the variable definition before stating the answer so the number receives the correct contextual unit.

A bicycle rental costs nine dollars plus four dollars per hour. The total is twenty-nine dollars. Find the rental time and check the cost.

This reviewed reading order covers the lesson explanation, equations, example, and checks without claiming that a video exists.

1. Part 1

   Lesson overview

   A word problem becomes manageable when the unknown, units, quantities, and equality relationship are named before arithmetic begins. Translation starts by deciding what one symbol represents. A variable without a unit or contextual meaning makes later expressions difficult to interpret and easy to misuse. Define an unknown variable with a precise contextual meaning and unit.

2. Part 2

   Visual model and equations

   Each phrase should describe a quantity before those quantities are connected. Words such as total, difference, per, more than, and remaining indicate operations, but their order depends on the sentence. Place the unknown in a labeled center box, connect each sentence phrase to an expression with units, and join the two expressions that the story says are equal. Total equals fixed amount plus rate times count. The units on the left must match the units on the right.

   • Total equals fixed amount plus rate times count.
   • The units on the left must match the units on the right.
3. Part 3

   Worked example

   A museum charges a twelve-dollar booking fee plus seven dollars per student. A class pays one hundred ten dollars. How many students attended? Define the student count, write fixed plus per-student cost equals total cost, solve the linear equation, and verify the bill. Let s be the number of students. The count must be a nonnegative whole number because partial students are not meaningful. The booking fee contributes twelve dollars once, and the student charge contributes seven s dollars. Their sum equals the paid total. Subtract twelve to get ninety-eight, then divide by seven. The result fourteen is a valid whole-number count. Fourteen students attended the museum visit. Seven dollars times fourteen students is ninety-eight dollars; adding the twelve-dollar booking fee gives one hundred ten dollars, matching the stated total.

   • s equals the number of students.
   • Twelve plus seven s equals one hundred ten.
   • Seven s equals ninety-eight, so s equals fourteen.
4. Part 4

   Checks, practice, and scope

   The equation expresses one equality relationship from the story. Solving produces a candidate, and substituting that value back into every quantity checks both the algebra and the interpretation. Multiplying the fixed booking fee by the student count. A fixed fee occurs once. Only the phrase per student indicates a quantity that scales with the count. Reporting fourteen dollars instead of fourteen students. Return to the variable definition before stating the answer so the number receives the correct contextual unit. A bicycle rental costs nine dollars plus four dollars per hour. The total is twenty-nine dollars. Find the rental time and check the cost. Let h be hours and write nine plus four h equals twenty-nine. The bicycle was rented for five hours because nine plus four times five equals twenty-nine. A precise variable definition anchors every later expression and final unit. Translate quantities first, then connect the relationship stated by the problem. Contextual substitution checks modeling choices in addition to algebra. This lesson uses a single linear relationship; multistep rates, systems, probability, and changing-rate models require additional structures. Real situations may include rounding, taxes, capacity limits, or measurement uncertainty not stated in a simplified exercise.

Scope and limitations

• This lesson uses a single linear relationship; multistep rates, systems, probability, and changing-rate models require additional structures.
• Real situations may include rounding, taxes, capacity limits, or measurement uncertainty not stated in a simplified exercise.