How can I solve a word problem about a fixed amount plus a repeating amount?

Separate the story into two parts: what is already there at the beginning, and what is added or changed each time. Let x count the number of repeated steps. If the starting amount is b and each step adds m, the total after x steps is y = b + mx. In this model, b is the initial value and m is the constant rate of change. [S1]

For example, suppose a jar already contains 12 stickers, and someone adds 4 stickers each week. These are example numbers, not facts about your situation. The starting amount is 12 stickers; the repeated change is 4 stickers per week. If x is the number of weeks, the model is y = 12 + 4x, where y is the number of stickers. Keep the units visible: the 4 is per week, while y is a sticker count.

Use x for the count of repeated steps—weeks in this example—and multiply that count by the amount added per step. The starting amount is not multiplied, because it is there before any steps happen. If the question asks for the total after 5 weeks, substitute x = 5: y = 12 + 4(5) = 12 + 20 = 32 stickers. At zero weeks, the model gives 12, which matches the assumed starting amount. That is a useful quick check on whether the constant term is in the right place.

Keep the same model, put the known total in for y, and solve for x. Using the example, if the jar has 32 stickers, write 32 = 12 + 4x. Subtract 12 from both sides: 20 = 4x. Divide by 4: x = 5 weeks. Check by putting 5 back into the original relationship: 12 + 4(5) = 32.

Then answer in a sentence with the requested quantity and units: “The jar reached 32 stickers after 5 weeks,” assuming the example’s starting amount and weekly addition. A word-problem strategy is to identify the quantity sought, name it with a variable, translate the relationship into an equation, solve, check, and answer in a complete sentence. [S2]

Copy and fill in this template:

“Let x = the number of [steps and units]. The starting amount is [b and units]. Each step changes the amount by [m and units per step]. So the total is y = [b] + [m]x, where y is [total and units]. The question asks for [unknown]. Substitute [known value] and solve: [equation and steps]. Check by [substituting the result or testing the starting point]. Therefore, [complete answer with units].”

For a result that seems odd, check what x counts, whether the initial amount was included once, and whether the repeated change has the right sign and units. Also ask whether the story allows a fractional number of steps: a calculation can produce one, but a context involving whole weeks or visits may require a whole-number answer. The equation captures the stated fixed amount and equal repeated change; it does not establish whether those assumptions are realistic outside the problem.