5.3 Modeling From Tables and Word Problems
Key Takeaways
- ACT modeling questions often combine statistics with rates, percents, averages, or algebra, so translate the table before calculating.
- Use differences to recognize linear patterns and ratios to recognize exponential percent-change patterns.
- When a table reports percentages, convert to counts only after identifying the correct total group.
- A good model answer should match the requested unit, the direction of change, and a reasonable real-world size.
Modeling means translating before solving
ACT's Math Test Description treats Modeling as a cross-category skill: producing, interpreting, evaluating, and improving models can appear inside algebra, functions, geometry, statistics, probability, or essential skills. For this chapter, modeling usually means turning a table, graph, survey, or word problem into a compact rule you can use.
The best first step is not a formula. It is a translation sentence: x represents what, y represents what, and one unit of x means what. If you cannot say that sentence, you are likely to misuse the table.
Table-to-model workflow
| Step | Question to ask | Why it matters |
|---|---|---|
| 1. Label variables | What does each row or column measure? | Prevents using the wrong total or unit |
| 2. Check pattern | Are differences constant or ratios constant? | Distinguishes linear from exponential models |
| 3. Build the rule | What starting value and change rate fit? | Converts the display into an equation or prediction |
| 4. Answer the prompt | Is it asking for count, percent, rate, or comparison? | Keeps the final unit aligned |
| 5. Sanity-check | Is the size and direction reasonable? | Catches swapped variables and percent errors |
Linear and percent-change tables
A table with t = 0, 1, 2, 3 and y = 120, 150, 180, 210 has a constant difference of 30. That is a linear model: y = 120 + 30t. If the question asks for t = 5, the prediction is 120 + 30(5) = 270.
A table with t = 0, 1, 2, 3 and y = 50, 60, 72, 86.4 has a constant ratio of 1.2. That is a 20% growth model: y = 50(1.2)^t. The values do not rise by the same amount each time; they rise by the same percent. ACT answer choices often include a linear answer for an exponential table, so check differences and ratios before choosing.
Weighted averages from tables
A weighted average is a model of a combined group. If Group A has 18 students with an average of 76 and Group B has 12 students with an average of 91, the combined mean is not (76 + 91)/2. The groups are different sizes. Compute total points first: 18(76) + 12(91) = 1368 + 1092 = 2460. Divide by all 30 students to get 82.
This structure appears in ACT word problems about test scores, prices, speeds, surveys, and mixtures. The safer expression is always total amount / total count. For average speed, that means total distance / total time, not the average of two speeds unless the times are equal.
Percent tables and sample spaces
A table may give percentages instead of counts. If 35% of 240 surveyed students chose online tutoring, the count is 0.35(240) = 84. But if the prompt says 35% of juniors and there are 80 juniors, use 80, not the full 240. The base is the group named by the percent phrase.
Two-way tables add another layer. Suppose a transportation table for 200 students lists 72 bus riders, 108 car riders, and 20 walkers. If 15 car riders switch to the bus, the total stays 200, bus becomes 87, and car becomes 93. The new bus percentage is 87/200 = 43.5%. A common wrong answer is 87/185, which incorrectly removes the switched students from the total even though they are still in the survey.
Evaluating models, not just using them
ACT modeling questions may ask which conclusion is supported. A model is strong only for the data it describes. Interpolating inside the observed range is usually safer than extrapolating far beyond it. A line fitted to data from ages 12 to 18 should not automatically be used to predict age 60. A survey model based on volunteers may be biased if the target group is all students.
Also watch units. If a table gives dollars per month and the question asks for yearly cost, multiply by 12 after applying the model. If a graph gives distance in miles and time in minutes, the slope is miles per minute unless you convert.
Word-problem modeling checklist
- Underline the requested final unit before calculating.
- Convert percentages to decimals only after identifying the base.
- For tables, test differences first, ratios second.
- For averages, reconstruct totals whenever groups have different sizes.
- For probability from a table, condition on the group named by given, among, or of.
- Reject answers with the wrong direction: increases should not produce smaller totals unless the question describes a decrease.
The ACT often hides the hard part in one phrase: per month, among seniors, after the increase, or in the original group. Slow down on those phrases and the arithmetic becomes ordinary.
When the table is incomplete
Not every ACT table gives every total. If a row total or column total is missing, build it before answering the question. In a two-way table, the grand total must equal the sum of all row totals and also the sum of all column totals. That redundancy is useful because it lets you catch arithmetic errors before choosing an answer.
Modeling questions also test whether a prediction is reasonable. If a table shows a plant growing from 4 inches to 7 inches to 10 inches over equal time intervals, a linear model with a 3-inch increase per interval is reasonable. A prediction of 100 inches after one more interval should be rejected even before exact calculation because it violates the observed pattern. Conversely, if a population doubles each period, an added constant model is too small.
Treat each answer choice as a model claim. Ask whether it preserves the starting value, the rate or percent change, and the unit requested. That habit turns many word problems into quick elimination.
A table has x-values 0, 1, 2, 3 and y-values 6, 9, 13.5, 20.25. Which model type best fits the pattern?
A player scored an average of 8 points per game over 12 games and an average of 16 points per game over the next 4 games. What was the player's average over all 16 games?
In a survey of 200 students, 72 usually ride the bus, 108 usually ride by car, and 20 usually walk. If 15 car riders switch to the bus and the total number of students stays 200, what percent usually ride the bus now?