Translating multistep word problems
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1.1 Name the target before choosing variables
The first job is not to calculate. It is to state what the final response must represent. “How many were purchased?” and “How many remain?” can arise from the same story but require different answers. A problem may give an equation in x while asking for x + 3, a percentage, or a total over several stages.
Write a short target statement on scratch paper: original number of parts, total travel time in hours, or maximum planted area in square meters. Include the unit and any qualifier such as positive, integer, least, or greater.
A four-entry modeling ledger
Target: the quantity to report. Variables: named quantities with units. Relationships: equations, inequalities, or proportions. Restrictions: what values make sense in the original situation.
Question wording | A useful variable | The final target |
|---|---|---|
Original price before a discount | p | p, not the amount saved |
Least number sold for a profit | n | Least integer with revenue > cost |
Water added to a mixture | w | w, not the final total volume |
Average speed for a whole trip | Segment times or speeds | Total distance / total time |
Probability among volunteers | Counts in relevant groups | Favorable volunteers / all volunteers |
Choose variables that make the relationships short
A ratio of 3 : 5 often suggests 3k and 5k, not two unrelated unknowns. A rectangle whose length is 4 more than its width suggests w and w + 4. If a problem asks for a combined quantity and the equations already contain it, avoid splitting it into unnecessary pieces.
A variable is not automatically a count. It may represent dollars, hours, a scale factor, or an input to a function. Consequently, “x must be an integer” is justified for a number of tickets but not generally for the price of one ticket.
Read for relationships, then reread for the target
On the first pass, determine what the story is about. On the second, connect each number to its role. A price of $8 per item is a rate; a $20 delivery charge is a fixed amount. Treating both as per-item quantities creates a model that looks reasonable but answers a different question.
Success criterion. Before doing algebra, you can explain each term in your model in a complete sentence.
1.2 Build relationships instead of chaining keywords
Words such as “of,” “more,” and “per” are clues, not a substitute for understanding. “Five less than twice n” is 2n − 5, while “twice a number that is five less than n” is 2(n − 5). The difference is the grouping.
Underlying relationship | Reliable mathematical form |
|---|---|
A total made of parts | Total |
A constant rate over an interval | Amount |
A fixed amount plus variable use | Total |
A percentage applied to a base | Part |
A mixture or conserved ingredient | Ingredient before + ingredient added |
A mean | Sum of values |
A capacity or spending limit | Required amount ≤ allowed amount |
Track the base through successive changes
If a quantity decreases by p%, its multiplier is 1 − p/100. A later q% increase acts on the new quantity, so the overall multiplier is (1 − ) (1 + ).
Do not combine percentages by adding or subtracting them unless their bases and the operation justify it. A tax applied after a discount acts on the discounted price. A percentage of accepted products acts on the accepted group, not all products manufactured.
Track what is conserved
When water is added to a salt solution, the amount of salt remains unchanged but the volume changes. When a solid is remolded without loss, volume is conserved even though surface area need not be. When workers share a job, their rates add while they work simultaneously; their completion times do not.
The model must preserve the story
An equation can be algebraically solvable and still be wrong. Before solving, point to each operation and explain what in the situation licenses it. In particular, explain why a quantity is added, multiplied, or divided.
Use inequalities when equality is not promised
“At least” includes the boundary; “more than” does not. A break-even calculation solves revenue
1.3 Choose a representation that reduces work
A well-chosen table or sketch reduces the amount of information you must hold in memory. It is part of the mathematics, not an ornamental step.
Use a table when several groups have the same kinds of information. In a mixture table, columns for volume, concentration, and ingredient amount make the conserved quantity visible. In a trip table, columns for distance, speed, and time prevent averaging speeds incorrectly.
Trip segment | Distance | Speed | Time |
|---|---|---|---|
First segment | d1 | v1 | d1/v1 |
Second segment | d2 | v2 | d2/v2 |
Whole trip | d1 + d2 | d1/v1 + d2/v2 |
Use a sketch when a statement describes a border, a shared side, a radius, or repeated dimensions. Label what is given and what repeats. A border of width x on all four sides adds 2x to each overall dimension, not x.
Use an equation or graph when the requested quantity is an intersection, a threshold, a maximum, or an output at a specified input. Define the input’s starting point first: t
Use a ratio or invariant when the target is relative rather than absolute. Similarity problems may not require either actual side length. Two exponential models with the same growth factor may have a constant ratio. A change over 12 months can be found from the time difference without separately computing two enormous totals.
An efficient representation answers a specific question
A graph helps only after you know which coordinate matters. A table helps only after you know which totals belong in it. Choose the representation because it makes the target accessible, not because it is familiar.
1.4 Turn a mathematical result into an admissible answer
After constructing the model, solve it with the tools from the earlier volumes. Then return to the original situation. This second translation is as important as the first.
An algebraic solution satisfies a derived equation. An admissible solution also satisfies the original domain, signs, units, and contextual restrictions. The requested answer may be a function of that admissible solution rather than the solution itself.
What the algebra produces | What the context may require |
|---|---|
A negative and a positive width | Keep only a physically possible width. |
A break-even quantity of 20 | Positive profit starts at 21 whole items. |
A total final volume of 64 liters | Report only the 24 liters added, when that is the target. |
A meeting time of 2.4 hours after both workers start | Add any earlier work time if the question asks for total elapsed time. |
Two roots of a geometry equation | Verify positive side lengths and all geometric conditions. |
A conditional probability denominator | Count the group named after “given” or “among.” |
Use a quick independent check
Substitute into the story’s quantities, not just your rearranged equation. Rebuild revenue from actual ticket counts. Recompute the mixture’s concentration from its salt amount and final volume. Check both total distance and total time for average speed.
A rough check can reject an impossible result quickly: adding water must lower concentration; an average speed for two positive-duration segments must lie between the segment speeds; doubling every length of a similar solid multiplies its volume by 23, not 2.
Plan the algebra before entering the calculator
Write the relationship first. Numerical computation may then be useful for an inconvenient decimal, and graphing may locate an intersection. Neither tool decides which price is taxed or which time interval the exponent represents. Calculator permission is not a reason to replace a simple model with a long sequence of entries.[4]
Section 1 exit standard
You can write a model with named quantities and units, solve it, and state why the reported answer satisfies the original conditions. The next 15 examples move from straightforward totals to linked rates, conditional groups, and constrained optimization.
15 worked examples
1.01. Separate a fixed charge from the taxed subtotal
A caterer charges $8 per meal plus a $20 delivery fee. A 10% tax is applied to the entire subtotal. The final bill is $198. How many meals were purchased?
Show worked solutionHide worked solution for example 1.01
Recognize. The tax applies after the meal cost and delivery fee have been added. Let n be the number of meals.
Work. The subtotal is 8n + 20, so 1.10(8n + 20)
Answer: 20 meals.
Check. The meals cost $160; delivery brings the subtotal to $180. Tax is $18, and the final bill is $198.
Avoid the trap. The model 1.10(8n) + 20 taxes only the meals. The wording explicitly taxes the delivery fee as well. Drawing a subtotal bracket prevents that error.
1.02. Represent a ratio with one scale factor
A store initially has red and blue notebooks in the ratio 3 : 5. After receiving 12 additional red notebooks and no blue notebooks, it has equal numbers of the two colors. How many notebooks did it have initially?
Show worked solutionHide worked solution for example 1.02
Recognize. The initial counts can be written as 3k and 5k. The requested quantity is their original sum.
Work. Equality after the shipment gives 3k + 12
Answer: 48 notebooks.
Check. Adding 12 to the original 18 red notebooks gives 30 of each color. The new total is 60, but the question asks for the initial total.
Avoid the trap. Do not treat 3 and 5 as the actual counts. A ratio determines relative sizes; k supplies the scale.
1.03. Convert a target mean into a required sum
A student’s first three quiz scores are 78, 86, and 90. All four quizzes have equal weight. What score on the fourth quiz gives an average of 87?
Show worked solutionHide worked solution for example 1.03
Recognize. A mean is a statement about a total. Four scores with mean 87 must sum to 4(87).
Work. The current sum is 78 + 86 + 90
Answer: 94.
Check. The completed total is 348, and 348/4
Avoid the trap. Averaging 87 with the three given scores assumes the target mean is itself the missing score. Multiply mean by count before subtracting known scores.
1.04. Preserve the order of percentage operations
A jacket’s marked price is $240. A store applies a 25% discount, then charges 8% sales tax on the discounted price. What is the final price?
Show worked solutionHide worked solution for example 1.04
Recognize. Each percentage has a base. The discount uses $240; the tax uses the reduced amount.
Work. The discounted price is 240(0.75)
Answer: $194.40.
Check. The tax is 0.08(180)
Avoid the trap. A 25% decrease followed by an 8% increase is not a 17% decrease. Their combined multiplier is 0.81, which represents a 19% decrease from the marked price.
1.05. Distinguish breaking even from making a profit
A club pays a fixed $180 setup fee and $6 for each shirt it sells. Each shirt sells for $15. What is the least number of shirts the club must sell to make a positive profit?
Show worked solutionHide worked solution for example 1.05
Recognize. Profit is revenue minus cost, and “positive” requires a strict inequality. The number of shirts is an integer.
Work. For n shirts, profit is 15n − (180 + 6n)
Answer: 21 shirts.
Check. At 20 shirts, revenue and cost are both $300. At 21 shirts, revenue is $315 and cost is $306, leaving a $9 profit.
Avoid the trap. Solving the break-even equation correctly gives 20, but that is not the requested answer. The last step is an inequality-and-integer decision, not ordinary rounding.
1.06. Average rates through total amount and total time
A driver travels 120 miles at 40 miles per hour and another 120 miles at 60 miles per hour, with no stops. What is the average speed for the entire trip?
Show worked solutionHide worked solution for example 1.06
Recognize. Equal distances do not mean equal times. Average speed is total distance divided by total elapsed time.
Work. The first segment takes 120/40
Answer: 48 miles per hour.
Check. The result lies between 40 and 60 and is closer to 40 because more time was spent at that speed.
Avoid the trap. The arithmetic mean (40 + 60)/2
1.07. Conserve the ingredient, not the concentration
A container holds 40 liters of a solution that is 20% salt by volume. Water is added without removing solution, and volumes add. How many liters of water must be added to make the solution 12.5% salt by volume?
Show worked solutionHide worked solution for example 1.07
Recognize. Adding water changes the total volume but not the salt volume. Let w be the liters of water added.
Work. The salt volume is 0.20(40)
Therefore 40 + w
Answer: 24 liters.
Check. The final concentration is 8/64 = 1/8 = 12.5%. The final volume is 64 liters, but only 24 liters were added.
Avoid the trap. Do not average 20% and 0% without accounting for the amounts of the two liquids.
1.08. Translate a uniform border into both dimensions
A rectangular garden measures 8 meters by 12 meters. A path of uniform width surrounds it on all four sides. The combined area of the garden and path is 192 square meters. What is the path’s width?
Show worked solutionHide worked solution for example 1.08
Recognize. A width of x adds x at each end of both dimensions. Thus the outer dimensions are 8 + 2x and 12 + 2x.
Work. Solve (8+2x)(12+2x)
Answer: 2 meters.
Check. The outside dimensions are 12 by 16, with area 192. The path alone has area 192 − 96
Avoid the trap. The equation (8 + x)(12 + x)
1.09. Use the elapsed interval rather than two large outputs
A population is modeled by N(t) = 500 · 2t/6, where t is the number of months since measurements began. How many times as large is N(15) as N(3)?
Show worked solutionHide worked solution for example 1.09
Recognize. The target is a ratio. The initial amount cancels, and the time difference controls the change.
Work. Compute
= = 2(15−3)/6 = 22 = 4.
Answer: 4 times as large.
Check. The interval from month 3 to month 15 is 12 months, or two six-month doubling periods.
Avoid the trap. “Four times as large” is not a 400% increase. The new amount is 400% of the old amount, an increase of 300%.
1.10. Build the conditional group before finding probability
Of 200 students, 60% belong to a club. Of the club members, 25% volunteer. Of the students who do not belong to a club, 10% volunteer. A volunteer is selected at random. What is the probability that this student belongs to a club?
Show worked solutionHide worked solution for example 1.10
Recognize. The selection is from volunteers, not from all students or all club members.
Work. There are 0.60(200)
P(club | volunteer) = = .
Answer: 15/19.
Check. The denominator contains every volunteer exactly once. Most volunteers are club members, so a probability greater than 1/2 is plausible.
Avoid the trap. 30/120 answers a different question: the probability of volunteering given club membership.
1.11. Separate the solo phase from the shared-work phase
Pump A can fill an empty tank in 6 hours, and pump B can fill it in 9 hours. Their constant rates add, and no water drains. A runs alone for 2 hours; then B joins A. How many hours elapse from A’s start until the tank is full?
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Recognize. There are two time intervals. Let u be the hours when both pumps run; total elapsed time is 2 + u.
Work. A fills 2/6
+ u = 1 u = .
Total time is 2 + 12/5
Answer: 22/5 hours, or 4.4 hours.
Check. A runs 4.4 hours and B runs 2.4 hours; 4.4/6 + 2.4/9
Avoid the trap. Adding the completion times 6 + 9 or reporting only 2.4 hours ignores how the stages combine.
1.12. Identify the geometric quantity that is conserved
A solid sphere of radius 3 centimeters is melted and recast as a solid circular cylinder of radius 1.5 centimeters. No material is lost, and the material’s volume does not change. What is the cylinder’s height?
Show worked solutionHide worked solution for example 1.12
Recognize. The conserved quantity is volume, not surface area or radius. Let h be the height in centimeters.
Work. Equate the two volumes:
π(3)3 = π(1.5)2 h 36π = 2.25πh.
Cancel π and divide to obtain h
Answer: 16 centimeters.
Check. The cylinder volume is π(2.25)(16)
Avoid the trap. A smaller radius does not imply a smaller height or an unchanged surface area. State what the problem preserves before selecting a formula.
1.13. Use a baseline to simplify a two-price model
A theater sells 180 tickets. Adult tickets cost $12 each, and student tickets cost $8 each. Total ticket revenue is $1,760. How many adult tickets were sold?
Show worked solutionHide worked solution for example 1.13
Recognize. Start with a baseline in which all 180 tickets cost $8. Every adult ticket adds $4 beyond that baseline.
Work. The all-student baseline is 180(8)
Answer: 80 adult tickets.
Alternative. Let a be the adult count. Then 12a + 8(180 − a)
Check. The remaining 100 student tickets yield 80(12) + 100(8)
Avoid the trap. Dividing total revenue by the difference in ticket prices ignores the $8 that every ticket contributes.
1.14. Track a percentage of a remaining group
A factory makes a batch of parts. Ten percent are rejected. Of the accepted parts, one-third ship on Monday and the rest ship on Tuesday. Tuesday’s shipment contains 120 more parts than Monday’s. How many parts were manufactured?
Show worked solutionHide worked solution for example 1.14
Recognize. Both shipment fractions apply to the accepted group, which is 90% of the original batch.
Work. Let n be the original number. Monday’s shipment is (1/3)(0.90n)
Answer: 400 parts.
Check. There are 40 rejected parts and 360 accepted parts. Shipments of 120 and 240 differ by 120 and account for all accepted parts.
Avoid the trap. Treating Monday’s shipment as n/3 forgets the rejected parts. A phrase such as “of the accepted parts” resets the base.
1.15. Model a geometric constraint before optimizing
Three adjacent rectangular pens share a straight wall as one long side. The layout uses one long fence parallel to the wall and four equal perpendicular fence segments. Parallel fencing costs $8 per meter; perpendicular fencing costs $6 per meter. With exactly $480 to spend, what is the greatest possible total area of the pens?
Show worked solutionHide worked solution for example 1.15
Recognize. Let L be the total length parallel to the wall and w the common width. The four perpendicular segments, including the two interior dividers, cost 4(6w).
Work. The budget equation is 8L + 24w
A = wL = w(60 − 3w) = −3(w − 10)2 + 300.
Because a square is nonnegative, the maximum area is 300, attained at w
Answer: 300 square meters.
Check. The dimensions are positive, and the fence cost is 8(30) + 4(6)(10)
Avoid the trap. Counting only two perpendicular segments omits the dividers. Maximizing perimeter or making the entire outer rectangle a square is not justified when different segments have different costs.
Transfer. The difficult step is often building the correct constraint. Once one variable is removed, a familiar quadratic gives the optimum without calculus.
Key takeaway
The same sequence works for tickets, mixtures, travel, and geometry: identify the target, name the quantities, preserve the relationships, and check the final interpretation. A change of setting need not mean a new method.
1.6 Section exit check
Attempt these without reading the worked examples. Write a model before calculating. Explanations are linked from each question.
Review prompt
For every answer, distinguish the quantity used as a variable from the quantity requested. For E1.5, count the fence segments before optimizing.