6.3 Word problems (translating real-world situations)
Key Takeaways
- Assign a variable to the unknown, then translate the sentence phrase by phrase using keyword cues.
- 'Less than' and 'fewer than' reverse the order: '5 less than a number' is x - 5, not 5 - x.
- Undo two-step equations in reverse order -- add or subtract first, then multiply or divide.
- Use d = r . t for motion, I = P . r . t for interest, and rate-per-hour fractions for combined-work problems.
- Always check your answer by substituting it back into the original wording of the problem.
Translating Words into Algebra
Most PERT word problems are really just sentences that must be rewritten as equations. The skill is spotting keywords that map onto arithmetic operations and using a variable for the unknown. Read the problem twice: once to understand the situation, and once to assign a variable and translate the sentence phrase by phrase.
Keyword-Translation Table
| Words | Operation or symbol |
|---|---|
| sum, total, more than, increased by, plus | addition (+) |
| difference, less than, decreased by, fewer, minus | subtraction (-) |
| product, of, times, twice, doubled | multiplication (x) |
| quotient, per, ratio, divided by, split equally | division (/) |
| is, was, equals, results in, gives | equals (=) |
| a number, what, how many | the variable (x) |
Two phrases cause the most trouble because they reverse the order of the numbers: "5 less than a number" is x - 5 (not 5 - x), and "8 fewer than twice a number" is 2x - 8. Whenever you see "less than" or "fewer than," write the second quantity first and subtract the stated amount.
Worked example 1. "Seven more than three times a number is 22. Find the number." Translate piece by piece: three times a number = 3x; seven more than that = 3x + 7; "is 22" gives = 22. The equation is 3x + 7 = 22. Subtract 7 from both sides: 3x = 15. Divide by 3: x = 5.
One- and Two-Step Setups
A one-step problem needs a single operation to undo. A two-step problem contains both a multiplication/division and an addition/subtraction, so you undo the addition first and the multiplication second -- the reverse of the order of operations.
Worked example 2 (two-step). A taxi charges a $3 flat fee plus $2 per mile. If a ride costs $17, how many miles was it? Let m be the number of miles. Cost = 3 + 2m = 17. Subtract the flat fee: 2m = 14. Divide by 2: m = 7 miles.
Distance = Rate x Time
Motion problems use the formula d = r . t. Solve for whichever variable is missing by rearranging: r = d / t and t = d / r.
Worked example 3. A train travels 240 miles in 4 hours. What is its average speed? r = d / t = 240 / 4 = 60 miles per hour.
Worked example 4. How long does a 300-mile trip take at 50 miles per hour? t = d / r = 300 / 50 = 6 hours.
Age Problems
Age problems compare people's ages now (or at another time). Represent one age with a variable and write the other ages in terms of it.
Worked example 5. Maria is 4 years older than Ben. The sum of their ages is 30. How old is each? Let Ben = b, so Maria = b + 4. Their sum: b + (b + 4) = 30, which simplifies to 2b + 4 = 30. Subtract 4: 2b = 26. Divide by 2: b = 13. Ben is 13 and Maria is 17. Check: 13 + 17 = 30, which matches.
Mixture Problems
Mixture problems combine amounts at different rates or values. Track the total of the quantity being mixed, then divide to find the blended rate.
Worked example 6. A grocer mixes 10 pounds of nuts worth $6 per pound with 5 pounds worth $9 per pound. What is the price per pound of the mixture? Total value = 10 x 6 + 5 x 9 = 60 + 45 = $105. Total weight = 10 + 5 = 15 pounds. Price per pound = 105 / 15 = $7.
Work Problems
In work problems, express each worker's rate as a fraction of the job completed per unit of time, then add the rates. If one painter finishes a room in 4 hours, that rate is 1/4 of the room per hour.
Worked example 7. One pipe fills a tank in 6 hours; a second pipe fills it in 3 hours. Working together, how long does it take? Combined rate = 1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2 tank per hour. Time = 1 / (1/2) = 2 hours.
Consecutive-Integer Problems
Consecutive integers differ by 1, so name them x, x + 1, and x + 2. Consecutive even or odd integers differ by 2, so name them x, x + 2, and x + 4. Assigning names in terms of a single variable keeps the equation simple.
Worked example 8. The sum of three consecutive integers is 48. Find them. Let the integers be x, x + 1, and x + 2. Their sum is x + (x + 1) + (x + 2) = 3x + 3 = 48. Subtract 3: 3x = 45. Divide by 3: x = 15. The integers are 15, 16, and 17. Check: 15 + 16 + 17 = 48, which matches the problem.
Strategy checklist for every word problem:
- Identify the unknown and name it with a variable.
- Translate each phrase using the keyword table above.
- Write one equation that captures the whole sentence.
- Solve, and then check the answer in the original wording.
Which expression correctly translates '5 less than twice a number'?
A taxi charges a $3 flat fee plus $2 per mile. If a ride costs $17, how many miles was it?
One pipe fills a tank in 6 hours; a second fills it in 3 hours. Working together, how long does it take?