📝 Word Problems — Algebraic Models & Real-World Applications

Essential for translating real-world situations into algebraic equations and solving them.

🎯 Recognition Cues

  • “How many” — Counting problems
  • “What is the ratio” — Ratio and proportion problems
  • “Work together” — Rate and work problems
  • “Mixture” — Mixture and concentration problems
  • “Age” — Age-related problems

📚 Template Solutions

Common Problem Types

TypeKey VariablesEquationExample
Work/Rate$r = \frac{1}{t}$ (rate), $t$ (time)$r_1 + r_2 = r_{\text{total}}$$A$ can do job in $3$ hours, $B$ in $6$ hours
Mixture$c$ (concentration), $v$ (volume)$c_1v_1 + c_2v_2 = c_{\text{final}}v_{\text{final}}$Mix $20%$ acid with $50%$ acid
Age$a$ (age now), $y$ (years ago)$a - y = \text{age then}$“In $5$ years, John will be twice as old”
Distance$d$ (distance), $r$ (rate), $t$ (time)$d = rt$“Train travels $120$ miles in $2$ hours”

Translation Strategies

Word/PhraseTranslationExample
“is”$=$“John’s age is $x$” → $a = x$
“more than”$+$“$5$ more than $x$” → $x + 5$
“less than”$-$“$3$ less than $x$” → $x - 3$
“times”$\times$“$2$ times $x$” → $2x$
“of”$\times$“$20%$ of $x$” → $0.2x$

🎯 Worked Examples

Example 1: Work/Rate Problem

Problem: Alice can paint a room in $4$ hours, and Bob can paint the same room in $6$ hours. How long will it take them to paint the room together?

Solution:

  1. Alice’s rate: $r_A = \frac{1}{4}$ rooms per hour
  2. Bob’s rate: $r_B = \frac{1}{6}$ rooms per hour
  3. Combined rate: $r_{\text{total}} = r_A + r_B = \frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12}$
  4. Time together: $t = \frac{1}{r_{\text{total}}} = \frac{1}{\frac{5}{12}} = \frac{12}{5} = 2.4$ hours
  5. Answer: $2.4$ hours (or $2$ hours $24$ minutes)

Example 2: Mixture Problem

Problem: How many liters of a $20%$ acid solution must be mixed with $5$ liters of a $50%$ acid solution to get a $30%$ acid solution?

Solution:

  1. Let $x$ = liters of $20%$ solution
  2. Set up equation: $0.2x + 0.5(5) = 0.3(x + 5)$
  3. Simplify: $0.2x + 2.5 = 0.3x + 1.5$
  4. Rearrange: $2.5 - 1.5 = 0.3x - 0.2x$
  5. Solve: $1 = 0.1x$ → $x = 10$
  6. Answer: $10$ liters

Example 3: Age Problem

Problem: John is $3$ times as old as his son. In $5$ years, he will be twice as old as his son. How old is John now?

Solution:

  1. Let $s$ = son’s age now, $j$ = John’s age now
  2. From first sentence: $j = 3s$
  3. From second sentence: $j + 5 = 2(s + 5)$
  4. Substitute: $3s + 5 = 2s + 10$
  5. Solve: $3s - 2s = 10 - 5$ → $s = 5$
  6. Find John’s age: $j = 3(5) = 15$
  7. Answer: John is $15$ years old

⚠️ Common Pitfalls

Pitfall: Incorrect variable setup

  • Fix: Clearly define what each variable represents
  • Example: “Let $x$ = number of apples” not “Let $x$ = apples”

Pitfall: Forgetting units

  • Fix: Always include units in your answer
  • Example: “$10$ liters” not just “$10$”

Pitfall: Incorrect equation setup

  • Fix: Read the problem carefully and translate word by word
  • Example: “John is $3$ times as old as his son” means $j = 3s$, not $s = 3j$

🎯 AMC-Style Worked Example

Problem: A train leaves station A at $60$ mph and another train leaves station B at $80$ mph. If the stations are $280$ miles apart and the trains are traveling toward each other, how long will it take them to meet?

Solution:

  1. Let $t$ = time until they meet
  2. Distance covered by first train: $60t$ miles
  3. Distance covered by second train: $80t$ miles
  4. Total distance: $60t + 80t = 140t$ miles
  5. Set up equation: $140t = 280$
  6. Solve: $t = \frac{280}{140} = 2$ hours
  7. Answer: $2$ hours

Key insight: When objects move toward each other, their distances add up to the total distance between them.

🔗 Related Patterns

  • Systems of Equations — Many word problems require systems
  • Rate Problems — Often involve distance, time, and rate
  • Mixture Problems — Concentration and volume relationships
  • Age Problems — Time-based relationships

📝 Practice Checklist

  • Master common problem types
  • Practice translation strategies
  • Learn to set up equations correctly
  • Practice unit conversions
  • Understand rate and work problems
  • Work on speed and accuracy

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