Translating Word Problems
Translating Word Problems
Many competition problems are stated in words rather than symbols, and the hardest part is often turning the sentence into an equation. Once you have an equation, one- and two-step equation skills take over from there.
Translating Phrases into Symbols
| Phrase | Symbol |
|---|---|
| sum, more than, increased by, total | + |
| difference, less than, decreased by, fewer | - |
| product, times, of (with fractions/percents) | \times |
| quotient, divided by, per, ratio | \div |
| is, was, equals, results in | = |
Watch the word order carefully with subtraction: "5 less than a number" means x - 5, not 5 - x, because the number comes first and then 5 is taken away from it.
General Strategy
- Identify what's unknown and name it with a variable (e.g., let x be the unknown quantity).
- Translate each phrase of the sentence into symbols, one piece at a time.
- Write the full equation and solve it using one- or two-step methods.
- Answer the actual question being asked — sometimes the variable itself isn't the final answer!
Age Problems
"Maria is 3 years older than twice her brother's age. If Maria is 17, how old is her brother?" Let b be the brother's age. "Twice her brother's age" is 2b, and "3 years older than" that is 2b+3. So 2b + 3 = 17, giving 2b = 14 and b = 7.
Consecutive Integer Problems
Consecutive integers differ by 1 each, so three consecutive integers can be written as n, n+1, n+2. Consecutive even or odd integers differ by 2 each: n, n+2, n+4. For example, "three consecutive integers sum to 54" becomes n + (n+1) + (n+2) = 54, so 3n+3=54, 3n=51, n=17 — the integers are 17, 18, 19.
Coin/Value Problems
These problems track a quantity and a value per item separately, then combine them. "A jar has only dimes and quarters, 20 coins in all, worth \$3.05. How many dimes are there?" Let d be the number of dimes, so 20-d is the number of quarters. Dimes contribute 0.10d dollars and quarters contribute 0.25(20-d) dollars: 0.10d + 0.25(20-d) = 3.05. Expanding: 0.10d + 5 - 0.25d = 3.05, so -0.15d = -1.95, giving d=13.
Example
"Seven less than three times a number is 20. Find the number."
Step 1: Let the number be x. "Three times a number" is 3x, and "seven less than" that quantity is 3x - 7.
Step 2: Set up the equation: 3x - 7 = 20
Step 3: Solve: add 7 to both sides to get 3x = 27, then divide by 3 to get x = 9.
Step 4 (check): 3(9) - 7 = 27 - 7 = 20. Correct!
Key Takeaways
- Translate phrases into symbols piece by piece, watching word order carefully with subtraction ("less than" reverses the order).
- Always name your variable clearly, and re-read the question at the end — you may need to use your variable to compute a different final answer.
- Consecutive integers: n, n+1, n+2, \ldots; consecutive even/odd integers: n, n+2, n+4, \ldots.
- Coin/value problems combine a quantity equation and a value equation — often you only need the value equation once you've expressed all quantities in one variable.