Building Equations from Words
How to turn an everyday situation into an equation: name the unknown, let the wording pick the operation, then solve and check.
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Real problems do not arrive as equations. They arrive as sentences, and the hardest part is usually the translation. Once a situation is written as an equation, solving it is the easy step.
Concept
The route from words to answer
Three questions get you across: what do I know, what am I looking for, and what links them.
Concept
Phrases that name the operation
"more than", "greater by" — addition.
"less than", "younger by", "sold" — subtraction.
"times", "twice", "three times as many" — multiplication.
"shared between", "each" — division.
These phrases are the bridge. Spot the phrase and you have found the operation.
Example
A difference in grades
Khalid's grade is 3 marks higher than Ahmed's. Ahmed scored 90. What did Khalid score?
Known: Ahmed scored 90. Wanted: Khalid's grade, call it 
.
"higher than" signals addition.
⟹ 93
Example
A difference in ages
Ahmed is 27 years younger than his father, who is 50. How old is Ahmed?
Known: the father is 50. Wanted: Ahmed's age, call it 
.
"younger than" signals subtraction.
⟹ 23 years old
A quick sanity check: 23 is well below 50, which is what "younger" demands.
Example
A multiple
Faisal's father is three times Faisal's age. Faisal is 12. How old is his father?
Known: Faisal is 12. Wanted: his father's age, call it 
.
"three times" signals multiplication.
⟹ 36 years old
Example
Working backwards
A shop sold 16 balls and has 40 left. How many were there to begin with?
Let 
be the original number.
Selling removes 16, so 
.
Add 16 to both sides: 
.
Check: 
.
⟹ 56 balls
This one is different in an important way. The unknown is the starting amount, so the equation involves subtraction while the solution requires addition. Writing the equation first is what keeps this straight.
Note
Mistakes to avoid
Calculating before deciding what the unknown actually is.
Subtracting when the wording says "more than".
Assuming the operation in the story is the operation that solves it.
Losing track of which person or object each number refers to.
Skipping the check that the answer makes sense in the situation.
Summary
- Identify what is known and what is being asked for.
- Give the unknown a letter before doing any arithmetic.
- Let the wording point to the operation that links them.
- Write the equation, then solve it using the balance rule.
- Check the answer against the original situation.