Word Problems
Word problems — also called story problems or applied mathematics — are the bridge between abstract mathematical concepts and real-world situations. They require you to read a description of a situation, extract the relevant mathematical relationships, set up equations or calculations, and arrive at the correct answer. For the Post-UTME examination, a significant portion of the quantitative section consists of word problems.
The Key Skill: Translating Words to Mathematics
The most critical skill in solving word problems is the ability to convert English phrases into mathematical operations.
| English Phrase | Mathematical Operation |
|---|---|
| Sum of x and y | x + y |
| Difference of x and y | x − y |
| Product of x and y | x × y |
| x more than y | y + x |
| x less than y | y − x |
| x exceeds B by 3 | x = B + 3 |
| x is twice B | x = 2B |
| Speed = Distance/Time | v = d/t |
| Percentage increase | (change/original) × 100 |
Types of Word Problems
1. Age Problems
Key principle: The difference in ages between two people remains constant throughout their lives.
Example: A father is 4 times as old as his son. In 20 years, the father will be twice as old. Find their current ages.
Let son’s current age = x; Father’s current age = 4x In 20 years: 4x + 20 = 2(x + 20) 4x + 20 = 2x + 40 → 2x = 20 → x = 10 years (son), Father = 40 years
2. Distance, Speed, and Time Problems
Key formulas: Speed = Distance/Time; Distance = Speed × Time; Time = Distance/Speed
Example: A bus leaves Lagos at 8:00 AM at 60 km/h. Another bus leaves Abuja for Lagos at 10:00 AM at 80 km/h. Distance = 420 km. When do they meet?
- By 10:00 AM, Bus 1 has traveled: 60 × 2 = 120 km
- Remaining distance: 420 − 120 = 300 km
- Relative speed (toward each other) = 60 + 80 = 140 km/h
- Time to meet = 300/140 = 15/7 ≈ 2 hours 9 minutes
- Meeting time: 12:09 PM
3. Work and Rate Problems
Key principle: If a job can be done in x days, the work done in 1 day = 1/x of the job.
Example: A pipe fills a tank in 5 hours, another empties it in 8 hours. How long to fill if both open?
- Fill rate = 1/5 per hour; Empty rate = 1/8 per hour
- Net fill rate = 1/5 − 1/8 = (8−5)/40 = 3/40 per hour
- Time to fill = 1 ÷ (3/40) = 40/3 = 13 hours 20 minutes
4. Mixture Problems
Example: How many kg of peanuts at ₦800/kg must be mixed with 30 kg of cashew at ₦1,200/kg to make a mixture worth ₦960/kg?
0.30x + 0.60(100−x) = 0.45 × 100… wait, let me redo: Let x = kg of peanuts. 800x + 1200(30) = 960(x+30) 800x + 36,000 = 960x + 28,800 → 7,200 = 160x → x = 45 kg
5. Number Problems
Example: The sum of three consecutive integers is 72. Find them. Let n, n+1, n+2 → n + (n+1) + (n+2) = 72 → 3n + 3 = 72 → 3n = 69 → n = 23 → 23, 24, 25
Example: A two-digit number is three times the sum of its digits. If digits are x (tens) and y (units): 10x + y = 3(x + y) → 10x + y = 3x + 3y → 7x = 2y → x/y = 2/7 → Since x,y are digits (1-9): x=2, y=7 → Number = 27
Systematic Approach to Word Problems
- Read carefully: Read the problem twice
- Identify what you’re looking for: Let it be your variable (x)
- Form the equation: Translate the relationship into a mathematical equation
- Solve: Use appropriate algebraic methods
- Verify: Check that your answer makes sense
- State the answer: Give the answer in the units asked for
Key Post-UTME Exam Facts
- Age problems: Difference in ages is constant
- Distance problems: Relative speed when moving toward each other = sum of speeds
- Work problems: Rate = 1/time; Combined rate = sum of individual rates
- Profit/Loss: SP = CP(1 + profit%); SP = CP(1 − loss%)
- Number problems: Two-digit = 10(tens) + units
- ⚡ Exam tip: Always write out what your variable represents before forming equations. “Let x = …”
🟢 Lite — Quick Review (1h–1d)
Rapid summary for last-minute revision before your exam.
- Age problems: Use the constant difference in ages
- Distance problems: Remember t = d/v; when two objects move toward each other, add speeds
- Work problems: Rate = 1/time done per day; combined rate = sum of individual rates
- Number problems: Two-digit = 10(tens digit) + units digit; consecutive integers: n, n+1, n+2
- Profit/Loss: SP = CP(1 + profit/100); SP = CP(1 − loss/100)
- Always verify: Check that your answer is reasonable
- ⚡ Exam tip: “More/Less than” problems require careful attention. “x exceeds y by 3” → x = y + 3.
🟡 Standard — Regular Study (2d–2mo)
Standard content for students with a few days to months.
Relative Speed Problems (Trains)
Train passing a pole: Time = Length of train/Speed Train passing another train going in the opposite direction: Relative speed = sum of speeds; Distance = sum of lengths
Example: Two trains 200m and 150m long moving in opposite directions at 36 km/h and 54 km/h. How long to completely cross?
Convert speeds: 36 km/h = 10 m/s; 54 km/h = 15 m/s Relative speed = 25 m/s; Total distance = 200 + 150 = 350 m Time = 350/25 = 14 seconds
Tank/Flow Problems
Example: Two pipes fill a tank in 10h and 15h; drain empties in 20h. All opened at 8 AM. When full? Fill rate: 1/10 + 1/15 − 1/20 = (6+4−3)/60 = 7/60 per hour Time = 1 ÷ (7/60) = 60/7 hours = 8h 34m ≈ 4:34 PM
🔴 Extended — Deep Study (3mo+)
Comprehensive coverage for students on a longer study timeline.
Installment/Loan Problems
Example: A TV costs ₦120,000 cash or ₦15,000 deposit plus monthly installments. Total paid = 120,000. Deposit = ₦15,000; Balance = ₦105,000 If paid in 8 installments: 105,000/8 = ₦13,125 per month
Quadratic Word Problems
Example: A rectangle’s length is 4 cm more than its width. Area = 60 cm². Find dimensions. Let w = width; w(w+4) = 60 → w² + 4w − 60 = 0 (w+10)(w−6) = 0 → w = 6 cm (reject −10); Length = 10 cm; Area = 60 cm² ✓
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