Word Problems and Applications

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Word problems use everyday language to describe mathematical situations. Your task is to translate the words into mathematical operations and solve. The key is understanding what the problem is asking and identifying the numbers involved.

The Problem-Solving Steps:

1. Read the problem carefully — twice
2. Identify what the problem is asking (what do you need to find?)
3. Identify the relevant numbers and quantities
4. Decide which mathematical operation(s) to use
5. Solve the problem
6. Check your answer

Translating Word Phrases to Operations:

Word Phrase	Operation
sum, total, plus, added to, increased by	+
difference, minus, subtracted from, decreased by, less	−
product, multiplied by, times, of (in fractions)	×
quotient, divided by, shared equally, groups of	÷
equals, is, makes, results in	=

Word to Symbol Translation:

Phrase	Algebraic Form
A number	$x$
A number plus 5	$x + 5$
3 less than a number	$x - 3$
Twice a number	$2x$
Half of a number	$x/2$
The square of a number	$x^2$
4 more than twice a number	$2x + 4$

Basic Problem Types:

1. Addition Problems: “Tola has 45 books and Buys 20 more. How many books does she have?” $45 + 20 = 65$ books

2. Subtraction Problems: “Emeka had ₦500 and spent ₦180. How much does he have left?” $500 - 180 = ₦320$

3. Multiplication Problems: “A classroom has 8 rows with 6 chairs in each row. How many chairs?” $8 \times 6 = 48$ chairs

4. Division Problems: “60 sweets are shared equally among 12 children. How many sweets each?” $60 \div 12 = 5$ sweets each

⚡ Exam Tip (NCEE): The words “each,” “every,” “a,” and “per” often indicate multiplication or division. “6 in each group” means $6 \times \text{number of groups}$. “Shared equally among” means division.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding of problem-solving.

Two-Step Word Problems:

These require two operations.

Example: “A shopkeeper sells 3 bags of rice at ₦2,500 each. He also sells 2 bags at ₦3,000 each. How much money does he collect?”

Step 1: First type: $3 \times 2,500 = ₦7,500$ Step 2: Second type: $2 \times 3,000 = ₦6,000$ Step 3: Total: $7,500 + 6,000 = ₦13,500$

Example: “A farmer collects 120 eggs. He puts them in cartons of 12 eggs each. He sells 7 cartons. How many eggs are left?”

Step 1: Total cartons: $120 \div 12 = 10$ cartons Step 2: Eggs sold: $7 \times 12 = 84$ eggs Step 3: Eggs left: $120 - 84 = 36$ eggs

Worked Examples:

Example 1 — Age problems: “Tunde is 15 years old. His sister is 4 years younger. How old will both be in 5 years?”

Tunde in 5 years: $15 + 5 = 20$ Sister now: $15 - 4 = 11$ Sister in 5 years: $11 + 5 = 16$

Example 2 — Distance: “A car travels 60 km/h for 3 hours. How far does it travel?”

Distance = Speed × Time = $60 \times 3 = 180$ km

Example 3 — Sharing with remainder: “55 students are grouped into teams of 6. How many complete teams are there? How many students are left over?”

$55 \div 6 = 9$ remainder $1$ 9 complete teams, 1 student left over

Rate Problems:

Example: “A tap fills a tank in 4 hours. How much of the tank is filled in 1 hour?” In 1 hour: $1/4$ of the tank

Example: “If 5 books cost ₦2,500, how much does 1 book cost?” $2,500 \div 5 = ₦500$ per book

Percentage Word Problems:

Example: “In a class of 40 students, 25% passed with distinctions. How many students is that?” $25%$ of $40 = \frac{25}{100} \times 40 = 10$ students

⚡ Common NCEE Error: In problems with “of” (meaning multiplication), students sometimes treat it as addition. “25% of 40” means $0.25 \times 40$, not $0.25 + 40$.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Mixture and Ratio Problems:

Example: “A merchant mixes 20 kg of rice at ₦200 per kg with 30 kg at ₦300 per kg. What is the price per kg of the mixture?”

Value of first rice: $20 \times 200 = ₦4,000$ Value of second rice: $30 \times 300 = ₦9,000$ Total value: $₦4,000 + ₦9,000 = ₦13,000$ Total weight: $20 + 30 = 50$ kg Price per kg: $13,000 \div 50 = ₦260$ per kg

Ratio Problems:

Example: “The ratio of boys to girls in a school is 3:2. If there are 450 boys, how many girls are there?”

Ratio boys:girls = 3:2 For every 3 boys → 2 girls For 450 boys → $450 \div 3 \times 2 = 150 \times 2 = 300$ girls

Work and Time Problems:

Example: “If Adaeze can paint a room in 8 hours and Chidi can paint the same room in 6 hours, how long will they take working together?”

Adaeze’s rate: $1/8$ room per hour Chidi’s rate: $1/6$ room per hour Combined rate: $1/8 + 1/6 = (3+4)/24 = 7/24$ room per hour Time: $1 \div (7/24) = 24/7 = 3\frac{3}{7}$ hours $\approx 3$ hours 26 minutes

Simple Interest Word Problems:

Interest = Principal × Rate × Time / 100

Example: “Find the simple interest on ₦5,000 for 3 years at 8% per annum.”

$I = P \times R \times T / 100 = 5,000 \times 8 \times 3 / 100 = ₦1,200$

Profit and Loss Word Problems:

Example: “A trader buys a television for ₦80,000 and sells it for ₦95,000. Find the profit and profit percentage.”

Profit = Selling Price − Cost Price = $95,000 - 80,000 = ₦15,000$ Profit % = (Profit/Cost Price) × 100 = $(15,000/80,000) \times 100 = 18.75%$

Distance Problems:

Example: “A cyclist travels 45 km at 15 km/h and then another 40 km at 10 km/h. What is the average speed for the whole journey?”

Total distance = $45 + 40 = 85$ km Time for first part = $45 \div 15 = 3$ hours Time for second part = $40 \div 10 = 4$ hours Total time = $3 + 4 = 7$ hours Average speed = Total Distance ÷ Total Time = $85 \div 7 = 12.14$ km/h

⚡ Extended Tip — Perimeter and Area Word Problems: For rectangular shapes: Perimeter = $2(l + w)$; Area = $l \times w$. If a fence costs ₦500 per metre, and a rectangular garden is 20 m by 15 m, the perimeter is $2(20+15) = 70$ m, so the fence cost is $70 \times 500 = ₦35,000$. The area (for planting) would be $20 \times 15 = 300$ m².

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