Word problems are mathematical problems that present a situation in a narrative format, requiring students to translate the text into mathematical expressions and equations to find a solution. They are crucial for developing problem-solving skills, critical thinking, and the ability to interpret information.

Key Concepts in Solving Word Problems

  1. Understanding the Problem: Read the problem carefully to identify what is being asked. Look for keywords that indicate mathematical operations (e.g., “total” suggests addition, “difference” suggests subtraction).
  2. Identifying the Information: Extract the relevant data from the problem. Highlight or underline key numbers and terms.
  3. Setting Up the Equation: Translate the word problem into a mathematical equation using the information identified.
  4. Solving the Equation: Perform the necessary calculations to solve the equation.
  5. Interpreting the Solution: Once you have an answer, make sure to interpret it in the context of the problem. Check if the answer makes sense logically.

Common Types of Word Problems

  • Addition and Subtraction Problems: Involve finding totals or differences.
  • Multiplication and Division Problems: Often related to groups or equal sharing.
  • Rate Problems: Involve speed, distance, or time.
  • Proportion Problems: Deal with ratios and comparisons.
  • Money Problems: Involve currency, budgeting, and spending.

Practice Questions on Word Problems

Easy Level

  1. Sarah has 5 apples. She buys 3 more. How many apples does she have now?
  2. Tom has 10 marbles. He gives 4 to his friend. How many marbles does Tom have left?
  3. A book costs £7. If you buy 2 books, how much do you spend?
  4. There are 12 cookies in a jar. If 3 cookies are eaten, how many are left?
  5. Lucy has 4 red balloons and 6 blue balloons. How many balloons does she have in total?
  6. A pencil costs £2. How much do 5 pencils cost?
  7. There are 20 students in a classroom. If 5 students leave, how many students are still in the classroom?
  8. Emma has 3 dogs and 2 cats. How many pets does she have in total?
  9. A cake is cut into 8 pieces. If 3 pieces are eaten, how many pieces remain?
  10. James had £15. He spent £5 on a toy. How much money does he have left?
  11. A pack of cards has 52 cards. If 5 cards are removed, how many cards are left?
  12. There are 6 chairs in a room. If 2 more chairs are added, how many chairs are there in total?
  13. A fruit basket contains 10 bananas. If you eat 2 bananas, how many bananas are left?
  14. A classroom has 4 tables. If each table seats 3 students, how many students can sit at the tables?
  15. A gardener plants 10 flowers in the morning and 5 in the afternoon. How many flowers did he plant in total?
  16. There are 15 books on a shelf. If 3 books are taken away, how many books remain on the shelf?
  17. A jar contains 30 jellybeans. If 10 jellybeans are taken out, how many are left?
  18. A farmer has 25 cows and 10 sheep. How many animals does the farmer have in total?
  19. A toy costs £8. If you buy 3 toys, how much do you spend in total?
  20. If a sandwich costs £3 and you buy 4 sandwiches, how much do you spend?

Medium Level

  1. A train travels at a speed of 60 km/h. How far does it travel in 2 hours?
  2. There are 40 students in a class. If 8 students are absent, how many students are present?
  3. A box contains 24 chocolates. If you eat 6 chocolates, what fraction of the box is left?
  4. A car can hold 5 people. How many cars are needed for 25 people?
  5. A recipe calls for 2 cups of flour. If you want to make half the recipe, how much flour do you need?
  6. A book has 300 pages. If you read 50 pages each day, how many days will it take to finish the book?
  7. A pack of 10 pencils costs £3. How much do 4 packs cost?
  8. There are 60 minutes in an hour. How many minutes are there in 3 hours?
  9. A school has 250 students. If 50 students join the school, how many students are there now?
  10. A pizza is cut into 12 slices. If 4 slices are eaten, what fraction of the pizza is left?
  11. A bottle holds 1.5 litres of water. How much water is in 4 bottles?
  12. A car travels 180 km in 3 hours. What is its average speed?
  13. A bag contains 40 marbles. If 10 marbles are blue, what percentage of the marbles are blue?
  14. A farmer has 200 apples. He sells 75 apples. How many apples does he have left?
  15. A movie ticket costs £10. If a family of 4 goes to the movies, how much do they spend?
  16. A train leaves the station at 9:00 AM and arrives at 10:30 AM. How long is the journey?
  17. A book costs £12, and a magazine costs £4. If you buy 2 books and 3 magazines, how much do you spend?
  18. A bicycle costs £150. If it is on sale for 20% off, what is the sale price?
  19. A bag contains 50 candies. If you take out 15 candies, how many candies are left?
  20. A jar can hold 2 litres of juice. How many litres can 5 jars hold?

Hard Level

  1. A rectangular garden measures 15 m by 10 m. What is its area?
  2. A car travels 240 km in 4 hours. What is its average speed?
  3. A box contains 120 items. If 25% are defective, how many items are not defective?
  4. A recipe requires 3 tablespoons of sugar for 4 servings. How much sugar is needed for 10 servings?
  5. A school has 300 students. If 60% are girls, how many boys are there?
  6. A cyclist travels at a speed of 15 km/h. How long will it take to travel 45 km?
  7. A shop sells pencils for £0.80 each. If you buy 7 pencils, how much do you pay?
  8. A classroom has 24 desks. If each desk can hold 2 students, what is the maximum number of students that can be seated?
  9. A field is 50 m long and 30 m wide. What is its perimeter?
  10. A person runs 5 km every day. How far will they run in 2 weeks?
  11. A factory produces 150 toys every hour. How many toys does it produce in 8 hours?
  12. A bag of flour weighs 2.5 kg. If you buy 3 bags, how much flour do you have?
  13. A car uses 6 litres of fuel to travel 100 km. How many litres are needed to travel 250 km?
  14. A train leaves the station at 2:30 PM and arrives at its destination at 4:15 PM. How long is the journey?
  15. A rectangle has a length of 12 cm and a width of 5 cm. What is its area?
  16. A recipe requires 2 cups of milk. If you have 1.5 cups, how much more do you need?
  17. A farmer has 120 sheep. If he sells 30% of them, how many sheep does he have left?
  18. A cyclist completes a race in 3 hours at an average speed of 20 km/h. How far did the cyclist travel?
  19. A pack of cookies costs £2.50. If you buy 4 packs, how much do you spend?
  20. A swimming pool is filled with water at a rate of 200 litres per hour. How long will it take to fill a pool with a capacity of 8000 litres?

Answers and Explanations

Easy Level

  1. 5 + 3 = 8
    apples.
  2. 10 – 4 = 6
    marbles left.
  3. 2 \times 7 = 14
    pounds.
  4. 12 – 3 = 9
    cookies left.
  5. 4 + 6 = 10
    balloons.
  6. 5 \times 2 = 10
    pounds.
  7. 20 – 5 = 15
    students.
  8. 3 + 2 = 5
    pets.
  9. 8 – 3 = 5
    pieces left.
  10. 15 – 5 = 10
    pounds left.
  11. 52 – 5 = 47
    cards left.
  12. 6 + 2 = 8
    chairs.
  13. 2 – 1 = 1
    banana left.
  14. 4 \times 3 = 12
    students.
  15. 10 + 6 = 16
    books.
  16. 2 \times 10 = 20
    dollars.
  17. 5 + 2 = 7
    items left.
  18. 8 + 6 = 14
    days.
  19. 20 + 10 = 30
    cookies.
  20. 2 + 4 = 6
    litres.

Medium Level

  1. Distance = 60 \times 2 = 120 \text{ km}
    .
  2. Students present = 40 – 8 = 32
    .
  3. Remaining = 24 – 6 = 18
    chocolates.
  4. Cars needed = \frac{25}{5} = 5
    cars.
  5. Flour needed = 2 \times \frac{1}{2} = 1 \text{ cup}
    .
  6. Days to finish = \frac{300}{50} = 6 \text{ days}
    .
  7. Total cost = 3 \times 4 = 12 \text{ pounds}
    .
  8. Minutes = 60 \times 3 = 180 \text{ minutes}
    .
  9. Students now = 250 + 50 = 300
    .
  10. Remaining pizza = 12 – 4 = 8
    slices.
  11. Total = 4 \times 3 = 12 \text{ litres}
    .
  12. Average speed = \frac{180}{3} = 60 \text{ km/h}
    .
  13. Percentage = \frac{10}{40} \times 100 = 25\%
    .
  14. Apples left = 200 – 75 = 125
    .
  15. Total = 4 \times 10 = 40 \text{ pounds}
    .
  16. Travel time = 10 – 2 = 8 \text{ hours}
    .
  17. Total = 60 \times 10 = 600 \text{ minutes}
    .
  18. Travel time = \frac{45}{15} = 3 \text{ hours}
    .
  19. Percentage = \frac{9}{15} \times 100 = 60\%
    .
  20. Total = 4 + 3 = 7 \text{ litres}
    .

Hard Level

  1. Area = \frac{1}{2} \times (12 + 8) \times 5 = 50 \text{ cm}^2
    .
  2. Volume = \frac{4}{3}\pi(3)^3 \approx 113.04 \text{ cm}^3
    .
  3. Surface area = 2(3 \times 4 + 4 \times 5 + 5 \times 3) = 94 \text{ cm}^2
    .
  4. Circumference = \pi(12) \approx 37.68 \text{ cm}
    .
  5. Area = \sqrt{s(s-a)(s-b)(s-c)} \approx 84 \text{ cm}^2
    .
  6. Diagonal = 5\sqrt{2} \approx 7.07 \text{ cm}
    .
  7. Volume = \frac{1}{3} \times 20 \times 9 = 60 \text{ cm}^3
    .
  8. Area = \frac{60}{360} \times \pi(10)^2 \approx 52.36 \text{ cm}^2
    .
  9. Using Pythagorean theorem: \text{hypotenuse} = \sqrt{6^2 + 8^2} \approx 10 \text{ cm}
    .
  10. Area = \frac{5}{4} \sqrt{5} \approx 14.72 \text{ cm}^2
    .
  11. Volume = \pi(4)^2(10) \approx 502.4 \text{ cm}^3
    .
  12. Surface area = \pi(3)^2 + \pi(3)(5) \approx 75.36 \text{ cm}^2
    .
  13. Area = \frac{1}{2}(8)(12) = 48 \text{ cm}^2
    .
  14. Length of arc = \frac{90}{360} \times 2\pi(5) \approx 3.93 \text{ cm}
    .
  15. Area = 10 \times 6 \sin(60°) \approx 51.96 \text{ cm}^2
    .
  16. Volume = \frac{1}{2}(3)(4)(10) = 60 \text{ cm}^3
    .
  17. Angle = \tan^{-1}\left(\frac{12}{16}\right) \approx 36.87°
    .
  18. Perimeter = 6 \times 4 = 24 \text{ cm}
    .
  19. Area = \frac{120}{360} \times \pi(7)^2 \approx 51.73 \text{ cm}^2
    .
  20. Surface area = 4\pi(4)^2 \approx 201.06 \text{ cm}^2
    .

These questions and answers provide a comprehensive overview of word problems relevant to the 11+ exam, covering various difficulty levels and encouraging students to develop their understanding and problem-solving skills.