Algebra Word Problems (Class 6)
You have learned about variables, constants, and how to form algebraic expressions. Now it is time to put these skills to work by solving simple word problems using algebra.
In algebra word problems, you translate English sentences into mathematical expressions or equations, and then find the unknown value. The unknown is represented by a letter (usually x, n, or any variable).
This is one of the most important skills in mathematics — it connects words to numbers and helps you solve real-life problems logically.
What is Algebra Word Problems?
Key concepts:
- A variable is a letter that stands for an unknown number (x, y, n, etc.).
- An expression is a combination of variables, numbers, and operations (e.g., 2x + 5).
- An equation is a statement that two expressions are equal (e.g., 2x + 5 = 15).
Steps to solve algebra word problems:
- Read the problem carefully.
- Identify the unknown. Assign it a variable (let x = ...).
- Translate the words into an algebraic expression or equation.
- Solve the equation (if asked).
- Check by substituting the answer back into the original problem.
Common translations:
- "5 more than a number" → x + 5
- "3 less than a number" → x − 3
- "Twice a number" → 2x
- "One-third of a number" → x/3
- "A number increased by 7" → x + 7
Types and Properties
1. Forming Expressions from Words
Translate a sentence into an algebraic expression.
- "Ravi's age is 5 years more than Sita's age." If Sita's age = x, Ravi's age = x + 5.
2. Forming Equations from Words
Translate a sentence into an equation (with =).
- "A number added to 8 gives 15." → x + 8 = 15.
Find the value of the variable.
- x + 8 = 15 → x = 15 − 8 = 7.
4. Age Problems
Relate ages of people using variables.
5. Number Problems
Find unknown numbers from given conditions.
Solved Examples
Example 1: Example 1: Forming an expression
Problem: Priya has some marbles. She gets 12 more. Write an expression for her total marbles.
Solution:
- Let Priya's original marbles = x.
- Total = x + 12.
Answer: x + 12
Example 2: Example 2: Forming an equation
Problem: A number added to 7 gives 20. Find the number.
Solution:
- Let the number = x.
- Equation: x + 7 = 20.
- x = 20 − 7 = 13.
Answer: The number is 13.
Example 3: Example 3: Twice a number
Problem: Twice a number is 24. Find the number.
Solution:
- Let the number = x.
- 2x = 24.
- x = 24 ÷ 2 = 12.
Answer: The number is 12.
Example 4: Example 4: Age problem
Problem: Anu is 4 years older than Binu. The sum of their ages is 28. Find their ages.
Solution:
- Let Binu's age = x. Then Anu's age = x + 4.
- Sum: x + (x + 4) = 28.
- 2x + 4 = 28.
- 2x = 24 → x = 12.
- Binu = 12 years, Anu = 16 years.
Answer: Binu = 12 years, Anu = 16 years.
Example 5: Example 5: Perimeter problem
Problem: The perimeter of a square is 48 cm. Find the side.
Solution:
- Let side = s.
- Perimeter = 4s = 48.
- s = 48 ÷ 4 = 12.
Answer: Side = 12 cm.
Example 6: Example 6: Cost problem
Problem: A pen costs Rs. x. A book costs Rs. 3 more than the pen. Together they cost Rs. 75. Find the cost of each.
Solution:
- Pen = x. Book = x + 3.
- x + (x + 3) = 75 → 2x + 3 = 75 → 2x = 72 → x = 36.
- Pen = Rs. 36, Book = Rs. 39.
Answer: Pen = Rs. 36, Book = Rs. 39.
Example 7: Example 7: Number decreased
Problem: When 9 is subtracted from 3 times a number, the result is 18. Find the number.
Solution:
- Let the number = x.
- 3x − 9 = 18.
- 3x = 27 → x = 9.
Answer: The number is 9.
Example 8: Example 8: Consecutive numbers
Problem: The sum of two consecutive numbers is 47. Find them.
Solution:
- Let the numbers be x and x + 1.
- x + (x + 1) = 47 → 2x + 1 = 47 → 2x = 46 → x = 23.
Answer: The numbers are 23 and 24.
Example 9: Example 9: Dividing marbles
Problem: Ravi has twice as many marbles as Sita. Together they have 36 marbles. How many does each have?
Solution:
- Let Sita = x. Ravi = 2x.
- x + 2x = 36 → 3x = 36 → x = 12.
- Sita = 12, Ravi = 24.
Answer: Sita = 12, Ravi = 24.
Example 10: Example 10: Checking the answer
Problem: A number multiplied by 5 and then reduced by 3 gives 22. Find and verify.
Solution:
- 5x − 3 = 22 → 5x = 25 → x = 5.
- Check: 5 × 5 − 3 = 25 − 3 = 22 ✓.
Answer: The number is 5.
Real-World Applications
Shopping: "I bought x notebooks at Rs. 30 each and paid Rs. 210. How many did I buy?" → 30x = 210 → x = 7.
Age Puzzles: "Father is 30 years older than son. In 5 years, father will be twice the son's age." These classic puzzles use algebra.
Sports: "A team scored twice in the second half compared to the first. Total goals = 9." → x + 2x = 9.
Sharing: Dividing money, sweets, or tasks equally or in a given ratio uses algebraic expressions.
Geometry: Finding unknown sides from perimeter or area formulas uses algebra (e.g., 2(l + w) = 40).
Key Points to Remember
- Assign a variable (x) to the unknown quantity.
- Translate words to math: "more than" = +, "less than" = −, "times" = ×, "divided by" = ÷.
- Form an expression (no = sign) or an equation (has = sign) as needed.
- Solve by performing inverse operations: if x + 5 = 12, then x = 12 − 5.
- Always check your answer by substituting back.
- "Consecutive numbers" means x, x+1, x+2, etc.
- "Twice a number" = 2x. "Thrice" = 3x. "Half" = x/2.
- Read the problem at least twice before writing the equation.
Practice Problems
- A number added to 15 gives 32. Find the number.
- Three times a number minus 4 equals 20. Find the number.
- Anu has x chocolates. She gives 5 to her friend and has 12 left. Find x.
- The sum of three consecutive numbers is 36. Find them.
- A rectangle has length (x + 3) cm and width x cm. Its perimeter is 30 cm. Find x.
- Father is 25 years older than his son. Their combined age is 45. Find both ages.
- A bag has red and blue balls. There are 3 times as many red balls as blue balls. Total = 32. Find each.
- If you subtract 7 from half a number, you get 8. Find the number.
Frequently Asked Questions
Q1. What is the difference between an expression and an equation?
An expression is a mathematical phrase with variables and numbers (like 3x + 5). An equation has an equals sign and says two expressions are equal (like 3x + 5 = 20).
Q2. How do I choose the variable?
Pick any letter (x, n, y). Let it represent the unknown quantity. For example, 'let x = the number of apples'. Choose what makes the problem easiest to set up.
Q3. What does 'more than' mean in algebra?
'More than' means addition. '5 more than x' = x + 5. Be careful: '5 more than 3' = 3 + 5 = 8.
Q4. What does 'less than' mean?
'Less than' means subtraction. '7 less than x' = x − 7. Note the order: it is x minus 7, not 7 minus x.
Q5. How do I solve x + 8 = 15?
Use the inverse operation. Since 8 is added to x, subtract 8 from both sides: x = 15 − 8 = 7.
Q6. How do I check my answer?
Substitute the value back into the original equation. If x = 7 and the equation was x + 8 = 15, check: 7 + 8 = 15 ✓. If both sides are equal, the answer is correct.
Q7. Can there be two unknowns?
Yes, but at Class 6 level, most problems have one unknown. If there are two unknowns, they are usually related (e.g., 'Anu is 4 years older than Binu' — both ages depend on one variable).
Q8. Are algebra word problems important for exams?
Yes. Forming expressions and simple equations from word problems is a key topic in Class 6 NCERT/CBSE Maths.
