Forming Equations from Statements
In everyday life, we often hear statements like: "I am thinking of a number. If I add 5 to it, I get 12. What is the number?" You can easily guess it is 7. But how do we write this using maths?
We write: x + 5 = 12. Here, x is the unknown number. This statement with an equal sign is called an equation. Forming equations means converting word sentences into mathematical sentences using variables and the equals sign.
Equations are one of the most powerful tools in mathematics. Once you write an equation, you can solve it to find the unknown value. In this chapter, we will learn how to translate English statements into algebraic equations.
What is Forming Equations from Statements - Grade 6 Maths (Algebra)?
Definition: An equation is a mathematical statement that says two things are equal. It always has an equals sign (=).
Parts of an equation:
- Left-Hand Side (LHS) — the expression on the left of the equals sign.
- Right-Hand Side (RHS) — the expression on the right of the equals sign.
- Variable — the unknown number, usually written as x, y, n, or any letter.
Examples:
- x + 3 = 10 is an equation (LHS = x + 3, RHS = 10).
- 2y = 14 is an equation.
- x + 5 is NOT an equation (no equals sign — it is just an expression).
Key words and their mathematical meaning:
- "sum of" or "added to" → use +
- "difference" or "subtracted from" → use −
- "product of" or "times" → use ×
- "divided by" or "quotient" → use ÷
- "is", "equals", "gives", "results in" → use =
Forming Equations from Statements Formula
Steps to form an equation:
Read → Identify unknown → Use a variable → Translate → Write equation
- Read the word statement carefully.
- Identify the unknown quantity (what you need to find).
- Choose a variable (like x) to represent the unknown.
- Translate each part of the sentence into mathematical operations.
- Write the equation using the equals sign.
Derivation and Proof
Let us form an equation from this statement step by step:
"A number increased by 8 gives 15."
- Read: A number increased by 8 gives 15.
- Identify unknown: "a number" — this is what we do not know.
- Choose variable: Let the number be x.
- Translate:
- "increased by 8" means + 8.
- "gives 15" means = 15.
- Write equation: x + 8 = 15.
Another example:
"Three times a number is 21."
- Unknown: "a number" → let it be x.
- "Three times" means 3 × x = 3x.
- "is 21" means = 21.
- Equation: 3x = 21.
Checking: To verify, substitute x = 7: 3 × 7 = 21. Correct!
Types and Properties
Types of equations you can form:
- Type 1: Addition equations — "A number plus 6 equals 14" → x + 6 = 14.
- Type 2: Subtraction equations — "7 less than a number is 20" → x − 7 = 20.
- Type 3: Multiplication equations — "Twice a number is 18" → 2x = 18.
- Type 4: Division equations — "A number divided by 4 gives 5" → x/4 = 5.
- Type 5: Two-step equations — "Three times a number plus 2 is 17" → 3x + 2 = 17.
- Type 6: Age problems — "Rina is 5 years older than Sita. Rina is 13." → x + 5 = 13 (where x = Sita's age).
- Type 7: Money problems — "After spending Rs 45, Ravi has Rs 30 left." → x − 45 = 30.
Solved Examples
Example 1: Example 1: Addition Equation
Statement: A number added to 9 gives 16.
Solution:
- Let the number be x.
- "added to 9" → x + 9
- "gives 16" → = 16
Equation: x + 9 = 16.
Example 2: Example 2: Subtraction Equation
Statement: 12 subtracted from a number is 25.
Solution:
- Let the number be x.
- "12 subtracted from a number" → x − 12
- "is 25" → = 25
Equation: x − 12 = 25.
Example 3: Example 3: Multiplication Equation
Statement: Five times a number is 45.
Solution:
- Let the number be x.
- "Five times a number" → 5x
- "is 45" → = 45
Equation: 5x = 45.
Example 4: Example 4: Division Equation
Statement: A number divided by 3 equals 7.
Solution:
- Let the number be x.
- "divided by 3" → x/3
- "equals 7" → = 7
Equation: x/3 = 7.
Example 5: Example 5: Two-Step Equation
Statement: If you double a number and add 3, you get 19.
Solution:
- Let the number be x.
- "double a number" → 2x
- "add 3" → 2x + 3
- "you get 19" → = 19
Equation: 2x + 3 = 19.
Example 6: Example 6: Age Problem
Statement: Meera is 4 years younger than her brother. Meera is 9 years old. What is her brother's age?
Solution:
- Let the brother's age be x.
- "4 years younger" means Meera's age = x − 4.
- Meera is 9 → x − 4 = 9
Equation: x − 4 = 9.
(Solving: x = 9 + 4 = 13. Brother is 13 years old.)
Example 7: Example 7: Money Problem
Statement: Ravi had some money. After spending Rs 120, he has Rs 80 left. How much did he have?
Solution:
- Let Ravi's original money be x.
- "After spending Rs 120" → x − 120
- "has Rs 80 left" → = 80
Equation: x − 120 = 80.
(Solving: x = 80 + 120 = Rs 200.)
Example 8: Example 8: Sharing Equally
Statement: A bag of sweets is shared equally among 6 children. Each child gets 8 sweets. How many sweets were in the bag?
Solution:
- Let total sweets be x.
- "shared equally among 6" → x/6
- "each gets 8" → = 8
Equation: x/6 = 8.
(Solving: x = 8 × 6 = 48 sweets.)
Example 9: Example 9: Consecutive Numbers
Statement: The sum of two consecutive numbers is 35. Find the numbers.
Solution:
- Let the smaller number be x.
- The next consecutive number is x + 1.
- "sum is 35" → x + (x + 1) = 35
Equation: 2x + 1 = 35.
(Solving: 2x = 34, x = 17. The numbers are 17 and 18.)
Example 10: Example 10: Perimeter Problem
Statement: The length of a rectangle is twice its width. The perimeter is 36 cm. Find the width.
Solution:
- Let the width be x cm.
- Length = twice the width = 2x cm.
- Perimeter = 2(length + width) = 2(2x + x) = 2(3x) = 6x.
- "Perimeter is 36" → 6x = 36
Equation: 6x = 36.
(Solving: x = 6. Width = 6 cm, Length = 12 cm.)
Real-World Applications
Where do we form equations?
- Solving puzzles — "I think of a number" problems are equations in disguise.
- Shopping — "I bought 3 pens and spent Rs 45. How much does each pen cost?" → 3x = 45.
- Age problems — Finding someone's age using clues about relationships.
- Geometry — Finding unknown lengths when perimeter or area is given.
- Science — Physics formulas like distance = speed × time are equations with unknowns.
- Everyday decisions — "If I save Rs 50 per day, how many days to save Rs 600?" → 50d = 600.
Key Points to Remember
- An equation has an equals sign (=) and shows that two expressions are equal.
- To form an equation: read the statement, identify the unknown, use a variable, and translate.
- "Sum", "added to", "plus" → use +.
- "Difference", "less than", "subtracted from" → use −.
- "Times", "product", "twice", "thrice" → use ×.
- "Divided by", "quotient", "shared equally" → use ÷.
- "Is", "equals", "gives", "results in" → use =.
- Always define what your variable represents before writing the equation.
- After forming the equation, you can solve it to find the unknown value.
- Check your answer by putting it back into the original word statement.
Practice Problems
- Form an equation: "A number added to 11 gives 20."
- Form an equation: "8 less than a number is 15."
- Form an equation: "Four times a number equals 32."
- Form an equation: "A number divided by 5 is 9."
- Form an equation: "If you triple a number and subtract 4, you get 23."
- Priya has some pencils. She gives 7 to her friend and has 12 left. Form an equation for the total number of pencils.
- The perimeter of a square is 48 cm. Form an equation for the side length.
- Form an equation: "The sum of three consecutive numbers is 36."
Frequently Asked Questions
Q1. What is the difference between an expression and an equation?
An expression is a combination of numbers and variables using operations (like 3x + 5). An equation has an equals sign and says two expressions are equal (like 3x + 5 = 20). An expression does NOT have an equals sign.
Q2. What does 'forming an equation' mean?
It means converting a word statement into a mathematical sentence using a variable (like x) and an equals sign. For example, 'a number plus 7 is 15' becomes x + 7 = 15.
Q3. How do I know which operation to use?
Look for key words. 'Added to' or 'sum' means +. 'Less than' or 'subtracted from' means −. 'Times' or 'product' means ×. 'Divided by' or 'shared' means ÷. 'Is' or 'equals' means =.
Q4. What does 'twice a number' mean?
'Twice a number' means 2 times the number. If the number is x, then twice the number is 2x. Similarly, 'thrice' means 3 times, so thrice x = 3x.
Q5. What does '5 less than a number' mean?
It means the number minus 5. If the number is x, it is x − 5. Be careful: '5 less than x' is x − 5, NOT 5 − x. The phrase 'less than' reverses the order.
Q6. Can I use any letter for the variable?
Yes. You can use x, y, n, a, p, or any letter. The letter x is the most common. Always state what your variable represents (e.g., 'Let x be the number of apples').
Q7. How do I check if my equation is correct?
Solve the equation to get a value. Put that value back into the original word statement and check if it makes sense. If it does, your equation is correct.
Q8. What is a two-step equation?
An equation that needs two steps to solve. Example: 2x + 3 = 11. First subtract 3 (2x = 8), then divide by 2 (x = 4). The statement 'double a number and add 3 gives 11' leads to this equation.
