Pair of Linear Equations – Word Problems
Class 10 Maths me Pair of Linear Equations in Two Variables ke word problems students ke liye initially confusing lag sakte hain. Problem usually calculation ki nahi hoti, balki question ki language ko mathematical equations me convert karne ki hoti hai.
Word problem solve karne ka sabse important rule hai: pehle story ko samjho, phir variables assume karo, statements ko equations me convert karo aur finally equations solve karo.
Word Problem → Understand Story → Assume Variables → Form Equations → Solve → Interpret Answer
Is chapter me hum numbers, ages, cost, digits, speed-distance, fractions, geometry aur money/notes jaise important word problems ko step-by-step samjhenge.
Word Problem Solve Karne Ka Basic Method
Har word problem ko solve karne ke liye ek fixed approach follow karna bahut useful hai.
Step 1 – Question Ko Dhyan Se Padho
Question ko kam se kam do baar read karo. Pehli reading me story samjho aur second reading me important information identify karo.
Step 2 – Unknown Quantities Identify Karo
Jo quantities unknown hain unhe variables se represent karo.
Let x = first unknown
Let y = second unknown
Step 3 – Statements Ko Equations Me Convert Karo
Question me diye gaye mathematical relationships ko equations ke form me likho.
Step 4 – Equations Solve Karo
Aap Elimination Method ya Substitution Method ka use kar sakte ho.
Step 5 – Answer Ko Interpret Karo
Sirf x aur y ki values likhna enough nahi hai. Final answer ko question ke context ke according likhna chahiye, jaise Father = 37 years, Pen = ₹16 ya Distance = 100 km.
Type 1 – Sum and Difference Problems
Numbers wale word problems me sum aur difference commonly use hote hain.
Example
The sum of two numbers is 50 and their difference is 10. Find the numbers.
Step 1 – Variables
Let:
First number = x
Second number = y
Step 2 – Equations Form Karo
Sum of numbers:
x + y = 50 ...(1)
Difference of numbers:
x − y = 10 ...(2)
Step 3 – Eliminate
Dono equations ko add karo:
x + y + x − y = 50 + 10
2x = 60
x = 30
Step 4 – Find y
Equation (1) me x = 30 put karo:
30 + y = 50
y = 20
Answer → Numbers = 30 and 20
Translation Trick
Sum → +
Difference → −
Product → ×
Quotient → ÷
Type 2 – Age Problems
Age problems me sabse important concept hai ki present, past aur future age ko correctly represent karna.
Example
The sum of the ages of a father and son is 50 years. Five years ago, the father's age was four times the son's age. Find their present ages.
Step 1 – Variables
Let:
Father's present age = x
Son's present age = y
Step 2 – Present Age Equation
Their present ages ka sum 50 hai:
x + y = 50 ...(1)
Step 3 – Five Years Ago
Five years ago father ki age:
x − 5
Five years ago son ki age:
y − 5
Question ke according:
x − 5 = 4(y − 5)
Expand:
x − 5 = 4y − 20
x − 4y = −15 ...(2)
Step 4 – Elimination
Equation (1):
x + y = 50
Equation (2):
x − 4y = −15
Subtract equation (2) from equation (1):
(x + y) − (x − 4y) = 50 − (−15)
5y = 65
y = 13
Step 5 – Find x
x + 13 = 50
x = 37
Answer → Father = 37 years, Son = 13 years
Age Problem Shortcut
Present → x, y
5 Years Ago → x − 5, y − 5
5 Years Later → x + 5, y + 5
Important → Never subtract or add years to only one person's age unless the question specifically requires it.
Type 3 – Cost Problems
Cost-based questions me generally quantity aur price ko variables se represent kiya jata hai.
Example
2 pens and 3 notebooks cost ₹110. 3 pens and 2 notebooks cost ₹100. Find the cost of each.
Step 1 – Variables
Let:
Cost of one pen = ₹x
Cost of one notebook = ₹y
Step 2 – Form Equations
First statement:
2x + 3y = 110 ...(1)
Second statement:
3x + 2y = 100 ...(2)
Step 3 – Eliminate x
Multiply equation (1) by 3:
6x + 9y = 330
Multiply equation (2) by 2:
6x + 4y = 200
Subtract:
5y = 130
y = 26
Step 4 – Find x
Equation (1):
2x + 3(26) = 110
2x + 78 = 110
2x = 32
x = 16
Answer → Pen = ₹16, Notebook = ₹26
Type 4 – Digit Problems
Digit-based questions me students ko sabse zyada confusion two-digit number ki formation me hota hai.
Two-Digit Number
Agar:
Tens digit = x
Units digit = y
To two-digit number hoga:
10x + y
Reversed Number
Digits reverse karne par:
10y + x
Example
The sum of the digits of a two-digit number is 9. When the digits are reversed, the new number is 27 more than the original number. Find the number.
Step 1 – Variables
Let:
Tens digit = x
Units digit = y
Step 2 – Sum of Digits
x + y = 9 ...(1)
Step 3 – Number Formation
Original number:
10x + y
Reversed number:
10y + x
Question ke according:
10y + x = 10x + y + 27
Simplify:
9y − 9x = 27
y − x = 3 ...(2)
Step 4 – Solve
Equation (1): x + y = 9
Equation (2): −x + y = 3
Add:
2y = 12
y = 6
Therefore:
x + 6 = 9
x = 3
Answer → Number = 36
Digit Trick
Two-Digit Number → 10 × Tens + Units
Three-Digit Number → 100 × Hundreds + 10 × Tens + Units
Type 5 – Speed and Distance Problems
Speed-distance problems me formula aur unit conversion dono important hote hain.
Basic Formula
Distance = Speed × Time
Speed = Distance ÷ Time
Time = Distance ÷ Speed
Example
A car travels a certain distance at 40 km/h and takes 30 minutes more than when travelling at 50 km/h. Find the distance.
Step 1 – Variable
Let distance = x km.
Step 2 – Time at 40 km/h
Time = Distance ÷ Speed
Time = x/40
Step 3 – Time at 50 km/h
Time = x/50
Difference in time is 30 minutes.
30 minutes = 1/2 hour
Therefore:
x/40 − x/50 = 1/2
LCM of 40 and 50 = 200.
5x − 4x = 100
x = 100
Answer → Distance = 100 km
Important Time Conversions
30 minutes = 1/2 hour
15 minutes = 1/4 hour
45 minutes = 3/4 hour
Important → Speed km/h me hai to time ko hours me convert karo.
Type 6 – Fraction Problems
Fraction-based word problems me numerator aur denominator ko variables ke through represent karna easy approach hai.
Example
The denominator of a fraction is 3 more than its numerator. If 1 is added to both numerator and denominator, the fraction becomes 1/2. Find the fraction.
Step 1 – Variables
Let:
Numerator = x
Denominator = y
Denominator numerator se 3 more hai:
y = x + 3
Therefore:
x − y = −3 ...(1)
Step 2 – Add 1 to Both
Question ke according:
(x + 1)/(y + 1) = 1/2
Cross multiplication:
2(x + 1) = y + 1
2x + 2 = y + 1
2x − y = −1 ...(2)
Step 3 – Eliminate
Equation (2) minus equation (1):
(2x − y) − (x − y) = −1 − (−3)
x = 2
Using y = x + 3:
y = 5
Answer → Fraction = 2/5
Type 7 – Geometry-Based Word Problems
Geometry ke word problems me rectangle, perimeter aur dimensions se related questions commonly pooche ja sakte hain.
Rectangle – Important Formulas
Let:
Length = x cm
Breadth = y cm
Perimeter:
2(x + y)
Area:
xy
Example
The perimeter of a rectangle is 50 cm. Its length is 5 cm more than its breadth. Find its dimensions.
Step 1 – Variables
Length = x
Breadth = y
Step 2 – Perimeter Equation
2(x + y) = 50
Divide by 2:
x + y = 25 ...(1)
Step 3 – Length Relation
Length breadth se 5 cm more hai:
x − y = 5 ...(2)
Step 4 – Solve
Add equations:
2x = 30
x = 15
Now:
15 + y = 25
y = 10
Answer → Length = 15 cm, Breadth = 10 cm
Type 8 – Money, Notes and Coins Problems
Money problems me quantity aur denomination ko multiply karke total value obtain ki jati hai.
Basic Idea
Agar:
x = number of ₹10 notes
y = number of ₹20 notes
To total number of notes:
x + y
Total value:
10x + 20y
Example
There are 20 notes of ₹10 and ₹20. Their total value is ₹300. Find the number of each type.
Step 1 – Variables
Let:
₹10 notes = x
₹20 notes = y
Step 2 – Total Number
x + y = 20 ...(1)
Step 3 – Total Value
10x + 20y = 300
Divide by 10:
x + 2y = 30 ...(2)
Step 4 – Solve
Subtract equation (1) from equation (2):
(x + 2y) − (x + y) = 30 − 20
y = 10
Therefore:
x = 10
Answer → ₹10 notes = 10, ₹20 notes = 10
Important Translation Table
| Words in Question | Mathematical Form |
|---|---|
| Sum | x + y |
| Difference | x − y |
| Twice x | 2x |
| Thrice x | 3x |
| 5 more than x | x + 5 |
| 5 less than x | x − 5 |
| x is 3 times y | x = 3y |
| Product | xy |
| Two-digit number | 10x + y |
| Reversed number | 10y + x |
| Perimeter of rectangle | 2(l + b) |
| Distance | Speed × Time |
| Total cost | Price × Quantity |
Most Important Language Trap
Word problems me language ko correctly interpret karna bahut important hai.
A is 5 More Than B
Correct:
A = B + 5
Incorrect:
B = A + 5
A is 5 Less Than B
Correct:
A = B − 5
A is 3 Times B
Correct:
A = 3B
A is 3 Less Than Twice B
Correct:
A = 2B − 3
Board Tip → "more than", "less than", "times", "twice" jaise words ko carefully translate karo.
Word Problem Master Flow
READ QUESTION
↓
IDENTIFY UNKNOWN
↓
LET x, y
↓
TRANSLATE SENTENCES
↓
FORM 2 EQUATIONS
↓
CHOOSE METHOD
↓
SOLVE
↓
CHECK
↓
WRITE ANSWER WITH UNITS
Ye flow almost har pair of linear equations word problem me apply kiya ja sakta hai.
Board Exam Strategy – L E T S A
Word problems ke liye ek simple five-step memory trick yaad rakho:
L → E → T → S → A
L → Let variables
E → Equations form karo
T → Transform/Solve equations
S → Substitute
A → Answer with meaning and units
Common Mistakes in Word Problems
Students word problems me kuch mistakes repeatedly karte hain. In mistakes ko avoid karna board exam me marks secure karne ke liye important hai.
- ❌ x aur y ko define na karna.
- ❌ "More than" ko reverse kar dena.
- ❌ Final answer me units na likhna.
- ❌ Past ya future age me incorrect adjustment karna.
- ❌ Two-digit number ko x + y likhna.
- ❌ Minutes ko hours me convert na karna.
- ❌ x aur y solve karne ke baad unka actual meaning na batana.
- ❌ Original question me answer verify na karna.
Important Board Questions
Question 1 – Numbers
The sum of two numbers is 40 and their difference is 8. Find the numbers.
Answer → 24 and 16
Question 2 – Ages
The sum of ages of two persons is 60 years. One is twice as old as the other. Find their ages.
Answer → 40 years and 20 years
Question 3 – Cost
2 pencils and 3 pens cost ₹35. 3 pencils and 2 pens cost ₹30. Find the cost of each.
Answer → Pencil = ₹4, Pen = ₹9
Question 4 – Digits
The sum of digits of a two-digit number is 11. The number exceeds the reversed number by 27. Find the number.
Answer → 74
Question 5 – Geometry
The perimeter of a rectangle is 64 cm. Its length is 4 cm more than its breadth. Find its dimensions.
Answer → Length = 18 cm, Breadth = 14 cm
How to Check Your Word Problem Answer?
Equations solve karne ke baad answer ko blindly accept nahi karna chahiye. Ek quick verification zarur karo.
Check 1 – Original Equation
Apni x aur y values ko first equation me substitute karo.
Check 2 – Second Equation
Same values ko second original equation me bhi substitute karo.
Check 3 – Context
Answer practical hona chahiye.
For example, age negative nahi ho sakti, number of notes fraction me nahi hona chahiye aur distance negative nahi ho sakta.
Check 4 – Units
Final answer me correct unit likho:
Age → years
Distance → km
Length → cm
Cost → ₹
Quick Revision – Different Types of Word Problems
| Problem Type | Important Representation |
|---|---|
| Numbers | Sum / Difference → x + y, x − y |
| Ages | Past → x − years, Future → x + years |
| Cost | Price × Quantity |
| Digits | Two-digit number → 10x + y |
| Speed-Distance | Distance = Speed × Time |
| Fractions | Numerator = x, Denominator = y |
| Rectangle | Perimeter = 2(l + b) |
| Notes/Coins | Value = Denomination × Number |
5-Second Memory Tricks
Sum → Plus
Difference → Minus
Twice → 2×
Thrice → 3×
More → Plus
Less → Minus
Two-Digit Number → 10x + y
Reversed Number → 10y + x
Distance → Speed × Time
Rectangle Perimeter → 2(l + b)
Total Cost → Price × Quantity
Exam Hall Checklist
Word problem solve karne ke baad ye checklist quickly revise karo:
- ✔ Kya maine x aur y define kiye?
- ✔ Kya maine dono equations correctly form ki?
- ✔ Kya "more than" aur "less than" correctly interpret kiya?
- ✔ Kya age problem me correct years add/subtract kiye?
- ✔ Kya two-digit number ko 10x + y represent kiya?
- ✔ Kya speed-distance question me units consistent hain?
- ✔ Kya equations correctly solve ki?
- ✔ Kya answer ko original question ke context me interpret kiya?
- ✔ Kya final answer ke saath correct units hain?
- ✔ Kya answer verify kiya?
Frequently Asked Questions (FAQs)
Pair of Linear Equations ke word problems kaise start karein?
Sabse pehle question ko carefully read karein, unknown quantities identify karein aur unhe x aur y se represent karein. Uske baad statements ko mathematical equations me convert karein.
Word problems me x aur y define karna zaruri hai?
Haan. Exam solution me clearly batana chahiye ki x aur y kya represent karte hain. Isse solution understandable aur logically complete hota hai.
Two-digit number ko x + y kyu nahi likh sakte?
Agar tens digit x aur units digit y hai, to number 10x + y hota hai. Tens digit ki place value 10 hoti hai, isliye x ko 10 se multiply kiya jata hai.
Age problems me 5 years ago kaise represent karte hain?
Agar present age x hai, to 5 years ago age x − 5 hogi. Similarly, 5 years later age x + 5 hogi.
Speed-distance problems me minutes ko hours me kyu convert karte hain?
Agar speed km/h me given hai, to time hours me hona chahiye. Isliye 30 minutes ko 1/2 hour me convert karna zaruri hai.
Word problem solve karne ke liye Elimination ya Substitution kaunsa method use karein?
Dono methods valid hain. Aap wahi method choose kar sakte hain jisme calculation simple ho. Agar kisi variable ke coefficients easily equal ya opposite ban sakte hain, to elimination method convenient hota hai.
Kya final answer me sirf x aur y ki values likhna enough hai?
Better approach ye hai ki x aur y ka actual meaning likhein. For example, agar x pen ki cost hai aur y notebook ki cost hai, to final answer me Pen = ₹16, Notebook = ₹26 likhna chahiye.
Word problems me sabse common mistake kya hoti hai?
Sabse common mistakes variables ko incorrectly define karna, statements ko equations me incorrectly translate karna, "more than" ko reverse karna, units ignore karna aur final answer verify na karna hain.
PDF Notes
Pair of Linear Equations – Word Problems ke detailed revision notes ko PDF format me directly yahan read kar sakte hain:
Final Thoughts
Pair of Linear Equations – Word Problems ko master karne ka secret sirf equations solve karna nahi hai. Sabse important skill hai question ki language ko mathematics ki language me convert karna.
Har question ke liye ek fixed process follow karo:
Read → Let x, y → Form Equations → Solve → Substitute → Check → Interpret Answer
Numbers ke questions me sum aur difference ko identify karo. Age problems me present, past aur future age ko carefully represent karo. Cost problems me price × quantity ka concept use karo. Digit problems me two-digit number ko 10x + y aur reversed number ko 10y + x likhna yaad rakho.
Speed-distance problems me Distance = Speed × Time formula aur correct unit conversion important hai. Fraction problems me numerator aur denominator ko variables se represent karke cross multiplication use ki ja sakti hai. Geometry problems me perimeter aur dimensions ke formulas ko correctly apply karo. Money aur notes problems me number of notes aur total value ki equations form karo.
Board exam me ek aur important habit develop karo: answer ko context ke saath likho. Agar x = 16 hai aur x pen ki cost represent karta hai, to final answer Pen = ₹16 likhna better hai. Isi tarah ages, dimensions, distance aur number of notes ko actual meaning ke saath mention karo.
Finally, answer ko original equations aur question ke context se verify karna mat bhoolo. Thodi si careful reading aur systematic approach ke saath Pair of Linear Equations ke word problems easily solve kiye ja sakte hain.
Master Formula:
Understand Story → Assume Variables → Form 2 Equations → Solve → Verify → Write Meaningful Answer
Quick Revision → L → E → T → S → A
L = Let Variables → E = Equations → T = Transform/Solve → S = Substitute → A = Answer