Word problems using linear equations

2.1.8 · Maths › Algebra — Introduction & Intermediate

The Five-Step Method

Har word problem is systematic approach ko follow karta hai:

Kya: Choose karo kya $x$ se represent karna hai.
Kaise: Woh quantity choose karo jo problem tum se dhundne ko bol rahi hai, YA woh sabse simple quantity jisse tum answer compute kar sako.
Kyu: Unknown ko label karna invisible ko visible banata hai. $x$ ke bina, tum vague ideas juggle kar rahe ho. $x$ ke saath, tumhare paas ek handle hai.

Step 2: Express Other Quantities in Terms of $x$

Kya: Har doosri relevant quantity ko $x$ ka use karke likho.
Kaise: Problem mein di gayi relationships use karo. Agar John ke paas Mary se 5 zyada apples hain aur Mary ke paas $x$ apples hain, to John ke paas $x + 5$ hain.
Kyu: Ye step saari information ko ek symbol ke neeche unify karta hai, jisse tum ek equation likh sako instead of kai disconnected facts ke.

Step 3: Set Up the Equation

Kya: Main relationship ko ek equation mein translate karo.
Kaise: Woh sentence dhundo jo quantities ko "is," "equals," "total," ya "same as" se relate karta hai. Wahi tumhari equation hai.
Kyu: Equation constraint ka mathematical form hai. Wahin problem ki logic rehti hai.

Step 4: Solve the Equation

Kya: Algebraic techniques use karke $x$ ko isolate karo.
Kaise: Inverse operations apply karo: agar $x$ kisi cheez mein add hai, to use dono sides se subtract karo; agar $x$ multiply hai, to dono sides ko divide karo.
Kyu: Solve karne se woh numerical value milti hai jo saari relationships simultaneously satisfy karti hai.

Step 5: Verify and Interpret

Kya: Check karo ki tumhara answer original problem mein sense karta hai.
Kaise: Apni value wapas problem ki conditions mein substitute karo. Kya ye saare statements satisfy karta hai? Kya units sahi hain? Kya number reasonable hai?
Kyu: Ye arithmetic errors aur bekar answers (jaise negative age ya fractional log) pakad leta hai.

Figure — Word problems using linear equations

Common Problem Types & Translation Patterns

Type 1: Age Problems

Pattern: Alag-alag times par ages ko $+$ ya $-$ use karke time shifts ke liye relate karo.

Example Structure: "3 saal baad, Sarah ki age Tom ki current age se do guni hogi."

Let Sarah ki current age = $x$
Sarah 3 saal baad = $x + 3$
Equation: $x + 3 = 2 (Tom’s current age)$

Type 2: Number Problems

Pattern: Consecutive integers, digit problems, ya numbers ke beech relationships.

Key Translations:

Consecutive integers: $x$ , $x + 1$ , $x + 2$ , ...
Consecutive even/odd: $x$ , $x + 2$ , $x + 4$ , ...
"Sum of" → terms ke beech $+$ use karo
"Product of" → terms ke beech $\times$ use karo

Type 3: Money/Mixture Problems

Pattern: Alag-alag rates par alag-alag quantities, total value = individual values ka sum.

Formula Pattern: Agar tumhare paas quantity $q_{1}$ rate $r_{1}$ par hai aur quantity $q_{2}$ rate $r_{2}$ par: $Total Value = q_{1} \cdot r_{1} + q_{2} \cdot r_{2}$

Kyu: Value extensive hoti hai (add hoti hai). Har component independently contribute karta hai.

Type 4: Distance/RateTime Problems

Pattern: $Distance = Rate \times Time$ use karo

Key Insight: Agar kai objects travel karte hain, to unki distances/times/rates ko constraint ke basis par equal set karo (meeting point, same time, etc.).

Type 5: Work Problems

Pattern: Agar koi $t$ hours mein koi kaam complete karta hai, to unki rate $\frac{1}{t}$ jobs per hour hoti hai.

Combined Work Formula: Agar person A kaam $a$ hours mein complete karta hai aur person B $b$ hours mein: $Combined rate = \frac{1}{a} + \frac{1}{b} = \frac{1}{t _{together}}$

Derivation: Rate "fraction of job done per unit time" measure karta hai. Agar A $\frac{1}{a}$ per hour karta hai aur B $\frac{1}{b}$ per hour, to saath mein wo $\frac{1}{a} + \frac{1}{b}$ per hour karte hain. 1 kaam complete karne ka time $t$ dhundne ke liye: $t \cdot (\frac{1}{a} + \frac{1}{b}) = 1$ , isliye $t = \frac{1}{\frac{1}{a} + \frac{1}{b}} = \frac{ab}{a + b}$ .

Worked example Worked Example 1: Age Problem

Problem: Ek baap apne bete se 3 guna bada hai. 12 saal baad, wo apne bete se do guna bada hoga. Abhi beta kitne saal ka hai?

Step 1 — Identify Unknown:
Let bete ki current age = $x$ saal.
$x$ bete ke liye kyu? Problem bete ki age pooch rahi hai, isliye wo hamara target hai.

Step 2 — Express Other Quantities:
Baap ki current age = $3 x$ saal (baap 3 guna bada hai)
Bete ki age 12 saal baad = $x + 12$ saal
Baap ki age 12 saal baad = $3 x + 12$ saal
Ye expressions kyu? "12 saal baad" ka matlab hai current age mein 12 add karo. "3 guna bada" ka matlab hai 3 se multiply karo.

Step 3 — Set Up Equation:
"12 saal baad, baap bete ki age se do guna hoga" — ye relationship deti hai: $3 x + 12 = 2 (x + 12)$ Ye equation kyu? Left side = baap ki future age, right side = bete ki future age ka do guna. "Hoga" ka matlab equals hai.

Step 4 — Solve: $3 x + 12 = 2 x + 24$ Dono sides se $2 x$ kyu subtract karein? $x$ terms ek side par collect karne ke liye. $3 x - 2 x = 24 - 12$ $x = 12$ Dono sides se 12 kyu subtract karein? $x$ ko isolate karne ke liye. Step 5 — Verify:
Beta abhi: 12 saal
Baap abhi: $3 (12) = 36$ saal (✓ teen guna)
12 saal baad: Beta = 24, Baap = 48 (✓ 48 = 2×24)

Answer: Beta abhi 12 saal ka hai.

Worked example Worked Example 2: Money Problem

Problem: Ek piggy bank mein ₹5 aur ₹10 ke notes mein ₹155 hain. Total 23 notes hain. Har type ke kitne notes hain?

Step 1 — Identify Unknown:
Let ₹5 ke notes ki sankhya = $x$
Kyu? Hum koi bhi note type choose kar sakte the; ₹5 arbitrary hai lekin theek hai.

Step 2 — Express Other Quantities:
₹10 ke notes ki sankhya = $23 - x$ (kyunki total notes = 23)
Kyu? Agar $x$ notes ₹5 ke hain, to baaki ₹10 ke honge.

Step 3 — Set Up Equation:
Total value equation: $5 x + 10 (23 - x) = 155$ Ye equation kyu? (value per note) × (quantity) har type ke liye, total value mein sum hoti hai.
"Number of notes" use kyu nahi kiya? Hum pehle hi usse quantities express karne ke liye use kar chuke hain. Value hamari independent constraint hai.

Step 4 — Solve: $5 x + 230 - 10 x = 155$ Expand kyu karein? 10 ko parentheses mein distribute karo. $- 5 x + 230 = 155$ $- 5 x = 155 - 230 = - 75$ $x = \frac{- 75}{- 5} = 15$ -5 se divide kyu karein? $x$ ko isolate karne ke liye.

₹10 ke notes ki sankhya = $23 - 15 = 8$ Step 5 — Verify:
Total notes: $15 + 8 = 23$ ✓
Total value: $5 (15) + 10 (8) = 75 + 80 = 155$ ✓

Answer: ₹5 ke 15 notes aur ₹10 ke 8 notes.

Worked example Worked Example 3: Work Problem

Problem: Pipe A ek tank ko 6 ghante mein bhar sakta hai. Pipe B use 9 ghante mein bhar sakta hai. Agar dono pipes saath kholi jaayein, tank bharne mein kitna time lagega? Step 1 — Identify Unknown:
Let saath mein bharne ka time = $t$ ghante
Kyu? Wohi to problem seedha pooch rahi hai.

Step 2 — Understand Rates:
Pipe A ki rate = $\frac{1}{6}$ tank per hour (har ghante $\frac{1}{6}$ tank bharta hai)
Pipe B ki rate = $\frac{1}{9}$ tank per hour
Rates kyu? Work problems mein contributions add karne ke liye rates chahiye hoti hain.

Step 3 — Set Up Equation:
Combined rate = $\frac{1}{6} + \frac{1}{9}$ tank per hour
1 complete tank bharne ke liye: $t (\frac{1}{6} + \frac{1}{9}) = 1$ Rate ko time se multiply kyu karein? Rate × Time = Work done. Hum chahte hain work done = 1 tank.

Step 4 — Solve: $\frac{1}{6} + \frac{1}{9} = \frac{3}{18} + \frac{2}{18} = \frac{5}{18}$ Common denominator kyu dhundein? Fractions add karne ke liye. LCM(6,9) = 18. $t \cdot \frac{5}{18} = 1$ $t = \frac{18}{5} = 3.6 hours = 3 hours 36 minutes$ Invert aur multiply kyu karein? Fraction se divide karna = uske reciprocal se multiply karna.

Step 5 — Verify:
3.6 ghante mein, Pipe A bharta hai: $3.6 \times \frac{1}{6} = 0.6$ tank
3.6 ghante mein, Pipe B bharta hai: $3.6 \times \frac{1}{9} = 0.4$ tank
Total: $0.6 + 0.4 = 1.0$ tank ✓

Answer: 3 ghante 36 minute (ya $\frac{18}{5}$ ghante).

Worked example Worked Example 4: Consecutive Integer Problem

Problem: Teen consecutive odd numbers ka sum 63 hai. Numbers dhundo.

Step 1 — Identify Unknown:
Let pehla odd number = $x$
Odd kyu, even nahi? Problem odd specify karti hai. Hum verify karenge ki answer odd hai.

Step 2 — Express Other Quantities:
Doosra odd number = $x + 2$ (odd numbers mein 2 ka fark hota hai)
Teesra odd number = $x + 4$
2 add kyu karein, 1 nahi? Consecutive odd: 1,3,5,... ya 7,9,11,... Gap 2 ka hota hai.

Step 3 — Set Up Equation:
Sum = 63: $x + (x + 2) + (x + 4) = 63$ Ye equation kyu? "Teen numbers ka sum 63 hai" ka seedha translation.

Step 4 — Solve: $3 x + 6 = 63$ Combine kyu karein? $x + x + x = 3 x$ aur $2 + 4 = 6$ . $3 x = 57$ $x = 19$ Teen numbers: 19, 21, 23 Step 5 — Verify:
Kya wo odd hain? 19, 21, 23 sab odd hain ✓
Kya wo consecutive hain? 21-19=2, 23-21=2 ✓
Sum: $19 + 21 + 23 = 63$ ✓

Answer: Numbers hain 19, 21, aur 23.

Formula Master Translation Table

English Phrase	Mathematical Symbol	Example
is, equals, is the same as	$=$	"Age is 15" → $x = 15$
more than, increased by, sum	$+$	"5 more than $x$ " → $x + 5$
less than, decreased by, difference	$-$	"5 less than $x$ " → $x - 5$
times, product of (multiplication)	$\times$ or $\cdot$	"3 times $x$ " → $3 x$
per, ratio, quotient	$\div$ or $/$	"km per hour" → $\frac{km}{hour}$
total, combined, altogether	$+$ (sum all)	"Total of $x$ and $y$ " → $x + y$
consecutive integers	$x, x + 1, x + 2, \dots$	Teen consecutive: $x$ , $x + 1$ , $x + 2$
consecutive odd/even	$x, x + 2, x + 4, \dots$	Consecutive odd: $x$ , $x + 2$ , $x + 4$

Recall Ek 12-Saal-Ke Bachche Ko Explain Karo (Feynman Technique)

Socho tumhare paas ek mystery box hai jisme kuch marbles hain. Tum nahi jaante kitne hain, lekin koi tum ko clues deta hai: "Agar tum 5 aur marbles add karo, to tumhare paas total 20 ho jaayenge."

Guessing ki jagah, mystery number ke liye ek nickname banao: use $x$ marbles kaho. Ab clue translate karo: $x + 5 = 20$ . $x$ dhundne ke liye, 5 add karna "undo" karo dono sides se 5 subtract karke: $x = 15$ . Tumhare paas 15 marbles the!

Word problems bhi aise hi hain. Problem tum ko English mein clues deti hai. Tum:

Unknown number ke liye ek nickname ( $x$ ) choose karo
Clues ko ek math equation mein translate karo
Equation ko puzzle ki tarah solve karo (opposite operations karo)
Check karo ki tumhara answer saare clues fit karta hai

Ye ek detective banne jaisa hai: equation pehle se words mein chupi hui hai, tum bas use reveal karte ho. Har "is" ek equals sign ban jaata hai. Har "more than" ek plus ban jaata hai. Pattern dikha to, word problems sirf translation practice hain.

Connections

Linear Equations in One Variable — foundational equation form
Algebraic Expressions — " $x$ ke terms mein express karo" step ke liye expressions banana
Inverse Operations — solving techniques (add/subtract, multiply/divide)
Ratio and Proportion — kaafi word problems proportional relationships involve karte hain
Applications of Algebra — real situations model karne ka broader context
Simultaneous Linear Equations — jab ek equation kaafi nahi hoti

#flashcards/maths

Linear equations use karke word problems solve karne ke paanch steps kya hain? :: 1) Unknown identify karo ( $x$ choose karo), 2) Doosri quantities ko $x$ ke terms mein express karo, 3) Equation set up karo, 4) Equation solve karo, 5) Answer verify aur interpret karo.

x

se 5 kam" ko kaise represent karein?

x - 5

(NOT

5 - x

). "Less than" ka matlab hai number SE subtract karo.

x

se shuru hote consecutive odd integers ka formula kya hai?

x

x + 2

x + 4

, ... (inka fark 2 hota hai, 1 nahi).

Work problems mein, agar koi kaam 8 ghante mein complete hota hai, to rate kya hai?

\frac{1}{8}

jobs per hour (rate time ka reciprocal hai).

Agar Pipe A tank ko

a

ghante mein bharta hai aur Pipe B

b

ghante mein, to saath mein bharne ka time kya hai?

t = \frac{ab}{a + b}

ghante. Derive karo: combined rate

= \frac{1}{a} + \frac{1}{b} = \frac{a + b}{ab}

, isliye time

= \frac{1}{rate} = \frac{ab}{a + b}

"is, equals, is the same as" ko math mein kaise translate karein?

Equals sign

=

. Ye dikhata hai ki do expressions ki same value hai.

Age problem mein, "5 saal baad" current age

x

par kya operation karta hai?

Add 5:

x + 5

. Future age = current age + beete hue saal.

Word problems mein variables define karte waqt kya key galti hoti hai?

x

define karo phir khud se contradict karo same quantity ko alag tarah se re-express karke. Apni original definition ke saath consistent raho.

Money problem mein jahan

n

items ke do types hain, agar

x

Type A ke hain, to Type B kitne hain?

n - x

. Sum total ke equal hona chahiye, isliye Type B = Total - Type A.

"Teen guna bada" mathematically kya translate hota hai?

3 se multiply. Agar bete ki age

x

hai, "teen guna bada" ka matlab

3 x

hai.