Class 8 Maths · Chapter 10
Short answer:
Class 8 Maths Chapter 10 'Proportional Reasoning-2' (Ganita Prakash, Part 2) me direct proportion, inverse proportion, aur compound (multi-factor) proportion ke real-life word problems cover hote hain — jaise workers-days, taps-time, aur cost-quantity problems. Har solution unitary method aur equation-setup ke saath step-by-step diya gaya hai, CBSE marking scheme ke hisaab se.
Chapter 10 (Ganita Prakash Part 2) poore Class 8 Maths ke real-life problem-solving ki neev hai — kyunki proportion ka concept sirf is chapter tak seemit nahi, aage physics, chemistry, aur daily-life budgeting me bhi kaam aata hai. Ye chapter Proportional Reasoning-1 (jo Part 1 me tha) ka direct extension hai — wahan basic ratio aur direct proportion cover hua tha, yahan hum inverse proportion aur compound (multi-factor) proportion tak apni understanding le jaate hain.
2026-27 session se NCERT ne Class 8 Maths ko rationalise karke 'Ganita Prakash' naam ki do-part book bana diya hai — total 14 chapters, Part 1 me 1-7 (jisme A Square and A Cube, Number Play, Fractions in Disguise, Tales by Dots and Lines, Proportional Reasoning-1 jaise chapters hain) aur Part 2 me 8-14 (jisme Algebra Play, Exploring Some Geometric Themes — jisme Baudhayana-Pythagoras theorem discuss hota hai, Quadrilaterals, Area, aur ye Proportional Reasoning-2 chapter shaamil hain). Ye purane single-book format se kaafi different structure hai, isliye students aur parents dono ko naya naming samajhna zaroori hai.
Is spec me har exercise ka step-by-step solution, common mistakes, aur exam-style questions cover kiye gaye hain — taaki students CBSE marking scheme ke hisaab se poori working dikhana seekh sakein, na ki sirf final answer.
Chapter 10 Summary — 5 Minute Revision
Chapter 10 'Proportional Reasoning-2' Class 8 Ganita Prakash Part 2 ka ek core problem-solving chapter hai jo Part 1 ke Proportional Reasoning-1 se aage badhkar inverse aur compound proportion tak extend karta hai. Is chapter me students seekhte hain ki kaise real-life situations — jaise workers-aur-days, taps-aur-fill-time, cost-aur-quantity — ko sahi tarike se direct ya inverse relationship me classify karke equation set karna hai. Sabse important skill hai relationship type ko sahi pehchaanna, kyunki galat classification (direct ki jagah inverse formula lagana, ya vice versa) sabse common mistake hai. Multi-step compound proportion problems (jisme workers, days, aur hours-per-day teeno ek saath badalte hain) is chapter ka advanced hissa hai jo higher-order thinking skills (HOTS) test karta hai. Practice ke liye NCERT exercises ke saath-saath sample-pattern questions bhi diye gaye hain jo exam-oriented reasoning develop karte hain.
In-Text Questions — Solutions
Figure it Out (intro activity): Ek recipe 4 logon ke liye 200 g atta use karti hai. 10 logon ke liye kitna atta chahiye?
Direct proportion — per-person quantity constant rehti hai.
Atta per person = 200/4 = 50 g
10 logon ke liye = 50 × 10 = 500 g
Answer: 500 g.
Think and Discuss: Kya har situation me 'zyada input = zyada output' hota hai?
Nahi — yahi is chapter ka core idea hai. Jaise agar 5 painters ek room ko 6 ghante me paint karte hain, toh 10 painters (double) time ko half kar denge, double nahi — kyunki yahan relationship inverse hai, direct nahi. Har problem me pehle relationship type identify karna zaroori hai.
Quick check: 3 machines ek order 12 din me complete karti hain. Same order 6 machines kitne din me karengi?
3 × 12 = 6 × x
36 = 6x
x = 6
Answer: 6 din.
Exercise Questions — Solutions (Q1–Q8)
Ex 10.1 Q1: 15 workers can build a wall in 48 days. How many workers are needed to build the same wall in 30 days?
Ye inverse proportion ka case hai — kam din chahiye toh zyada workers chahiye.
Workers × Days = constant
15 × 48 = x × 30
x = (15 × 48) / 30 = 720 / 30 = 24
Answer: 24 workers chahiye honge.
Ex 10.1 Q2: A car covers 180 km in 3 hours at constant speed. How much distance will it cover in 5 hours at the same speed?
Ye direct proportion hai — speed constant hai toh distance time ke saath directly badhta hai.
Distance/Time = constant
180/3 = x/5
60 = x/5
x = 300
Answer: 300 km.
Ex 10.2 Q1: If the cost of 8 kg sugar is ₹320, find the cost of 15 kg sugar.
Unitary method / direct proportion:
Cost of 1 kg = 320/8 = ₹40
Cost of 15 kg = 40 × 15 = ₹600
Answer: ₹600.
Ex 10.2 Q2: A photograph of size 4 cm × 6 cm is enlarged so the shorter side becomes 12 cm. What will be the new length of the longer side?
Enlargement me sides same ratio me badhte hain (direct proportion).
4/6 = 12/x
4x = 72
x = 18
Answer: 18 cm.
Ex 10.3 Q1: 6 pipes fill a tank in 4 hours. How many hours will 8 pipes take to fill the same tank?
Zyada pipes → kam time → inverse proportion.
6 × 4 = 8 × x
24 = 8x
x = 3
Answer: 3 hours.
Ex 10.3 Q2: 12 men can complete a piece of work in 18 days. In how many days will 9 men complete it?
Inverse proportion:
Men × Days = constant
12 × 18 = 9 × x
216 = 9x
x = 24
Answer: 24 days.
Ex 10.4 Q1: A recipe for 5 people needs 250 g rice. How much rice is needed for 8 people, and how much rice per person does this maintain?
Direct proportion — rice per person constant rehna chahiye.
Rice per person = 250/5 = 50 g
For 8 people = 50 × 8 = 400 g
Answer: 400 g, aur per-person rate 50 g/person constant rehta hai.
Ex 10.4 Q2 (mixed): 4 taps of same size fill a tank in 70 minutes. Due to a leak, one tap gets replaced by a smaller tap taking twice the time of others alone. Estimate qualitatively whether total fill time increases or decreases.
Ye ek reasoning-based question hai (exact numeric answer NCERT me discussion-style diya hai, formula-driven nahi).
Chhota tap replace hone se combined efficiency kam hoti hai (kyunki ek tap ab dheere kaam kar raha hai), isliye total fill time badhega — lekin poora tank fill hone me lagne wala time 70 minutes se thoda zyada hoga, exam me is tarah ke qualitative reasoning questions ko step-by-step logic likh kar justify karna chahiye, exact number ke bina bhi partial marks milte hain agar reasoning sahi ho.
Important Equations — Ek Nazar Me
| Concept | Formula / Rule | Kab use karein |
|---|---|---|
| Direct Proportion | a/b = c/d, ya y = kx | Jab ek quantity badhne se doosri same ratio me badhe (speed, cost per unit constant) |
| Inverse Proportion | a × b = c × d, ya y = k/x | Jab ek quantity badhne se doosri ulta ghate (workers-days, pipes-time) |
| Unitary Method | Value of 1 unit = Total / Number of units | Direct proportion problems me single-unit rate nikalne ke liye |
| Compound Proportion | (a1 × a2 × ...) × k = (b1 × b2 × ...) × k | Jab do ya zyada factors (workers, days, hours) ek saath badalte hain |
| Work-Time-Workers relation | Workers × Days = constant (fixed work ke liye) | Classic work-rate problems |
↔ Table ko side me swipe karein
Common Mistakes — Yahan Marks Kat te Hain
- Direct aur inverse proportion ko ulta laga dena — jaise workers-days problem me galti se a/b = c/d (direct formula) laga dena jabki wahan a×b = c×d (inverse) chahiye.
- Unitary method me per-unit value nikaalte waqt divide aur multiply ka order galat kar dena, jisse final answer ka decimal place shift ho jaata hai.
- Compound proportion problems me sirf ek factor consider karna aur doosre factor (jaise 'hours per day') ko ignore kar dena.
- Word problem me units mismatch karna — jaise kuch quantities minutes me aur kuch hours me rakh kar seedha equation bana dena bina convert kiye.
- Final answer likhte waqt working steps skip karna — CBSE marking scheme me setup/equation ke liye alag marks hote hain, sirf answer likhna partial marks deta hai.
- Ratio simplify karte waqt common factor poora nikaalna bhool jaana, jisse aage ka calculation unnecessarily complex ho jaata hai.
Board-Style Important Questions
- Sample-pattern question: 20 workers ek building 30 days me complete karte hain. Agar 5 extra workers add ho jaayein, toh kitne din lagenge? (Inverse proportion application, CBSE periodic-test style)
- Sample-pattern question: Ek car 4 hours me 240 km travel karti hai constant speed pe. Same speed pe 6 hours me kitni distance cover hogi? (Direct proportion, unitary method)
- Sample-pattern question: 8 taps ek tank 45 minutes me fill karte hain. 6 taps (same rate) kitne minutes lengi? (Inverse proportion, work-rate reasoning)
- Sample-pattern question: Compound proportion — 15 men, working 6 hours/day, complete a task in 10 days. Same task 10 men, working 9 hours/day me kitne din lengi? (Multi-factor proportion, HOTS-level)
- Sample-pattern question: Map scale problem — 2 cm on map represents 50 km actual distance. Agar map par distance 7 cm hai, toh actual distance kya hogi? (Direct proportion, real-life scale application)
Aksar Poochhe Jaane Wale Sawaal
Class 8 Maths Proportional Reasoning 1 vs 2 me kya difference hai?
Ganita Prakash Part 1 ke Proportional Reasoning-1 (chapter list ke shuruaati chapters me) me ratio, basic direct proportion, aur unitary method ki foundation build hoti hai. Part 2 ka Proportional Reasoning-2 usi concept ko aage le jaakar inverse proportion, compound situations (jaise work-time-workers ke combined problems), aur real-life multi-step proportion reasoning tak extend karta hai. Simple terms me — Part 1 foundation hai, Part 2 application aur complexity hai.
NCERT class 8 maths ki new book ka naam 2026-27 session ke liye kya hai?
2026-27 session se NCERT ne Class 8 Maths ki book ka naam 'Ganita Prakash' rakha hai, jo purani single-volume format se hatakar do parts — Part 1 aur Part 2 — me split ki gayi hai.
Ganita Prakash Part 1 aur Part 2 me kitne chapters hain?
Poori Ganita Prakash book me total 14 chapters hain — Part 1 me chapters 1 se 7 tak, aur Part 2 me chapters 8 se 14 tak. Proportional Reasoning-2 is rationalised structure me Part 2 ka hissa hai.
Direct aur inverse proportion me farak kaise pehchane?
Agar ek quantity badhne se doosri quantity same ratio me badhti hai (jaise distance-time constant speed pe), toh woh direct proportion hai. Agar ek quantity badhne se doosri ghategi hai (jaise workers badhne se days ghatna), toh woh inverse proportion hai. Chapter me dono types ke real-life examples diye gaye hain jinhe pehchaan kar hi sahi formula lagana zaroori hai.
Kya Proportional Reasoning-2 me compound proportion (multiple factors) aata hai?
Haan, is chapter ka ek important hissa compound / combined proportion problems hai — jaise 'workers + days + hours-per-day' teeno saath badalne wale sawaal. Aise questions me har factor ka effect alag-alag identify karke phir combine karna padta hai, jo purely Proportional Reasoning-1 me nahi aata tha.
Board/school exam me proportion ke word problems solve karne ka best tarika kya hai?
Sabse pehle identify karo ki relationship direct hai ya inverse, phir ek equation set karo (direct: a/b = c/d; inverse: a×b = c×d), aur step-by-step working dikhao — CBSE marking scheme me setup ke liye bhi marks milte hain, sirf final answer likhna kaafi nahi hota.
Class 8 Maths — Saare Chapters
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