How Quantities Combine: Understanding Data (Quantities Kaise Judti Hain: Data Ko Samajhna): Class 9 Maths Chapter 10 Hinglish mein

Ganita Manjari Part 2 · Hinglish notes, simulations aur practice · Official NCERT chapter PDF ↗

Published 13 Oct 2026 · Updated 13 Oct 2026

60 min
33 practice questions
19 MCQs

Is chapter mein aap kya seekhenge

Samjhao ki groups ka size alag ho to group means ka mean kyun gumraah kar sakta hai, aur sahi combined mean nikaalo
Groups ko jodne aur values add ya remove karne ke liye mean × number of values = total ka use karo
Mixture ka concentration nikaalo, aur target concentration tak pahunchne ke liye kitni quantity chahiye woh bhi
Aisa weighted mean calculate karo jiske weights size ya importance dikhate hain, jaise kisi ratio mein jode gaye marks
Jo sawaal poocha gaya hai uske hisaab se clustered, stacked aur 100% stacked bar charts mein se sahi chart chuno
100% stacked bar chart banane ke liye amounts ko percentages mein badlo, aur tay karo ki woh kaun se conclusions support karta hai

NCERT Mathematics Class 9, Chapter 10, pp. 8–36 par based original companion text. Yeh NCERT se juda nahi hai aur NCERT ne ise endorse nahi kiya hai.

10.1 Cheezon Ko Jodna

Section 9A ka class test mein mean 68 aaya aur Section 9B ka 74. Kya poori Class 9 ka mean bas 71 hai, dono ke beech mein? Sirf do averages se yeh pata nahi chal sakta. Jo baat missing hai woh yeh ki har section mein kitne students the.

Yaad karo ki values ke set ka mean (माध्य), yaani average, unka sum divided by values ki ginti hota hai. Is chapter mein hum pehle data ko sahi tareeke se combine karna seekhenge, jisse hum weighted mean tak pahunchenge. Phir hum un charts ki taraf badhenge jo dikhate hain ki ek whole kaise parts mein bantta hai.

10.1.1 Averages Ka Average

Hamare school ka Eco Club recycling ke liye plastic waste ikattha karta hai. Iske do groups hain: Class 10 ke 7 members aur Class 6 ke 3 members. Term ke end mein har member ka collection weigh kiya gaya.

Example 10.1NCERT Example 1 · p. 8

Eco Club ka average

Class 10 ke 7 members ne 32, 28, 35, 30, 26, 33 aur 26 kg ikattha kiya. Class 6 ke 3 members ne 12, 15 aur 18 kg ikattha kiya. Poore club ke liye per member mean collection kitna hai?

Pehle khud try karo. Phir solution ek-ek step karke kholo.

3/3Sabhi 10 members ko ek saath lene par 25.5 kg aata hai, jo Class 10 ke 7 members ki taraf khincha hua hai.

Fig. 10.1Sahi mean bade group ke zyada paas hota hai.

Ek number line 10 se 40 kilograms tak jaati hai. Saat circles Class 10 ke members ko 26 aur 35 ke beech dikhate hain, aur teen squares Class 6 ke members ko 12, 15 aur 18 par dikhate hain. Pehle step mein dashed lines 15 aur 30 par group means dikhati hain. Doosre step mein ek diamond 22.5 par hai, dono means ke beech mein, jis par mean of means likha hai. Teesre step mein ek solid line aur ek filled triangle 25.5 par true mean dikhate hain, jo bade Class 10 group ke zyada paas hai, aur halfway marker fade ho jaata hai.

Isko theek karne ke liye ek aisi baat kaam aati hai jo tum pehle se jaante ho. Agar $n$ values ke group ka mean $a$ hai, to group ka total $a \times n$ hota hai. To Class 10 ka total $30 \times 7 = 210$ kg hai aur Class 6 ka total $15 \times 3 = 45$ kg. Totals ko jodo aur members ki ginti se divide karo: $255 \div 10 = 25.5$ kg. Humein kisi ek member ki value ki zaroorat hi nahi padi.

General roop mein, maan lo ek collection mein $n$ values hain jinka mean $a$ hai, aur doosre mein $m$ values hain jinka mean $b$ hai. Unke totals $a n$ aur $b m$ hain, isliye sabhi $n + m$ values ka mean hai

Do combined groups ka mean

\overset{ˉ}{x} = \frac{a n + b m}{n + m}

(10.1)

jahan

$a, n$: pehle group ka mean aur size
$b, m$: doosre group ka mean aur size

Dekho, bada group combined mean ko apne mean ki taraf kaise kheenchta hai. Class 10 group mein 10 mein se 7 members hain, isliye 25.5 kg, 15 kg ke bajaye 30 kg ke kaafi paas hai.

Example 10.2NCERT Example 2 · p. 9

Shortcut kab kaam karta hai

Bhola hafte mein chhe din samosa stall chalata hai. Table dikhata hai ki teen hafton tak usne har din kitni plates bechi. Per day bechi gayi plates ka mean nikaalo.

Table 10.1Bhola ke stall par bechi gayi plates

Week	Mon	Tue	Wed	Thu	Fri	Sat	Weekly mean
Week 1	42	45	48	50	55	60	50
Week 2	48	50	52	55	60	65	55
Week 3	50	52	54	58	62	66	57

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Shortcut Bhola ke liye kyun chala par Eco Club ke liye kyun nahi? Teen collections lo jinke means $a$ , $b$ aur $c$ hain aur sizes $p$ , $q$ aur $r$ . Unka combined mean hai

Teen combined groups ka mean

\overset{ˉ}{x} = \frac{a p + b q + c r}{p + q + r}

(10.2)

jahan

$a, b, c$: teen collections ke means
$p, q, r$: unke sizes

To jab har collection ka size same ho, to means ka mean hi true mean hota hai. Jab sizes alag hon, jaise Eco Club mein, to jodne se pehle har mean ko uske group size se multiply karna padta hai.

Quick check

3 mein se 1

Section 9A mein 40 students hain jinka mean score 62 hai. Section 9B mein 20 students hain jinka mean score 74 hai. Sabhi 60 students ka mean score kya hai?

68666470

Answer dekho

Answer (B) 66

Totals

40 \times 62 = 2480

aur

20 \times 74 = 1480

hain, isliye mean

3960 \div 60 = 66

hai. Lalchaane wala 68 bas beech ka point hai, jo is baat ko ignore karta hai ki 9A mein do guna students hain.

10.1.2 Mixtures

Do drinks ko mila do aur poochho ki mixture kitna meetha hai. Yeh wahi sawaal hai, bas naye roop mein. Kisi drink mein sugar ka concentration (सांद्रता) drink ka woh fraction, ya percentage, hai jo sugar hai. Kisi bhi mixture ke liye,

c = \frac{s}{q}

jahan

$c$: mixture ka concentration
$s$: sugar (ya salt, ya fat …) ki total amount
$q$: mixture ki total amount

Example 10.3NCERT Example 3 · p. 11

Do barabar jugs

Jug A mein 1 litre aam panna hai jismein 8% sugar hai. Jug B mein 1 litre hai jismein 14% sugar hai. Dono ko ek bade jar mein daal diya jaata hai. Mixture ka kitna percentage sugar hai?

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Barabar jugs Bhola ke barabar hafton jaise behave karte hain: har jug ko barabar hissa milta hai. Ab amounts badlo.

Example 10.4NCERT Example 4 · p. 11

Alag-alag jugs

Is baar Jug A mein 1.5 litres (8% sugar) hai aur Jug B mein 0.5 litre (14% sugar). Calculate karne se pehle guess karo: mixture mein 11% sugar hai, 11% se zyada ya kam?

Pehle khud try karo. Phir solution ek-ek step karke kholo.

2/2Alag amounts: mixture 9.5% hai, bade glass ke 8% ke zyada paas.

Fig. 10.2Bada glass mixture ke concentration ko apni taraf kheenchta hai.

Left mein drink ke do glass hain, pehle mein 8% sugar aur doosre mein 14% sugar, aur har glass ke neeche sugar ek band ke roop mein dikhayi gayi hai. Right mein ek lamba jar 2 litre mixture rakhta hai. Neeche 8% se 14% tak ek scale hai jis par ek pointer hai. Pehle step mein dono glass mein 1 litre hai aur pointer 11% par hai, theek beech mein. Doosre step mein pehla glass badhkar 1.5 litres ho jaata hai aur doosra ghatkar 0.5 litre; mixture ka sugar band patla ho jaata hai aur pointer 9.5% ki taraf khisak jaata hai, 8% ke zyada paas.

Har jug ko aise socho jaise woh mixture ko apni mithaas ki taraf kheench raha hai, aur jitna zyada daaloge utna zor se kheenchega. Example 10.1 mein group sizes ne bhi yahi kaam kiya tha. Hum is idea ko ulta chala kar missing amount bhi nikaal sakte hain.

Example 10.5NCERT Example 5 · p. 12

Missing amount nikaalna

Ek dairy doodh ke teen lots mix karti hai: 6% fat wala 40 L, 3% fat wala 60 L, aur 5% fat wala kuch aur doodh. Blend mein 4.5% fat hona chahiye. 5% wala kitne litres doodh chahiye?

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Ab tak har calculation ki shape same rahi hai: har value ko uski quantity se multiply karo, sab jodo, aur total amount se divide karo. Values ko $x_{1}, x_{2}, \dots, x_{n}$ kaho aur amounts ko $w_{1}, w_{2}, \dots, w_{n}$ . Tab weighted mean, jise $\overset{ˉ}{x}$ likhte hain, hai

Weighted mean

\frac{w_{1} x_{1} + \dots + w_{n} x_{n}}{w_{1} + \dots + w_{n}}

jahan

$x_{1}, \dots, x_{n}$: values: marks, concentrations, group means …
$w_{1}, \dots, w_{n}$: unke weights: har ek kitna hai, ya har ek kitna important hai

Definition

Weighted mean

Har value ko uske weight se multiply karo, products ko jodo, aur weights ke sum se divide karo. Ise weighted average bhi kehte hain.

Example90 aur 60 marks, weights 1 aur 2 ke saath: $(90 + 120) \div 3 = 70$ .

Weight (भार) ka kilograms mein mass hona zaroori nahi. Yeh koi bhi number hai jo batata hai ki ek value kitna count honi chahiye: group ka size, jug ka volume, ya koi test kitna important hai. Agar har weight same ho, to weighted-mean formula ordinary mean ban jaata hai.

InteractiveWeighted-mean balance

Fig. 10.3Koi value ya uska weight badlo aur balance point, yaani weighted mean, ko khisakte dekho. Dashed line simple mean hai.

Quick check

3 mein se 1

10% salt solution ke 300 mL ko 30% salt solution ke 100 mL ke saath mix kiya jaata hai. Mixture ka concentration kya hai?

20%25%15%40%

Answer dekho

Answer (C) 15%

Salt: 400 mL mein

30 + 30 = 60

mL, aur

60 \div 400 = 15 %

. 20% sirf barabar volumes ke liye sahi hota; 40% galti se do percentages ko jod deta hai.

Practice · 5 questions

Practice 10.1: answers ke saath

NumericalEasyNCERT Exercise Set 10.1, Q1
Coaching batch P mein 36 students hain jinka mean mock-test score 68 hai. Batch Q mein 24 students hain jinka mean 78 hai. Sabhi 60 students ka mean score nikaalo.
Answer dekho
Answer 72
$(36 \times 68 + 24 \times 78) \div 60 = (2448 + 1872) \div 60 = 4320 \div 60 = 72$ .
NumericalEasyNCERT Exercise Set 10.1, Q2
Ek gardener teen potting mixes ki barabar amounts mix karta hai. Unmein compost ka fraction $\frac{1}{6}$ , $\frac{1}{4}$ aur $\frac{1}{12}$ hai. Final mix ka kitna fraction compost hai?
Answer dekho
Answer 0.1667
Amounts barabar hain, isliye plain mean kaam karta hai: $(\frac{2}{12} + \frac{3}{12} + \frac{1}{12}) \div 3 = \frac{6}{12} \div 3 = \frac{1}{6}$ .
3
NumericalMediumNCERT Exercise Set 10.1, Q3
(Śrīdharācārya ki style mein, lagbhag 750 CE.) Ek sunar 15 varṇa ke 4 units gold, 12 varṇa ke 10 units aur 13 varṇa ke 6 units saath pighlaata hai. Pure gold 16 varṇa ka hota hai.
1. Pighle hue gold ki purity kitni hai, varṇa mein?
  varṇa
  Answer dekho
  Answer 12.9 varṇa
  $(4 \times 15 + 10 \times 12 + 6 \times 13) \div 20 = 258 \div 20 = 12.9$ varṇa.
2. Aaj purity karats mein naapi jaati hai, aur pure gold 24 karat ka hota hai. Apne answer ko karats mein badlo.
  karat
  Answer dekho
  Answer 19.35 karat
  16 varṇa = 24 karat, isliye $\frac{24}{16} = 1.5$ se multiply karo: $12.9 \times 1.5 = 19.35$ karat.
4
Short answerMediumNCERT Exercise Set 10.1, Q4
Ek gaon ki dairy ne February 2026 mein roz mean 42 L doodh becha, March mein roz 38 L aur April mein roz 35 L.
1. Teen mahinon mein per day bechi gayi mean amount ke liye ek expression likho.
  Solution dikhao
  February 2026 mein 28 din hain, March mein 31 aur April mein 30. Har monthly mean ko uske dinon ki ginti se weight karo:
  $\overset{ˉ}{x} = (42 \times 28 + 38 \times 31 + 35 \times 30) \div (28 + 31 + 30)$
2. Ise ek decimal place tak evaluate karo.
  L
  Answer dekho
  Answer 38.2 L
  $(1176 + 1178 + 1050) \div 89 = 3404 \div 89 \approx 38.2$ L per day.
5
NumericalMediumNCERT Exercise Set 10.1, Q5
Ek mithai ki dukaan sugar syrup ke teen pans ko mix karti hai.
1. 20% sugar wala 1 L, 30% wala 2 L aur 40% wala 3 L. Mixture ka concentration nikaalo.
  %
  Answer dekho
  Answer 33.33 %
  Sugar: 6 L mein $0.2 + 0.6 + 1.2 = 2$ L, isliye $2 \div 6 \approx 33.3 %$ .
2. Ab volumes badal diye gaye: 20% wala 3 L, 30% wala 2 L aur 40% wala 1 L.
  %
  Answer dekho
  Answer 26.67 %
  Sugar: 6 L mein $0.6 + 0.6 + 0.4 = 1.6$ L, isliye $1.6 \div 6 \approx 26.7 %$ .
3. Wahi teen concentrations use hone ke baawajood answers alag kyun hain?
  Solution dikhao
  (a) mein zyaadatar syrup 40% wala hai, isliye mixture upar khinchta hai; (b) mein zyaadatar 20% wala hai, isliye neeche khinchta hai. Sirf barabar volumes se simple mean, 30%, milta.

10.1.3 Apne Weights Chunna

Ab tak weights sizes the: members, litres, days. Weights importance bhi dikha sakte hain. School tay kar sakta hai ki final exam class test se zyada count hona chahiye, kyunki usmein poora saal cover hota hai.

Example 10.6NCERT Example 6 · p. 14

Alag-alag importance wale marks

Ananya ne Science practicals mein 90%, periodic tests mein 70% aur final exam mein 62% score kiya. School inhe $1 : 3 : 6$ ke ratio mein combine karta hai. Uska final percentage kya hai?

Pehle khud try karo. Phir solution ek-ek step karke kholo.

2/21 : 3 : 6 ke ratio mein copies: das tiles ka mean 67.2 hai.

Fig. 10.4Weights ka ratio har value ki copies banane jaisa kaam karta hai.

Das tile spaces ki ek row mein Ananya ke scores hain: practical ka 90, tests ka 70 aur final exam ka 62. Pehle step mein har ek ki ek tile hai aur baaki halki outlines hain; neeche number line par ek pointer plain mean, 74, dikhata hai. Doosre step mein outlines bhar jaati hain: ek 90, teen 70 aur chhe 62, jin par times 1, times 3 aur times 6 likha hai, aur pointer left mein khisak kar 67.2 par aa jaata hai, jo sabhi das tiles ka mean hai.

Practice · 1 question

Practice 10.2: answers ke saath

1
Quick answerMediumNCERT Exercise Set 10.2, Q1
Karan ke Sanskrit marks tests mein 25 mein se 18, project mein 40 mein se 36 aur final exam mein 80 mein se 54 hain. Inhe $2 : 3 : 5$ ke ratio mein combine kiya jaata hai. Neeche diye gaye kaunse expressions uska final score percentage mein dete hain? Jo bhi sahi hon, sab chuno.
- (i) $(72 \times 2 + 90 \times 3 + 67.5 \times 5) \div 10$
- (ii) $(18 \times 2 + 36 \times 3 + 54 \times 5) \div 10$
- (iii) $(\frac{18}{25} \times 2 + \frac{36}{40} \times 3 + \frac{54}{80} \times 5) \div 10 \times 100$
- (iv) $(\frac{18}{100} \times 2 + \frac{36}{100} \times 3 + \frac{54}{100} \times 5) \div 10 \times 100$
(i)(ii)(iii)(iv)
Saare sahi options chuno
Answer dekho
Answer
(A) (i)
(C) (iii)
Har mark ko pehle percentage (i) ya apne total ka fraction (iii) banna hoga; dono se 75.15% aata hai. Expression (ii) 25, 40 aur 80 mein se raw marks ko mila deta hai, aur (iv) har mark ko aise maanta hai jaise woh 100 mein se ho. Dono se 41.4 aata hai, jo uska score nahi hai.

Group averages, mixtures, marks aur ratings, sab ek hi tool tak le jaate hain. Jab bhi kuch parts ko doosron se zyada say milna chahiye, unhe bade weights do.

Quick check

3 mein se 1

Agar har weight barabar ho, to weighted mean ordinary mean ke barabar hota hai.

SahiGalat

Answer dekho

Answer Sahi

Jab har weight

w

ke barabar ho, to upar

w (x_{1} + \dots + x_{n})

aata hai aur neeche

n w

w

cancel ho jaata hai, aur ordinary mean bachta hai.

Practice · 8 questions

Practice 10.3: answers ke saath

NumericalEasyNCERT Exercise Set 10.3, Q1
May mein ek juice stall ki sales juices se ₹12,000 thi, 25% profit ke saath, aur snacks se ₹3,000, 40% profit ke saath. Total sales ke percentage ke roop mein profit kitna tha?
%
Answer dekho
Answer 28 %
Profit $= 0.25 \times 12000 + 0.40 \times 3000 = 3000 + 1200 = 4200$ rupees, 15,000 rupees ki sales mein se, yaani 28%. Yeh 25% aur 40% ka weighted mean hai, jismein weights sales hain.
NumericalEasyNCERT Exercise Set 10.3, Q2
Ek delivery rider ne 24 dinon mein roz average 38.5 deliveries ki. 25th din usne 76 deliveries ki. Pehle guess karo: uska mean bahut badhega ya thoda sa? Phir 25 dinon ka uska mean nikaalo.
Answer dekho
Answer 40
24 dinon ka total $= 38.5 \times 24 = 924$ . Naya mean $= (924 + 76) \div 25 = 1000 \div 25 = 40$ . 25 dinon mein ek bada din mean ko bas thoda sa hilaata hai.
Quick answerMediumNCERT Exercise Set 10.3, Q3
6% salt wale solution ke 800 mL ko 9% sugar wale solution ke 400 mL ke saath mix kiya jaata hai. Mixed liquid mein salt ka percentage aur sugar ka percentage nikaalo.
salt 6%, sugar 9%salt 7.5%, sugar 7.5%salt 4%, sugar 3%salt 5%, sugar 4.5%
Answer dekho
Answer (C) salt 4%, sugar 3%
Salt: 1200 mL mein 48 mL yaani 4%. Sugar: 1200 mL mein 36 mL yaani 3%. Yahan 6% aur 9% ka average lena bekaar hai: woh alag-alag substances naapte hain.
NumericalMediumNCERT Exercise Set 10.3, Q4
Rose sherbet ke 12 L can mein 15% sugar hai. Sugar ko uske abhi ke concentration ka do-tihai karne ke liye kitne litres paani daalna hoga?
L
Answer dekho
Answer 6 L
Target: $\frac{2}{3} \times 15 % = 10 %$ . Sugar $0.15 \times 12 = 1.8$ L hi rahti hai, isliye $1.8 \div (12 + w) = 0.10$ . Tab $12 + w = 18$ aur $w = 6$ L.
5
Short answerMediumNCERT Exercise Set 10.3, Q5
Ek sports trial mein sprint, long jump aur endurance run ko $2 : 3 : 5$ ke ratio mein combine kiya jaata hai. Meera ne 80, 70 aur 60 score kiye. Nikhil ne 60, 70 aur 80 score kiye.
1. Calculate kiye bina batao, kiska final score zyada hai? Kyun?
  Solution dikhao
  Nikhil ka. Dono ke teen scores same hain, lekin Nikhil ka best score, 80, endurance run mein hai, jiska weight sabse bada hai.
2. Dono final scores nikaalo.
  Solution dikhao
  AnswerMeera 67, Nikhil 73.
  Meera: $(160 + 210 + 300) \div 10 = 67$ .
  Nikhil: $(120 + 210 + 400) \div 10 = 73$ .
NumericalHardNCERT Exercise Set 10.3, Q6
Ek homestay ke baarah guests ne teen cheezon ko 1 se 5 stars tak rate kiya. Table dikhata hai ki kitne guests ne har rating di. Owner cleanliness : food : location ko $5 : 3 : 2$ ke ratio mein weight karta hai. Overall rating nikaalo.
Table 10.2Har rating dene wale guests ki ginti
5★ 4★ 3★ 2★ 1★
Cleanliness 6 4 2 0 0
Food 3 5 3 1 0
Location 2 2 4 3 1
stars
Answer dekho
Answer 3.93 stars
Stars ke row totals: cleanliness $30 + 16 + 6 = 52$ , food $15 + 20 + 9 + 2 = 46$ , location $10 + 8 + 12 + 6 + 1 = 37$ , har ek 12 guests se. Overall $= (5 \times 52 + 3 \times 46 + 2 \times 37) \div (12 \times 10) = 472 \div 120 \approx 3.93$ stars.
7
NumericalHardNCERT Exercise Set 10.3, Q7
Ek sanctuary mein 50 spotted deer hain. Stags (males) ka mean weight 52 kg hai, does (females) ka 40 kg, aur poore herd ka 44.8 kg.
1. Calculate kiye bina batao, stags zyada hain ya does? Samjhao.
  Solution dikhao
  Does zyada hain. Herd ka mean, 44.8 kg, stags ke mean (52 kg) se zyada does ke mean (40 kg) ke paas hai; barabar ginti hoti to midpoint, 46 kg, aata.
2. Kitne stags hain?
  Answer dekho
  Answer 20
  $52 s + 40 (50 - s) = 44.8 \times 50 = 2240$ , isliye $12 s + 2000 = 2240$ aur $s = 20$ .
3. 46 kg ki ek doe herd mein shaamil hoti hai. Does ka naya mean weight kya hai, do decimal places tak?
  kg
  Answer dekho
  Answer 40.19 kg
  30 does hain jinka total $30 \times 40 = 1200$ kg hai. Naya mean $= 1246 \div 31 \approx 40.19$ kg.
4. Iske bajaye, 58 kg aur 50 kg ke do stags doosri sanctuary mein chale jaate hain. Stags ka naya mean weight kya hai, do decimal places tak?
  kg
  Answer dekho
  Answer 51.78 kg
  Stags ka total $20 \times 52 = 1040$ kg tha. Ab $(1040 - 58 - 50) \div 18 = 932 \div 18 \approx 51.78$ kg: jo do gaye woh average se bhaari the, isliye mean ghat jaata hai.
8
NumericalHardNCERT Exercise Set 10.3, Q8
Ek achaar banane wali ke paas strong brine hai jismein 25% salt hai aur weak brine jismein 5% salt hai.
1. Agar woh 1 L strong brine ko 3 L weak brine ke saath mix kare, to salt concentration kya hoga?
  %
  Answer dekho
  Answer 10 %
  $(0.25 + 0.15) \div 4 = 0.10$ , yaani 10%.
2. Kya in do brines ka koi mixture 30% salt wala ho sakta hai? Kyun ya kyun nahi?
  Solution dikhao
  Nahi. Mixture ka concentration 25% aur 5% ka weighted mean hai, isliye use in dono ke beech hi hona hoga.
3. 10% salt paane ke liye use 2 L strong brine mein kitna weak brine milana chahiye?
  L
  Answer dekho
  Answer 6 L
  $0.5 + 0.05 w = 0.10 (2 + w)$ , isliye $0.3 = 0.05 w$ aur $w = 6$ L.
4. Saade paani mein 0% salt hota hai. 5% brine banane ke liye use 1 L strong brine mein kitna paani daalna chahiye?
  L
  Answer dekho
  Answer 4 L
  $0.25 \div (1 + w) = 0.05$ , isliye $1 + w = 5$ aur $w = 4$ L.

	5★	4★	3★	2★	1★
Cleanliness	6	4	2	0	0
Food	3	5	3	1	0
Location	2	2	4	3	1

10.2 Data Ko Visualise Aur Interpret Karna

Table exact hota hai par padhne mein dheema. Chart thodi si precision chhod kar speed deta hai, kyunki aankh digits se kahin zyada tezi se lengths compare karti hai. Lekin har chart kuch comparisons aasaan aur kuch mushkil bana deta hai. Asli skill hai apne sawaal ke hisaab se sahi chart chunna.

10.2.1 Columns Ko Stack Karna

Ek residents' association ne teen households se kaha ki woh naapein ki ek aam din mein kitne litres paani use karte hain, purpose ke hisaab se.

Table 10.3Teen households ka daily paani use (litres)

	Bathing	Toilet	Laundry	Kitchen	Garden
Desai	180	120	90	60	50
Joseph	120	100	60	100	20
Rao	70	60	50	45	25

Clustered bar chart har number ke liye ek bar banata hai aur related bars ko aas-paas khada karta hai, har bar 0 se shuru hota hai. Is table ko do tarah se cluster kiya ja sakta hai: purpose ke hisaab se, har use mein households ko compare karne ke liye, ya household ke hisaab se, ek ghar ke uses compare karne ke liye. Fig. 10.5 dono dikhata hai.

1/2Purpose ke hisaab se clusters: har use mein teen households compare karo.

Fig. 10.5Wahi pandrah numbers, do tarah se cluster kiye gaye.NCERT Fig. 10.1 · p. 19

Daily litres ka ek column chart, 0 se 200 tak. Pehle step mein paanch clusters hain, har purpose ka ek (bathing, toilet, laundry, kitchen, garden), aur har ek mein teen columns hain jin par Desai, Joseph aur Rao households ke liye D, J aur R likha hai. Doosre step mein wahi pandrah columns khisak kar teen clusters mein aa jaate hain, har household ka ek, aur ab har column par uske purpose ke liye B, T, L, K ya G likha hai aur uska colour purpose ke hisaab se hai.

Ab kuch mushkil sawaal try karo. Kaunsa household kul milakar sabse kam paani use karta hai? Joseph household ek din mein lagbhag kitne litres use karta hai? Desai household ka lagbhag kitna percentage paani bathing mein jaata hai? Koi bhi cluster chart inka jaldi answer nahi deta, kyunki koi bhi household ka total nahi dikhata.

Iska hal hai har household ke bars ko ek ke baad ek jod dena. Tab ek lamba bar total dikhata hai, aur uske tukde parts dikhate hain. Yeh hai stacked bar chart.

2/2Ek ke baad ek jude paanch bars household ke total ka ek bar bana dete hain.

Fig. 10.6Household ke bars ko ek ke baad ek jodne se uska total dikhta hai.NCERT Fig. 10.2 · p. 20

Ek horizontal chart jismein 0 se 500 tak litre scale hai. Pehle step mein har household (Desai, Joseph, Rao) ke paanch patle bars hain, har purpose ka ek, sab 0 se shuru. Doosre step mein patle bars khisak kar ek ke baad ek judte hain aur mote hokar har household ka ek stacked bar ban jaate hain, bathing, toilet, laundry, kitchen, garden ke order mein, jin par B, T, L, K aur G likha hai, aur ends par totals 500, 400 aur 250 litres hain.

Definition

Stacked bar chart

Aisa bar chart jismein har whole ke parts ek ke baad ek jude hote hain, hamesha same order mein, taaki poora bar total dikhaye.

ExampleHar household ka ek bar, jo bathing, toilet, laundry, kitchen aur garden ke paani mein bata hai.

Ab totals saaf dikhte hain: Desai 500 L, Joseph 400 L, Rao 250 L. Desai household ke 500 L mein se bathing 180 L leta hai, yaani $\frac{180}{500} = 36 %$ . Iski keemat yeh hai ki har bar ka sirf pehla tukda 0 se shuru hota hai. Joseph household ka kitchen use padhne ke liye tumhe uska tukda jahan khatam hota hai (380 L) usmein se jahan shuru hota hai (280 L) woh ghatana hoga. Alag-alag jagah se shuru hone wale tukdon ko compare karna common base wale columns se zyada mushkil hai.

Short mein: agar tumhara sawaal wholes ke baare mein hai, to bars stack karo; agar alag-alag parts ke baare mein hai, to unhe cluster karo.

Quick check

3 mein se 1

Fig. 10.6 mein Rao household ka laundry tukda 130 L se 180 L tak jaata hai. Yeh kitne litres paani dikhata hai?

Answer dekho

Answer 50 L

End minus start padho:

180 - 130 = 50

Fig. 10.7Practice 10.4, Question 1 ke liye: ek town mein registered vehicles (thousands mein).

2010, 2015, 2020 aur 2025 ke liye chaar stacked columns, 0 se 70 thousand vehicles ke scale par. Har column neeche se two-wheelers, cars, autos, aur buses aur trucks mein bata hai. Columns ke upar likhe totals 28, 38, 50 aur 63 thousand hain. Two-wheelers aur cars bahut badhte hain; autos aur buses mushkil se badalte hain.

Practice · 2 questions

Practice 10.4: answers ke saath

1
Short answerEasyNCERT Exercise Set 10.4, Q1
Fig. 10.7 ka stacked column chart dikhata hai ki ek town mein type ke hisaab se kitne vehicles (thousands mein) register hue.
1. 2025 column ke upar likha number 63 kya dikhata hai?
  Solution dikhao
  2025 mein register hue chaaron types ke vehicles ki total ginti: lagbhag 63 thousand.
2. 2020 mein lagbhag kitni cars register hui, thousands mein?
  thousand
  Answer dekho
  Answer 14 thousand
  2020 column ka car wala tukda 30 se 44 thousand tak jaata hai, isliye lagbhag $44 - 30 = 14$ thousand.
3. 2010 aur 2025 ke beech kaun se types ke vehicles sabse kam badle?
  Solution dikhao
  Autos (lagbhag 3 → 5 thousand) aur buses aur trucks (lagbhag 1 → 2 thousand). Two-wheelers lagbhag 18 thousand badhe aur cars lagbhag 14 thousand.
2
DiagramMediumNCERT Exercise Set 10.4, Q2
Ek state ke athletes ne national school games mein yeh medals jeete.
Table 10.4Jeete gaye medals
Gold Silver Bronze
Boys 24 31 35
Girls 28 22 30
1. Ek stacked bar chart sketch karo jismein ek bar boys ka aur ek girls ka ho, medal type ke hisaab se bata hua.
  Solution dikhao
  Boys: total 90, tukde gold 0–24, silver 24–55, bronze 55–90. Girls: total 80, tukde gold 0–28, silver 28–50, bronze 50–80. Dono bars ke liye 0 se 100 tak ek hi scale use karo.
2. Ek doosra stacked bar chart sketch karo jismein har medal type ka ek bar ho, boys aur girls mein bata hua.
  Solution dikhao
  Gold: total 52 (boys 0–24, girls 24–52). Silver: total 53 (boys 0–31, girls 31–53). Bronze: total 65 (boys 0–35, girls 35–65).
3. Kaunsa chart ek nazar mein dikhata hai ki kul milakar kisne zyada medals jeete? Kaunsa sabse common medal type dikhata hai?
  Solution dikhao
  Chart (a): boys ka bar lamba hai (90 vs 80). Chart (b): bronze ka bar sabse lamba hai (65).

	Gold	Silver	Bronze
Boys	24	31	35
Girls	28	22	30

10.2.2 Pie Chart Ka Ek Alternative

Ab ek alag tarah ka sawaal: kaunsa household apne paani ka sabse chhota share kitchen mein use karta hai? Stacked chart litres dikhata hai, shares nahi. Desai household Rao household (45 L) se zyada kitchen ka paani (60 L) use karta hai, lekin uska total do guna bada hai, isliye uska kitchen share phir bhi chhota ho sakta hai.

Class 8 mein tum pie chart se mile the, jo har part ko poore circle ke ek slice ke roop mein dikhata hai. Fig. 10.8 har household ka ek pie banata hai.

2/2Kitchen shares: Desai 12%, Joseph 25%, Rao 18%.

Fig. 10.8Har household ka ek pie: har slice us household ke paani ka ek share hai.NCERT Fig. 10.3 · p. 22

Desai, Joseph aur Rao households ke liye teen pie charts aas-paas, har ek bathing, toilet, laundry, kitchen aur garden slices mein same order mein bata hai, upar se shuru hokar clockwise. Doosre step mein kitchen slice ke alawa har slice fade ho jaata hai, aur pies ke neeche labels kitchen shares dete hain: Desai ka 12%, Joseph ka 25% aur Rao ka 18%.

Pies shares to dikhate hain. Lekin teen kitchen slices ko rank karke dekho: woh alag-alag circles mein alag-alag angles par hain, aur hamari aankhein angles ka andaaza theek se nahi laga paati. Lengths ka andaaza hum kahin behtar lagaate hain. To har household ke stacked bar ko same length tak kheencho; tab har tukde ki length poore ka uska share hoti hai.

2/2Har bar 100% tak khincha hua: ab tukde percentages dikhate hain.

Fig. 10.9Har bar ko same length tak kheencho aur har tukda ek share ban jaata hai.NCERT Fig. 10.4 · p. 23

Desai, Joseph aur Rao households ke liye teen horizontal stacked bars. Pehle step mein woh 0 se 500 ke scale par litres mein bane hain, isliye unki lengths alag hain (500, 400 aur 250 litres). Doosre step mein har bar same poori length tak khinch jaata hai aur scale 0% se 100% ho jaata hai; ab tukde shares dikhate hain, aur kitchen tukdon par 12%, 25% aur 18% likha hai.

Definition

100% stacked bar chart

Aisa stacked bar chart jismein har bar ki length same hoti hai, jo apne total ke 100% ko dikhata hai, isliye har tukda amount ke bajaye percentage share dikhata hai.

ExampleDesai household: bathing 36%, toilet 24%, laundry 18%, kitchen 12%, garden 10%.

100% stacked bar chart ek strip mein khole gaye pie chart jaisa hai: jahan pie 360° ko baantta hai, wahan bar ek fixed length ko baantta hai. Ab hamare sawaal ka answer dikh raha hai. Desai household ka kitchen tukda sabse chhota (12%) hai, jabki woh Rao household (18%) se zyada kitchen litres use karta hai.

Phir bhi kuch comparisons mushkil rehte hain. Desai aur Rao ke toilet tukde dono theek 24% hain, lekin ek 36% se shuru hota hai aur doosra 28% se. Alag-alag positions par barabar length ke tukdon ko ek nazar mein milana mushkil hai.

Amounts se shares tak

Amounts se 100% stacked bar chart banane ke liye, har amount ko uske apne bar ke total ke percentage mein badlo.

Example 10.7NCERT Example 7 · p. 23

Ek family ka safar, tab aur ab

Table dikhata hai ki Sharma family ne 2016 aur 2026 mein ek mahine mein har mode se kitne kilometres travel kiye. Ek 100% stacked bar chart banao.

Table 10.5Ek mahine mein travel kiye gaye kilometres

	Bus	Metro	Auto	Two-wheeler
2016	450	50	150	350
2026	250	500	150	350

Pehle khud try karo. Phir solution ek-ek step karke kholo.

2/2Shares: two-wheeler ka tukda 35% se ghatkar 28% ho jaata hai.

Fig. 10.10Jab har bar 100% tak kheencha jaata hai to kilometres shares ban jaate hain.NCERT p. 24

2016 aur 2026 ke liye do horizontal stacked bars, bus, metro, auto aur two-wheeler mein bate hue. Pehle step mein woh 0 se 1250 ke scale par kilometres mein bane hain, 2026 ka bar lamba hai (1250 km vs 1000 km) aur tukdon mein kilometres likhe hain. Doosre step mein dono bars same length tak khinch jaate hain, scale 0% se 100% ho jaata hai, aur tukde percentages dikhate hain: 2016 ke liye 45, 15 aur 35 (metro 5% label ke liye bahut patla hai) aur 2026 ke liye 20, 40, 12 aur 28.

Two-wheeler ko dhyaan se dekho. Family ne dono saal usse 350 km chalaya, phir bhi uska share 35% se ghatkar 28% ho gaya. Amount same rehte hue bhi share ghat sakta hai, bas isliye ki total badh gaya. Metro travel, jo 2016 mein lagbhag kuch nahi tha, 2026 tak sabse bada tukda ban gaya.

100% chart kya bata sakta hai aur kya nahi

Kyunki 100% chart mein har bar ki length same hoti hai, chart totals chhupa leta hai. Isse yeh limit ho jaata hai ki tum usse kya conclude kar sakte ho.

Example 10.8NCERT Example 8 · p. 24

Mele mein do stalls

Asha aur Vikram ek hi mele mein snack stalls chalate hain. Ek 100% stacked bar chart dikhata hai ki har stall ki plates mein pani puri, chaat aur kulfi ka kitna share tha: Asha 45%, 35% aur 20%; Vikram 30%, 30% aur 40%. Inmein se kaunsi baatein sirf chart se nikalti hain?

(i) Vikram ke stall par kulfi ki plates chaat se zyada biki.
(ii) Asha ne Vikram se zyada pani puri ki plates bechi.
(iii) Dono stalls par sabse zyada bikne wala snack alag hai.
(iv) Vikram ke stall ne kul milakar zyada plates bechi.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

3/3Agar Asha ne 120 aur Vikram ne 300 bechi, to Vikram ne zyada pani puri bechi: 90 vs 54.

Fig. 10.11Same shares bahut alag-alag ginti ki plates se aa sakte hain.NCERT p. 25

Asha aur Vikram ke stalls ke liye do horizontal bars, pani puri, chaat aur kulfi mein bate hue. Pehle step mein dono bars ki length same hai aur woh shares dikhate hain: Asha ke 45, 35 aur 20 percent aur Vikram ke 30, 30 aur 40 percent. Doosre step mein bars ek plate scale par chhote ho jaate hain jahan dono stalls 200 plates bechte hain: Asha 90, 70 aur 40, Vikram 60, 60 aur 80. Teesre step mein Asha 120 plates (54, 42, 24) aur Vikram 300 (90, 90, 120) bechta hai, isliye ab Vikram ne zyada pani puri bechi.

Kiska typical din?

Log apne 24 ghante kaise bitaate hain, aise charts survey planners mein popular hain. India ka National Statistical Office ek Time Use Survey chalata hai jismein log poore din mein jo kiya woh record karte hain. Hamara survey ek school community ke 120 logon ka banaya hua survey hai: 30 students, 60 working-age adults aur 30 elders.

2/2Students, adults aur elders: teen bahut alag din.

Fig. 10.12Ek average din, phir wahi log groups mein bate hue (banaya hua survey).NCERT Fig. 10.5, 10.6 · p. 26

24 ghanton ke horizontal bars, jin par 0% se 100% bhi likha hai, sleep, personal care aur meals, learning, paid work, home aur care work, aur leisure aur travel mein bate hue, aur bade tukdon mein ghante likhe hain. Pehle step mein sirf "Sab log" wala bar dikhta hai: 8, 2.5, 2, 4, 2.5 aur 5 ghante. Doosre step mein teen aur bars aate hain. Students: 9 ghante sleep, 2 personal care, 8 learning, koi paid work nahi, 1 home work aur 4 leisure. Adults: 7, 2.5, koi learning nahi, 7.5 paid work, 3.5 aur 3.5. Elders: 9, 3, koi learning nahi, 1, 2 aur 9.

Example 10.9NCERT Example 9 · p. 26

Ek din, do tarah se padha gaya

Fig. 10.12 use karo. (a) Kya sach mein kisi ka din "Sab log" wale bar jaisa dikhta hai? (b) Dikhao ki "Sab log" ka sleep time teen groups ke sleep times ka weighted mean hai.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

120 logon ko groups mein baantne se humein single bar se kahin zyada pata chala. Doosre splits, jaise gaon aur sheher, income, ya ek decade vs doosra, aur bhi zyada bata sakte hain.

Practice · 1 question

Practice 10.5: answers ke saath

1
ActivityMediumNCERT Exercise Set 10.5, Q1
Ek fitness band ne record kiya ki kuch log har weekday mein mean kitne ghante lete hue, baithe hue, aur pairon par (khade, chalte ya kaam karte hue) bitaate the.
Table 10.6Har weekday ke ghante
Lete hue Baithe hue Pairon par
Gurpreet (kisaan) 7 3 14
Meenal (bank clerk) 7 12 5
Dadi (retired, 70) 9 8 7
Mystery person 7 14 3
Tum
Tumhari entries apne aap save hoti hain.
1. Ek typical school day ke liye apni row bharo. Tumhare teen numbers ka jod 24 hona chahiye.
  Solution dikhao
  Answers alag-alag honge. Ek typical student lagbhag 8.5 h lete hue, 9 h baithe hue (school, homework, khaana, travel) aur 6.5 h pairon par bita sakta hai.
2. Guess karo ki mystery person kis tarah ka kaam karta hoga. Apna guess kisi classmate ke guess se compare karo.
  Solution dikhao
  14 h baithe hue aur sirf 3 h pairon par, yeh desk work ya driving ki taraf ishaara karta hai: shayad software developer, call-centre worker ya long-distance bus driver.
3. Har vyakti ke ghanton ko din ke percentages mein badlo aur ek 100% stacked bar chart sketch karo, har vyakti ka ek bar.
  Solution dikhao
  Har ghanta din ka $\frac{100}{24} \approx 4.17 %$ hai. Gurpreet: 29.2%, 12.5%, 58.3%. Meenal: 29.2%, 50%, 20.8%. Dadi: 37.5%, 33.3%, 29.2%. Mystery person: 29.2%, 58.3%, 12.5%.
4. Agar Sundays bhi shaamil hote to kya Meenal ka bar badalta? Kya kisi aur bank clerk ka bar bilkul uske jaisa dikhta?
  Solution dikhao
  Shayad haan: chhutti ke din woh kam baith sakti hai aur zyada chal-phir sakti hai, isliye 7-day average pairon par bitaaye time ki taraf khisak jaata. Doosre clerk ka bar milta-julta hoga par bilkul same nahi, kyunki ek hi job ke log bhi alag-alag din jeete hain.

	Lete hue	Baithe hue	Pairon par
Gurpreet (kisaan)	7	3	14
Meenal (bank clerk)	7	12	5
Dadi (retired, 70)	9	8	7
Mystery person	7	14	3
Tum

Chapter ke end ki exercises

Fig. 10.13Question 7 ke liye: saal ke hisaab se library members.

2016, 2018, 2020, 2022, 2024 aur 2026 ke liye chhe stacked columns, 0 se 3500 members ke scale par. Har column ka neeche wala tukda adult members hai, jo dheere-dheere 900 se 1400 tak badhta hai; upar wala tukda student members hai, jo tezi se 300 se 2100 tak badhta hai. Column totals lagbhag 1200, 1450, 1700, 2200, 2800 aur 3500 hain.

1/2Har eatery ki ratings 100% ke shares ke roop mein.

Fig. 10.14Question 9 ke liye: ratings pehle shares ke roop mein, phir counts ke roop mein.

Eateries P, Q aur R ke liye teen horizontal bars, 5-star, 4-star, 3-star, 2-star aur 1-star ratings mein bate hue. Pehle step mein saare bars same length ke hain aur shares dikhate hain, 5-star shares par P ke liye 60%, Q ke liye 45% aur R ke liye 33% likha hai; R ka 1-star tukda bada hai. Doosre step mein bars 0 se 600 ke count scale par chhote ho jaate hain: P ki sirf 20 ratings hain aur woh ek patli si line ban jaata hai, Q ki 600 hain aur woh poora scale bhar deta hai, aur R ki 150 hain.

Practice · 16 questions

Practice questions answers ke saath (NCERT “Chapter ke end ki exercises”)

1
NumericalEasyNCERT End-of-chapter Q1
Cricket mein run rate per over banaye gaye runs ka mean hota hai.
1. 10 overs ke baad ek team ka run rate 7.2 hai. 11th over mein woh 15 runs banati hai. Ab run rate kya hai, do decimal places tak?
  Answer dekho
  Answer 7.91
  Ab tak ke runs $= 7.2 \times 10 = 72$ . Naya run rate $= (72 + 15) \div 11 = 87 \div 11 \approx 7.91$ .
2. 50-over match mein 49 overs ke baad run rate 5.4 hai. Aakhri over mein 20 runs aate hain. Final run rate kya hai?
  Answer dekho
  Answer 5.692
  $(5.4 \times 49 + 20) \div 50 = 284.6 \div 50 = 5.692$ .
2
Short answerMediumNCERT End-of-chapter Q2
Chaar doston ne apne paas ki dukaanon par chawal ki keemat (₹ per kg) note ki, aur har price par basmati ke liye B ya non-basmati ke liye N likha.
- Priya: 62 (N), 140 (B), 58 (N)
- Rahul: 150 (B), 65 (N), 160 (B), 55 (N)
- Sana: 75 (N), 135 (B)
- Tenzin: 145 (B), 60 (N), 155 (B)
1. Har dost ne apne prices ka mean nikaala: $a_{P}$ , $a_{R}$ , $a_{S}$ aur $a_{T}$ . Sabhi 12 prices ke mean ke liye ek expression likho.
  Solution dikhao
  Har dost ke mean ko unke note kiye gaye prices ki ginti (3, 4, 2 aur 3) se weight karo:
  $\overset{ˉ}{x} = (3 a_{P} + 4 a_{R} + 2 a_{S} + 3 a_{T}) \div 12$
2. Iske bajaye, woh prices ko type ke hisaab se group karte hain, jinke means $a_{B}$ aur $a_{N}$ hain. Ab expression likho.
  Solution dikhao
  Har type ke 6 prices hain, isliye $\overset{ˉ}{x} = (6 a_{B} + 6 a_{N}) \div 12$ . Kyunki groups ka size barabar hai, yeh bas $(a_{B} + a_{N}) \div 2$ hai.
3. Dono tareekon se mean price calculate karo. Kya dono tareekon ka answer same aata hai?
  rupees per kg
  Answer dekho
  Answer 105 rupees per kg
  Dost ke hisaab se: $(260 + 430 + 210 + 360) \div 12 = 1260 \div 12 = 105$ . Type ke hisaab se: $a_{B} = 147.5$ aur $a_{N} = 62.5$ , isliye $(885 + 375) \div 12 = 105$ . Dono tareeke sabhi 12 prices ke total ko 12 se divide karte hain.
4. Tenzin bas chaaron doston ke means ka average le leta hai. Use kya milta hai, aur yeh galat kyun hai?
  rupees per kg
  Answer dekho
  Answer 104.79 rupees per kg
  $(86.67 + 107.5 + 105 + 120) \div 4 \approx 104.79$ . Yeh galat hai kyunki yeh Sana ke 2 prices ko Rahul ke 4 prices jitna hi say deta hai.
3
NumericalMediumNCERT End-of-chapter Q3
Ek grocer ke paas 40 kg dal hai jiski use average ₹120 per kg cost padi.
1. Woh ₹100 per kg par 60 kg aur khareedta hai. Uski nayi average cost per kilogram kya hai?
  rupees per kg
  Answer dekho
  Answer 108 rupees per kg
  $(40 \times 120 + 60 \times 100) \div 100 = 10800 \div 100 = 108$ rupees per kg.
2. Ek aur grocer ke paas ₹120 per kg average par 20 kg hai. Woh ₹100 per kg par kuch dal khareedti hai aur uska average ghatkar ₹104 per kg ho jaata hai. Usne kitne kilograms khareede?
  kg
  Answer dekho
  Answer 80 kg
  $(2400 + 100 k) \div (20 + k) = 104$ , isliye $2400 + 100 k = 2080 + 104 k$ . Tab $4 k = 320$ aur $k = 80$ kg.
NumericalMediumNCERT End-of-chapter Q4
(Brahmagupta par Pṛthūdakasvāmī ki commentary ke ek problem par based, lagbhag 864 CE.) Hasta length ki ek purani unit hai, lagbhag ek forearm jitni. 24 hastas lambi ek khai chaar hisson mein khodi jaati hai jinki lengths 4, 6, 6 aur 8 hastas hain, aur depths 9, 7, 4 aur 3 hastas. Khai ki mean depth nikaalo.
hastas
Answer dekho
Answer 5.25 hastas
$(9 \times 4 + 7 \times 6 + 4 \times 6 + 3 \times 8) \div 24 = 126 \div 24 = 5.25$ hastas.
5
NumericalMediumNCERT End-of-chapter Q5
Ek dukaandaar ke paas ₹30 per kg par khareede gaye 10 kg pyaaz hain. Neeche di gayi har khareed ke liye, pehle number line par mark karo ki tumhe uski nayi average cost kahan aane ki ummeed hai. Phir use calculate karo.
1. Woh ₹50 per kg par 10 kg khareedta hai.
  rupees per kg
  Answer dekho
  Answer 40 rupees per kg
  Barabar amounts, isliye beech mein: $(300 + 500) \div 20 = 40$ rupees per kg.
2. Woh ₹50 per kg par 30 kg khareedta hai.
  rupees per kg
  Answer dekho
  Answer 45 rupees per kg
  Teen guna naya stock average ko teen-chauthai raste tak kheench leta hai: $(300 + 1500) \div 40 = 45$ .
3. Woh ₹20 per kg par 40 kg khareedta hai.
  rupees per kg
  Answer dekho
  Answer 22 rupees per kg
  $(300 + 800) \div 50 = 22$ , 20 ke paas kyunki ab zyaadatar stock saste wala hai.
4. Woh ₹30 per kg par 5 kg khareedta hai.
  rupees per kg
  Answer dekho
  Answer 30 rupees per kg
  Naye pyaaz ki keemat purane jitni hi hai, isliye average nahi hilta.
ProofMediumNCERT End-of-chapter Q6
Kuch values ke weights $w_{1}, w_{2}, \dots, w_{n}$ hain. Agar har weight ko same number $k$ se multiply kiya jaaye, jaise sab teen guna kar diye jaayein, to weighted mean ka kya hota hai? Ek example try karo, phir apna answer prove karo.
Solution dikhao
AnswerKuch nahi badalta: sirf weights ka ratio matter karta hai.
Example: 40 aur 80, weights 1 aur 3 ke saath, $(40 + 240) \div 4 = 70$ dete hain. Teen guna weights 3 aur 9 se $(120 + 720) \div 12 = 70$ aata hai, wahi.
General roop mein upar ke har term mein factor $k$ hai, aur neeche ke har term mein bhi:
$\frac{k (w_{1} x_{1} + \dots)}{k (w_{1} + \dots)}$
$k$ cancel ho jaata hai, isliye weighted mean nahi badalta.
7
Short answerMediumNCERT End-of-chapter Q7
Fig. 10.13 ka chart har saal ek town library ke members dikhata hai, students aur adults mein bata hua.
1. 2016 mein lagbhag kitne members the? Kis saal tak yeh lagbhag do guna ho gaya tha?
  Solution dikhao
  2016 mein lagbhag 1200. Do guna lagbhag 2400 hai. Total 2022 mein 2200 aur 2024 mein 2800 tha, isliye yeh 2022 aur 2024 ke beech do guna hua.
2. 2026 mein lagbhag kitne members students the, aur yeh sabhi members ka kitna share tha?
  Solution dikhao
  Student tukda lagbhag 1400 se 3500 tak jaata hai, isliye lagbhag 2100 students, jo members ka $2100 \div 3500 = 60 %$ hai.
3. Student aur adult members ki ginti kaise badli? Unke shares kaise badle?
  Solution dikhao
  Dono numbers badhe: students lagbhag 300 se 2100 tak, adults lagbhag 900 se 1400 tak. Students kahin zyada tezi se badhe, isliye unka share 25% se 60% ho gaya, jabki adults ka share 75% se ghatkar 40% ho gaya, halaanki adults pehle se kahin zyada the.
8
Short answerMediumNCERT End-of-chapter Q8
Table paanch districts mein tap-water connection wale households ka percentage deta hai (banaya hua data).
Table 10.7District ke hisaab se tap-water connections
District Tap water wale
Amli 64%
Barod 81%
Chirai 75%
Dhanpur 92%
Eklera 58%
1. (a)
  Inmein se tum kya conclude kar sakte ho? Jo bhi sahi hon, sab chuno.
  Barod mein Amli se zyada households ke paas tap water hai.Chirai mein har 4 mein se lagbhag 3 households ke paas tap water hai.Dhanpur mein sabse zyada households ke paas tap water hai.Agar Dhanpur mein Barod se zyada households hain, to wahan tap water wale households bhi zyada hain.Agar Eklera mein Amli se zyada households hain, to wahan tap water wale households bhi zyada hain.
  Saare sahi options chuno
  Answer dekho
  Answer
  (B) Chirai mein har 4 mein se lagbhag 3 households ke paas tap water hai.
  (D) Agar Dhanpur mein Barod se zyada households hain, to wahan tap water wale households bhi zyada hain.
  75% yaani 4 mein se 3 (B). D mein Dhanpur ke paas zyada households bhi hain aur percentage bhi zyada hai, isliye wahan tap water wale zyada hone hi chahiye. A aur C ke liye households ki ginti chahiye. E galat ho sakta hai: agar Eklera mein 1000 households (580 tap water wale) hain aur Amli mein 950 (608), to Amli mein zyada hain.
2. Kya tum paanchon districts ko milakar tap water wale households ka percentage nikaal sakte ho? Agar nahi, to aur kya chahiye?
  Solution dikhao
  Sirf percentages se nahi. Tumhe har district mein households ki ginti chahiye: overall percentage paanch percentages ka weighted mean hai, jismein weights yahi numbers hain.
9
Short answerMediumNCERT End-of-chapter Q9
Ek food app teen eateries, P, Q aur R, ki star ratings dikhata hai. Fig. 10.14 unhe pehle shares (100% chart) ke roop mein aur phir ratings ki ginti (stacked chart) ke roop mein dikhata hai.
1. Sirf 100% chart dekh kar, tum kaunsi eatery chunoge? Kyun?
  Solution dikhao
  P: uski 60% ratings 5 stars hain, sabse bada share. Uski mean rating, $(60 + 16 + 6 + 2 + 1) \div 20 = 4.25$ , bhi sabse zyada hai.
2. Ab ratings ki ginti dekho. Kya tum apni choice badloge? Discuss karo.
  Solution dikhao
  Bahut log Q par switch kar lenge. P ke shares sirf 20 ratings par tike hain, isliye kuch naye reviews unhe bahut badal sakte hain. Q ka 4.1 ka mean 600 ratings par tika hai aur kahin zyada reliable hai. R ka mean kam hai, lagbhag 2.93, aur uski 40% ratings 1 star hain.
10
Short answerMediumNCERT End-of-chapter Q10
Teen riders ne ek three-stage cycle rally poori ki. Unke times hours : minutes mein hain.
Table 10.8Stage times
Hill stage Plain stage City stage
Rider 1 2:10 3:05 1:20
Rider 2 2:25 2:35 1:30
Rider 3 2:05 3:20 1:35
1. Kya tum in times ko totals dikhane wale bars (stacked) se banaoge ya barabar length wale bars (100% stacked) se? Reason do.
  Solution dikhao
  Stacked bar chart. Total time winner tay karta hai, aur sirf stacked chart totals (6 h 35 min, 6 h 30 min aur 7 h) dikhata hai. 100% chart har rider ka bar same length ka banata.
2. (b)
  Agar 100% stacked bar chart banaya jaaye, to inmein se kis sawaal ka answer sirf usse de sakte ho?
  Sabse pehle kaun pahuncha?Rider 1 ka lagbhag kitna fraction time hill stage par laga?City stage par sabse zyada time kisne liya?
  Saare sahi options chuno
  Answer dekho
  Answer
  (B) Rider 1 ka lagbhag kitna fraction time hill stage par laga?
  Sirf B ek bar ke andar ka share hai: 395 mein se 130 minutes, lagbhag ek-tihai. Baaki do ke liye asli times chahiye. Dhyaan do: Rider 2 ka city share sabse bada hai (390 mein se 90 minutes, 23%), lekin sabse zyada time Rider 3 ne liya (95 minutes).
11
Short answerMediumNCERT End-of-chapter Q11
Ek school ne teen classes ke students se poochha ki woh aam taur par school kaise jaate hain. Table percentages deta hai.
Table 10.9Students school kaise jaate hain (%)
Paidal Cycle School bus Car ya two-wheeler
Class 6 40 10 35 15
Class 9 25 35 25 15
Class 12 15 30 20 35
1. Is data ka ek 100% stacked bar chart sketch karo.
  Solution dikhao
  Barabar length ke teen bars banao. Tukdon ke beech ki boundaries Class 6 ke liye 40%, 50% aur 85% par mark karo; Class 9 ke liye 25%, 60% aur 85% par; aur Class 12 ke liye 15%, 45% aur 65% par.
2. Do cheezein likho jo tumne notice ki aur do cheezein jinke baare mein tum sochte ho.
  Solution dikhao
  Possible noticings: students ke bade hone ke saath paidal jaana lagaataar ghatta hai (40% se 15%); Class 6 aur Class 9 ke beech cycling achaanak badhti hai; Class 12 sabse zyada cars aur two-wheelers use karti hai. Possible sawaal: kya bade students door rehte hain? Kya chhote students ko cycle chalane ki permission hai? Har class mein kitne students hain? Chart yeh nahi bata sakta.
12
ActivityEasyNCERT End-of-chapter Q12
Individual project. Inmein se ek karo.
1. Ek hafte tak apna screen time track karo, study, entertainment aur logon se baat karne mein baant kar. Har din ka ek bar lekar stacked bar chart banao.
  Solution dikhao
  Har shaam ek chhota sa log rakho. Har bar mein tukdon ka same order rakho aur scale ko minutes mein label karo. Phir do observations likho, jaise kis din ka total sabse bada tha, aur study ya entertainment mein se kiska time zyada badla.
2. Apni family ke saath, teen mahinon tak apne ghar ka kharcha 4 se 6 categories (food, travel, electricity waghairah) mein record karo. Har mahine ka ek bar lekar 100% stacked bar chart banao.
  Solution dikhao
  Pehle categories par sehmat ho jao. Har mahine ke liye, har category ko us mahine ke total se divide karo aur 100 se multiply karo. Aakhir mein, jo bhi share badla use note karo, aur tay karo ki us category par kharcha badla ya total badla.
13
ActivityMediumNCERT End-of-chapter Q13
Group project (3–4 students). Apni school canteen, kisi park, bus route ya clinic ke liye ek rating scheme design karo.
1. Rate karne ke liye 4 se 6 aspects chuno aur unke weights par sehmat ho jao. Apne weights ko justify karo.
  Solution dikhao
  Canteen ke liye sample: taste 4, hygiene 5, price 3, speed of service 2. Hygiene ko sabse zyada weight milta hai kyunki unsafe khaana kisi bhi cheez se zyada matter karta hai.
2. Kam se kam 10 logon se har aspect ko 1 se 5 tak score karne ko kaho. Har vyakti ki overall rating nikaalo, phir inka mean.
  Solution dikhao
  Ek vyakti jo taste ko 4, hygiene ko 5, price ko 3 aur speed ko 2 deta hai, uski overall rating $(16 + 25 + 9 + 4) \div 14 = 54 \div 14 \approx 3.86$ hai. Group ka mean individual overall ratings ka plain mean hai.
3. Ek 100% stacked bar chart banao jo har aspect ke liye 5-star, 4-star … 1-star ratings ke shares dikhaye. Likho ki tumhari scheme kya achhe se pakadti hai aur kya miss karti hai.
  Solution dikhao
  Har aspect ka ek bar, star rating ke hisaab se bata hua. Scheme woh sab miss karti hai jise rate karne ka kisi ne socha hi nahi, aur uske weights fact ke bajaye group ki raay dikhate hain.
NumericalMediumNCERT End-of-chapter Q14
Whole-class project. Har student ghar par per person mean daily water use aur ghar ke logon ki ginti nikaalta hai. Class apne sabhi gharon mein per person mean water use kaise nikaal sakti hai? Is sample ke saath try karo: 4, 6, 3 aur 5 logon wale ghar jo per person per day 110, 90, 130 aur 100 L use karte hain.
L
Answer dekho
Answer 103.9 L
Har ghar ko uske logon ki ginti se weight karo: $(440 + 540 + 390 + 500) \div 18 = 1870 \div 18 \approx 103.9$ L. Chaar gharon ke means ka plain mean, 107.5 L, bahut zyada hai kyunki sabse zyada use wala chhota ghar bade ghar jitna hi count hota hai.
ProofHardNCERT End-of-chapter Q15
(Challenge) Agar har weight mein same number $c$ jod do, to weighted mean ka kya hota hai? 40 aur 80 ko weights 1 aur 3 ke saath try karo, aur phir weights 3 aur 5 ke saath. Algebra se samjhao.
Solution dikhao
AnswerMean values ke simple mean ki taraf khisakta hai (jab tak woh pehle se uske barabar na ho).
Weights 1 aur 3 ke saath: $(40 + 240) \div 4 = 70$ . Weights 3 aur 5 ke saath: $(120 + 400) \div 8 = 65$ . Mean 70 se 60 ki taraf khisak gaya, jo 40 aur 80 ka simple mean hai.
General roop mein, maan lo weights ka jod $W$ hai aur weighted mean $m$ hai, to weighted total $W m$ hai. Har weight mein $c$ jodne se upar $c (x_{1} + \dots + x_{n}) = c n \overset{ˉ}{x}$ judta hai, jahan $\overset{ˉ}{x}$ simple mean hai, aur neeche $c n$ :
$\frac{W m + c n \overset{ˉ}{x}}{W + c n}$
Yeh $m$ aur $\overset{ˉ}{x}$ ka weights $W$ aur $c n$ ke saath weighted mean hai, isliye yeh in dono ke beech hai. Har weight mein jodne se result simple mean ki taraf khinchta hai.
16
Quick answerHardNCERT End-of-chapter Q16
(Challenge) Pichhle saal ek town ke vehicles mein 50% two-wheelers, 30% cars aur 20% autos the. Is saal har type ki ginti badhi. Shares ke saath inmein se kya ho sakta tha? Jo bhi possible hon, sab chuno.
Teeno shares badhe.Teeno shares ghate.Teeno shares same rahe.Two-wheeler share ghata, car share badha aur auto share same raha.Two-wheeler aur car shares badhe aur auto share ghata.
Saare sahi options chuno
Answer dekho
Answer
(C) Teeno shares same rahe.
(D) Two-wheeler share ghata, car share badha aur auto share same raha.
(E) Two-wheeler aur car shares badhe aur auto share ghata.
Shares ka jod hamesha 100% hota hai, isliye woh sab ek saath badh ya ghat nahi sakte (A aur B impossible hain). C tab hota hai jab har type same percentage se badhe. 500, 300 aur 200 se: 510, 350 aur 215 tak badhne par 47.4%, 32.6% aur 20% aata hai (D); 600, 360 aur 201 tak badhne par 51.7%, 31.0% aur 17.3% aata hai (E).

District	Tap water wale
Amli	64%
Barod	81%
Chirai	75%
Dhanpur	92%
Eklera	58%

	Hill stage	Plain stage	City stage
Rider 1	2:10	3:05	1:20
Rider 2	2:25	2:35	1:30
Rider 3	2:05	3:20	1:35

	Paidal	Cycle	School bus	Car ya two-wheeler
Class 6	40	10	35	15
Class 9	25	35	25	15
Class 12	15	30	20	35

Recap

Summary aur key points

Group means ka mean true mean ke barabar sirf tab hota hai jab groups same size ke hon (Section 10.1.1).
Mean × number of values = total, isliye do groups $\frac{a n + b m}{n + m}$ ke roop mein combine hote hain ((10.1)).
Mixture ka concentration hai total substance ÷ total mixture. Yeh mix kiye gaye concentrations ke beech hota hai, badi amount ke zyada paas (Section 10.1.2).
Weighted mean $\frac{w_{1} x_{1} + \dots + w_{n} x_{n}}{w_{1} + \dots + w_{n}}$ hai (formula). Weights sizes, amounts ya importance ho sakte hain (Section 10.1.3).
Barabar weights ordinary mean dete hain, aur har weight ko same number se multiply karne se kuch nahi badalta.
Values add ya remove karne ke liye total dobara banao, use badlo, aur nayi ginti se divide karo.
Clustered bar charts alag-alag parts compare karte hain; cluster ki choice tay karti hai ki kaun se comparisons aasaan honge (Section 10.2.1).
Stacked bar charts totals aur unke parts dikhate hain, lekin har bar ka sirf pehla tukda 0 se shuru hota hai.
100% stacked bar charts, pie charts ki tarah, shares dikhate hain, amounts nahi (Section 10.2.2).
Stacked bar chart se tum hamesha 100% chart bana sakte ho; wapas jaane ke liye totals chahiye (100% chart kya bata sakta hai).
Ek overall average groups ke beech bade differences chhupa sakta hai (kiska typical din?).

Apna NCERT exercise question dhoondo

Hamare practice questions ncert.ai ke liye likhe gaye hain aur NCERT exercises ko follow karte hain. Apni textbook ke question se match karne wala question dhoondo; uska answer uske saath hai.

Tumhari NCERT textbook mein	Yahan practice question
NCERT Exercise Set 10.1, Q1	Practice 10.1 · Q1
NCERT Exercise Set 10.1, Q2	Practice 10.1 · Q2
NCERT Exercise Set 10.1, Q3	Practice 10.1 · Q3
NCERT Exercise Set 10.1, Q4	Practice 10.1 · Q4
NCERT Exercise Set 10.1, Q5	Practice 10.1 · Q5
NCERT Exercise Set 10.2, Q1	Practice 10.2 · Q1
NCERT Exercise Set 10.3, Q1	Practice 10.3 · Q1
NCERT Exercise Set 10.3, Q2	Practice 10.3 · Q2
NCERT Exercise Set 10.3, Q3	Practice 10.3 · Q3
NCERT Exercise Set 10.3, Q4	Practice 10.3 · Q4
NCERT Exercise Set 10.3, Q5	Practice 10.3 · Q5
NCERT Exercise Set 10.3, Q6	Practice 10.3 · Q6
NCERT Exercise Set 10.3, Q7	Practice 10.3 · Q7
NCERT Exercise Set 10.3, Q8	Practice 10.3 · Q8
NCERT Exercise Set 10.4, Q1	Practice 10.4 · Q1
NCERT Exercise Set 10.4, Q2	Practice 10.4 · Q2
NCERT Exercise Set 10.5, Q1	Practice 10.5 · Q1
NCERT End-of-chapter Q1	Chapter ke end ki exercises · Q1
NCERT End-of-chapter Q2	Chapter ke end ki exercises · Q2
NCERT End-of-chapter Q3	Chapter ke end ki exercises · Q3
NCERT End-of-chapter Q4	Chapter ke end ki exercises · Q4
NCERT End-of-chapter Q5	Chapter ke end ki exercises · Q5
NCERT End-of-chapter Q6	Chapter ke end ki exercises · Q6
NCERT End-of-chapter Q7	Chapter ke end ki exercises · Q7
NCERT End-of-chapter Q8	Chapter ke end ki exercises · Q8
NCERT End-of-chapter Q9	Chapter ke end ki exercises · Q9
NCERT End-of-chapter Q10	Chapter ke end ki exercises · Q10
NCERT End-of-chapter Q11	Chapter ke end ki exercises · Q11
NCERT End-of-chapter Q12	Chapter ke end ki exercises · Q12
NCERT End-of-chapter Q13	Chapter ke end ki exercises · Q13
NCERT End-of-chapter Q14	Chapter ke end ki exercises · Q14
NCERT End-of-chapter Q15	Chapter ke end ki exercises · Q15
NCERT End-of-chapter Q16	Chapter ke end ki exercises · Q16

Quick check: MCQs

10.1.1 Averages Ka Average3 questions
10.1.2 Mixtures3 questions
10.1.3 Apne Weights Chunna3 questions
Charts padho4 questions
10.2.1 Columns Ko Stack Karna3 questions
Kiska typical din?3 questions

Glossary · 9

Key terms (English + Hindi)

100% stacked bar chart: Aisa stacked bar chart jiske saare bars ki length same hoti hai, jo 100% dikhati hai, isliye har tukda ek share dikhata hai.Chapter mein pehli baar yahan aaya →

clustered bar chart: Aisa bar chart jo related bars ko groups (clusters) mein aas-paas khada karta hai, har bar 0 se shuru.Chapter mein pehli baar yahan aaya →
concentrationसांद्रताsaandrata: Mixture ka woh fraction ya percentage jo ek khaas substance hai, jaise drink mein sugar.Chapter mein pehli baar yahan aaya →

meanमाध्यmadhya: Values ke set ka sum divided by values ki ginti; ise average bhi kehte hain.Chapter mein pehli baar yahan aaya →

pie chartपाई चार्टpai chart: Slices mein bata ek circle jinke angles whole ke parts ke proportion mein hote hain.Chapter mein pehli baar yahan aaya →

share: Koi part jo apne whole ke fraction ya percentage ke roop mein likha ho; 500 L mein se 180 L ek 36% share hai.Chapter mein pehli baar yahan aaya →
stacked bar chart: Aisa bar chart jismein har whole ke parts ek ke baad ek jude hote hain, taaki poora bar total dikhaye.Chapter mein pehli baar yahan aaya →

weightभारbhaar: Weighted mean mein woh number jo batata hai ki ek value kitna count hoti hai: group ka size, mix ki gayi amount, ya importance.Chapter mein pehli baar yahan aaya →
weighted meanभारित माध्यbharit madhya: Har value aur uske weight ke product ka sum, divided by weights ka sum.

10.1 Cheezon Ko Jodna

10.1.1 Averages Ka Average

10.1.2 Mixtures

10.1.3 Apne Weights Chunna

10.2 Data Ko Visualise Aur Interpret Karna

10.2.1 Columns Ko Stack Karna

10.2.2 Pie Chart Ka Ek Alternative

Amounts se shares tak

100% chart kya bata sakta hai aur kya nahi

Kiska typical din?

Chapter ke end ki exercises

Apna NCERT exercise question dhoondo

Quick check: MCQs

Tum chapter ke end tak pahunch gaye