NCERT Solutions Class 7 Maths Chapter 3: A Peek Beyond the Point (Hinglish)

Ganita Prakash Grade 7 Part 1 (Reprint 2026-27) ke Chapter 3, A Peek Beyond the Point, ke har sawaal ke jawab: text ke sawaal, Math Talk aur Try This, aur har Figure it Out. Page numbers kitaab ke hisaab se hain (pages 46–80). Tasveeron se padhi gayi lambaaiyan lagbhag hain. Chhote notes: Chapter 3 ke notes.

The Need for Smaller Units (pages 46–48)

Screws naapo. Pehla screw: 2 cm aur 3 cm ke beech; 2½ cm se zyaada par 3 cm se kam; 2 7/10 cm. Doosra screw: 3 cm aur 4 cm ke beech; 3 cm se zyaada par 3½ cm se kam; 3 2/10 cm.

Kis scale ne screws ko sateek naapa, aur kyun? Teesre ne, jisme centimetre ke tenths ke nishaan hain. Baaki do sirf batate hain ki screw kin poore ya aadhe centimetres ke beech hai; tenths theek batate hain ki wo kahan khatm hota hai.

2 7/10 cm ka matlab kya hai? 0 se 2 cm tak jaao, phir agle centimetre ke 10 barabar hisson mein se 7 lo. Ye 2 cm aur 7/10 cm hai, jise two and seven-tenth centimetres padhte hain.

Unit ko chhote hisson mein kyun baanta gaya? Dono screws mein ek centimetre se kam ka farq hai, isliye poore centimetres ye farq nahi dikha sakte. Chhote hisse zyaada sateek naap dete hain.

Pen, sharpener aur apni pasand ki ek cheez naapo. Jawab tumhari cheezon pe nirbhar hain, jaise 14 3/10 cm ka pen aur 2 5/10 cm ka sharpener. Pehle poore centimetres padho, phir tenths gino.

Tasveer ki cheezein (page 47). Tasveer se padhkar: rubber 2 4/10 cm, pencil 4 5/10 cm, chalk 1 4/10 cm.

A Tenth Part (pages 48–52)

Har lambaai do tarah likho (page 49). Tasveeron se padhkar, har ek ki apni unit mein: USB cable 4 8/10 = 48/10 units (udaaharan); pencil box ki oonchaai 2 8/10 = 28/10 units; angootha 1 3/10 = 13/10 units; patta 10 9/10 = 109/10 units. 2 8/10 ko two and eight-tenths aur 28/10 ko twenty-eight tenths padho.

Badhte kram mein rakho (page 49). 4/10 < 9/10 < 1 7/10 < 6 7/10 < 7 6/10 < 10 5/10 < 130/10 < 13 1/10. (130/10 matlab 13.)

4 1/10, 4/10, 41/10 aur 41 1/10 ko badhte kram mein rakho (page 50). 4/10 < 4 1/10 = 41/10 < 41 1/10. Dhyaan do ki 4 1/10 aur 41/10 ek hi lambaai hain.

Madhumakkhi (page 51). 2 3/10 + 5 4/10 + 7 5/10 = 14 12/10 = 15 2/10 units.

Ungli waale ghataav pe charcha (page 51). 4 tenths mein se 7 tenths nahi ghat sakte, isliye 12 ki ek unit ko 10 tenths mein baanta jaata hai, jisse 11 units aur 14 tenths bante hain. Phir 14 − 7 = 7 tenths aur 11 − 6 = 5 units: ungli 5 7/10 units ki hai. Tenths mein badalkar: 124/10 − 67/10 = 57/10 = 5 7/10.

Do machhliyan (page 52). 2 4/10 − 9/10 = 24/10 − 9/10 = 15/10 = 1 5/10 cm. Dono machhliyan ungli se bahut chhoti hain: danio lagbhag ungli ke upari jod jitni lambi hai, aur goby 1 cm se bhi chhoti.

Sequences aage badhaao (page 52).

• (a) Har baar 3/10 jodo: 4 9/10, 5 2/10, 5 5/10, 5 8/10.
• (b) 5/10 jodo: 9 7/10, 10 2/10, 10 7/10, 11 2/10.
• (c) 1 1/10 jodo: 9 8/10, 10 9/10, 12, 13 1/10.
• (d) 4/10 ghataao: 4 9/10, 4 5/10, 4 1/10, 3 7/10.
• (e) 5/10 ghataao: 12, 11 5/10, 11, 10 5/10.
• (f) 1 1/10 ghataao: 8 2/10, 7 1/10, 6, 4 9/10.

A Hundredth Part (pages 52–58)

Har chhote hisse ki lambaai kya hai, aur kitne milkar unit banaate hain? Har tenth ko 10 mein baantne pe unit ke 1/100 ke hisse milte hain, aur unke 100 milkar ek unit banaate hain. Muda kaagaz 4 4/10 5/100 pe khatm hota hai.

Ek tenth mein kitne hundredths? Kya lambaai 4 units aur 45 hundredths bhi hai? 10 hundredths ek tenth banaate hain. Haan: 4 tenths matlab 40 hundredths, aur 5 aur milkar 45 hundredths, isliye 4 4/10 5/100 = 4 45/100.

Khaali boxes bharo (page 54). 0 se hundredths mein naapkar, baayein se daayein: 55/100, 155/100, 174/100, 202/100 aur 240/100.

Teen roop ek hi lambaai kaise dikhaate hain? 1 1/10 4/100 = 1 14/100, kyunki 1 tenth aur 4 hundredths milkar 14 hundredths hain. Aur 1 14/100 = 114/100, kyunki 1 unit 100 hundredths hai.

Taaron ki lambaai likho aur padho (page 54). Tasveeron se padhkar: 5 37/100 (five and thirty-seven hundredths), 15 3/100 (fifteen and three hundredths), 7 52/100 (seven and fifty-two hundredths) aur 9 80/100 (nine and eighty hundredths, jo 9 8/10 hai).

Har samooh mein sabse lamba aur sabse chhota (pages 55–56).

• (a) 3/10, 3/100, 33/100: sabse lamba 33/100, sabse chhota 3/100.
• (b) 3 1/10, 30/10, 1 3/10: sabse lamba 3 1/10, sabse chhota 1 3/10.
• (c) 45/100, 54/100, 5/10, 4/10: sabse lamba 54/100, sabse chhota 4/10.
• (d) 3 6/10, 3 6/100, 3 6/10 6/100: sabse lamba 3 6/10 6/100, sabse chhota 3 6/100.
• (e) 8/10 2/100, 9/100, 1 8/100: sabse lamba 1 8/100, sabse chhota 9/100.
• (f) 7 3/10 5/100, 7 5/10, 7 41/100: sabse lamba 7 5/10, sabse chhota 7 3/10 5/100.
• (g) 6 5/10 15/100, 5 87/100, 5 7/100: sabse lamba 6 5/10 15/100, jo 6 65/100 hai; sabse chhota 5 7/100.

15 3/10 4/100 + 2 6/10 8/100 nikaalo. Kya dono tareeke alag hain? Jod 18 2/100 hai. Dono tareeke units ko units se, tenths ko tenths se aur hundredths ko hundredths se jodte hain, phir 10 hundredths ko ek tenth aur 10 tenths ko ek unit banaate hain. Farq sirf likhne ke dhang ka hai, bilkul 483 + 268 = 600 + 140 + 11 = 751 jaisa. Hundredths mein badalne pe bhi wahi milta hai: 1534/100 + 268/100 = 1802/100 = 18 2/100.

25 9/10 − 6 4/10 7/100 nikaalo, hundredths mein badalkar bhi. 25 9/10 − 6 4/10 7/100 = 19 4/10 3/100. Hundredths mein: 2590/100 − 647/100 = 1943/100, wahi jawab.

15 3/10 4/100 − 2 6/10 8/100 nikaalo. Ek tenth ko 10 hundredths aur ek unit ko 10 tenths ki tarah udhaar lo: antar 12 6/10 6/100 hai. Ye 653 − 268 = 385 jaisa hai, jahan ek ten aur ek hundred udhaar liye jaate hain.

Figure it Out (page 58)

• (a) 3/10 + 3 4/100 = 3 34/100.
• (b) 9 5/10 7/100 + 2 1/10 3/100 = 11 6/10 10/100 = 11 7/10.
• (c) 15 6/10 4/100 + 14 3/10 6/100 = 29 9/10 10/100 = 30.
• (d) 7 7/100 − 4 4/100 = 3 3/100.
• (e) 8 6/100 − 5 3/100 = 3 3/100.
• (f) Jaisa chhapa hai, 12 6/100 2/100 matlab 12 8/100, aur 12 8/100 − 9/10 9/100 = 11 9/100. Agar 12 6/10 2/100 kehna tha, to jawab 11 6/10 3/100 hai.

Decimal Place Value (pages 58–64)

Kya unit ko 4, 5 ya 8 barabar hisson mein baant sakte hain? Haan. Quarters, aur quarters ke quarters (sixteenths), se bhi lambaai naapi jaati hai. 10 se baantna isliye chuna jaata hai kyunki ye place value system ko aage badhaata hai, jisme har sthaan agle ka 10 guna hai.

1/100 ko 10 barabar hisson mein baantne pe kaunsa fraction milta hai? 1/1000, yaani ek thousandth; unke 1000 milkar ek unit banaate hain.

How Big? (page 61)

• (a) 1000 thousandths ek unit banaate hain.
• (b) 100 thousandths ek tenth banaate hain.
• (c) 10 thousandths ek hundredth banaate hain.
• (d) 100 tenths ek ten banaate hain.
• (e) 1000 hundredths ek ten banaate hain.

Aise aur sawaal: 10,000 hundredths ek hundred banaate hain, aur 10,000 thousandths ek ten.

Kya 4 2/10 ko 42 likh sakte hain? Nahi. 42 ka matlab pehle se 4 tens aur 2 units hai, to dono mein farq pata nahi chalega. Isiliye decimal point chahiye: 4 2/10 = 4.2.

Place value table (page 63).

• (a) 2 ones, 3 tenths aur 5 hundredths = 2.35, padho two point three five.
• (b) 1 ten aur 5 tenths = 10.5, padho ten point five.
• (c) 4 ones aur 6 hundredths = 4.06, padho four point zero six.
• (d) 1 hundred, 1 one aur 1 hundredth = 101.01, padho one hundred and one point zero one.
• (e) 8/100 aur 9/10 = 0.98, padho zero point nine eight.
• (f) 5/100 = 0.05, padho zero point zero five.
• (g) 1/10 = 0.1, padho zero point one.
• (h) 2 1/100 = 2.01, 4 1/10 = 4.1 aur 7 7/1000 = 7.007, padho two point zero one, four point one aur seven point zero zero seven.

Decimal mein likho (page 64). (a) 234 hundredths = 2 + 3/10 + 4/100 = 2.34. (b) 105 tenths = 10 + 5/10 = 10.5.

Units of Measurement (pages 64–69)

mm aur cm (page 65). 70 mm = 7 cm; 9 mm = 0.9 cm; 134 mm = 13.4 cm; 2036 mm = 203.6 cm.

Prakriti ki chhoti cheezein. Aankh se naap alag ho sakta hai; jaise 1 se 2 mm ka sarson ka daana 0.1 se 0.2 cm hai, aur 0.1 mm ka baal 0.01 cm.

cm aur m (page 66). 36 cm = 0.36 m; 50 cm = 0.5 m; 89 cm = 0.89 m; 4 cm = 0.04 m; 325 cm = 3.25 m; 207 cm = 2.07 m.

1 metre mein kitne mm? Kya 1 mm = 1/1000 m? 1 m = 100 cm = 100 × 10 mm = 1000 mm, isliye haan, 1 mm = 1/1000 m = 0.001 m.

g aur kg (page 68). 465 g = 0.465 kg; 68 g = 0.068 kg; 1560 g = 1.56 kg; 704 g = 0.704 kg; 560 g = 0.56 kg; 2500 g = 2.5 kg.

Chaawal ke dher. 10 kg + 1 kg + 100 g + 10 g + 1 g = 10 + 1 + 0.1 + 0.01 + 0.001 = 11.111 kg.

Rupaye aur paise (page 69). 10 p = ₹0.10; 5 p = ₹0.05; 36 p = ₹0.36; 50 p = ₹0.50; 99 p = ₹0.99; 250 p = ₹2.50.

Try This: purane daam. Jawab tumhare parivaar aur teachers se aayenge. Daamon ko rupaye aur paison mein likhna, jaise 1970s mein ek kilo chaawal ₹2.45, decimals ki achhi practice hai.

Locating and Comparing Decimals (pages 69–73)

1 aur 1.1 ke beech ke divisions ke naam likho. 1.01, 1.02, 1.03, 1.04, 1.05, 1.06, 1.07, 1.08 aur 1.09.

Letters ke decimals likho (page 70). Har chhota division 0.01 hai: A = 5.09, B = 5.13, C = 5.2, D = 5.31.

Zero ki uljhan: kaun sahi hai? Sonu. 0.2, 0.20 aur 0.200 teeno ka matlab 2 tenths hai, isliye barabar hain. Sabse bada 0.2 (= 0.20 = 0.200) aur sabse chhota 0.002 hai.

Inme se kaun barabar hain? 4.5 = 4.50 = 04.50, aur 4.05 = 4.050. 0.405 aur 4.005 baaki sabse alag hain.

Figure (b) mein ? waali sankhya. Lines 0–10 se 3–4 mein, phir 3–3.1 mein, phir 3.05–3.06 mein zoom karti hain. Aakhri line ke 10 nishaanon mein ? 9va hai, isliye ye 3.059 hai.

9.876 aur 0.407 ki number lines. 9.876 ke liye 0–10 se 9–10, phir 9.8–9.9, phir 9.87–9.88 mein zoom karo, aur 6va nishaan lo. 0.407 ke liye 0–10 se 0–1, phir 0.4–0.5, phir 0.40–0.41 mein zoom karo, aur 7va nishaan lo.

Boxes a, b aur c (page 71). 5 se 10 ki 5 units 10 hisson mein baanti hain, isliye har division aadhi unit hai: a = 6, b = 7.5, c = 9.5.

Boxes d se h (page 72). 8 se 8.1 tak 0.01 ke 10 divisions hain: d = 8.01, e = 8.05. 4.3 se 4.8 tak 0.05 ke 10 divisions hain: f = 4.35, g = 4.5, aur h, jo 4.8 ke ek division aage hai, 4.85 hai.

Kaun bada hai: 6.456 ya 6.465? 6.465. Units aur tenths barabar hain, aur hundredths mein 6 > 5.

Pehle alag digit pe tulna kyun rok sakte hain? Us sthaan ke baad ka har digit chhote sthaan ka hai, aur sab milkar bhi us sthaan ki ek unit se kam jodte hain. Jaise hundredths ke baad baaki sab 0.01 se kam jodta hai, jo poore ek hundredth ka farq poora nahi kar sakta.

Kaunsa decimal bada hai? (a) 1.32 (b) 13.800 (c) 1.090.

Sabse paas ke decimals (page 73). 1.09 ke sabse paas: 1.1, sirf 0.01 door. 4 ke sabse paas: 3.65, jo 0.35 door hai (3.56 hai 0.44 door aur 3.099 hai 0.901 door). 1 ke sabse paas: 1.08, jo 0.08 door hai (0.8 hai 0.2 door aur 0.69 hai 0.31 door).

4, 1, 8, 2 aur 5 se 25 ke jitna ho sake utna paas pahuncho.

• Point se pehle do boxes aur baad mein teen: 25.148, sirf 0.148 door. (24.851 hai 0.149 door.)
• Point se pehle ek box aur baad mein teen: aisi har sankhya 10 se kam hai, isliye sabse badi banao, 8.542. Boxes sirf chaar hain, isliye ek digit chhoot jaata hai.
• Point se pehle teen boxes aur baad mein do: aisi har sankhya 100 se zyaada hai, isliye sabse chhoti banao, 124.58.

Addition and Subtraction of Decimals (pages 74–76)

Priya aur Shylaja ka kapda. Kul 2.7 + 3.5 = 6.2 m; Shylaja ko 3.5 − 2.7 = 0.8 m zyaada chahiye. Decimal columns wahi jawab dete hain jo tenths: 2 7/10 + 3 5/10 = 6 2/10, aur 3 5/10 − 2 7/10 = 8/10.

84.691 − 77.345 ka place value hisaab. Thousandths: 1 − 5 nahi ho sakta, isliye 1 hundredth udhaar lo: 11 − 5 = 6 thousandths. Hundredths: 8 − 4 = 4. Tenths: 6 − 3 = 3. Units: 4 − 7 nahi ho sakta, isliye 1 ten udhaar lo: 14 − 7 = 7 units. Tens: 7 − 7 = 0. Chhota roop: 84.691 − 77.345 = 7.346.

Figure it Out (page 75)

1. Jod: (a) 5.3 + 2.6 = 7.9 (b) 18 + 8.8 = 26.8 (c) 2.15 + 5.26 = 7.41 (d) 9.01 + 9.10 = 18.11 (e) 29.19 + 9.91 = 39.10 (f) 0.934 + 0.6 = 1.534 (g) 0.75 + 0.03 = 0.78 (h) 6.236 + 0.487 = 6.723.
2. Antar: (a) 5.6 − 2.3 = 3.3 (b) 18 − 8.8 = 9.2 (c) 10.4 − 4.5 = 5.9 (d) 17 − 16.198 = 0.802 (e) 17 − 0.05 = 16.95 (f) 34.505 − 18.1 = 16.405 (g) 9.9 − 9.09 = 0.81 (h) 6.236 − 0.487 = 5.749.

Decimal sequences (pages 75–76). 4.4, 4.8, 5.2, 5.6, 6.0 mein har term 0.4 jodta hai, isliye agle teen 6.4, 6.8, 7.2 hain.

• (a) 0.05 jodo: 4.55, 4.6, 4.65.
• (b) 0.5 jodo: 27.25, 27.75, 28.25.
• (c) 0.11 jodo: 10.89, 11.00, 11.11.
• (d) 2.5 jodo: 21, 23.5, 26.
• (e) 0.9 jodo: 11.2, 12.1, 13.0.
• (f) 0.05 ghataao: 4.85, 4.80, 4.75.
• (g) 0.5 ghataao: 10.95, 10.45, 9.95.
• (h) 3.5 ghataao: 26, 22.5, 19.

Kya Sonu ka daava sahi hai? Haan. 25.936 + 8.202 = 34.138, jo 25 + 8 = 33 se zyaada aur 25 + 1 + 8 + 1 = 35 se kam hai. Ye kinhi bhi do decimals ke liye chalta hai, kyunki har decimal hissa 1 se kam hai, isliye dono milkar 2 se kam jodte hain. 25.93603259 + 8.202 ka jod 34.13803259 hai, phir se 33 aur 35 ke beech.

Antar ka daayra. Agar poore hisse A aur B hain, to antar A − B − 1 aur A − B + 1 ke beech hota hai. 84.691 − 77.345 ke liye ye 6 aur 8 ke beech hai, aur antar 7.346 hai.

More on the Decimal System (pages 76–78)

Bus station kab pahunchegi? Teeno mein se kisi samay nahi. 0.5 ghanta matlab 6-6 minute ke 5 hisse, yaani 30 minute, isliye bus 4:30 p.m. pe pahunchti hai.

Darwaaza band kyun nahi hua? Jagah 2 ft 5 in ki thi, par 2.5 ft matlab 2 ft 6 in, kyunki 0.5 ft matlab 12 inch ka aadha. Darwaaza 1 inch zyaada chauda tha.

5.5 overs ka matlab kya hai? 5 overs aur 5 balls: point ke baad ka digit balls ginta hai, aur ek over mein 6 balls hoti hain.

Aur kahan decimal jaisi dikhne waali gair-decimal sankhyaayein milti hain? 10.30 jaise ghadi ke samay, 15.08.2026 jaisi taareekh, kitaab ke section number (3.10, 3.9 ke baad aata hai), app ke version (2.10, 2.9 se naya hai), aur 5 ft 6 in ke liye boli gayi 5.6 jaisi oonchaai.

Figure it Out (pages 78–80)

1. (a) 5/100 = 0.05 (b) 16/1000 = 0.016 (c) 12/10 = 1.2 (d) 254/1000 = 0.254.
2. (a) 0.34 = 3/10 + 4/100 (b) 1.02 = 1 + 0/10 + 2/100 (c) 0.8 = 8/10 (d) 0.362 = 3/10 + 6/100 + 2/1000.
3. Har tenth 0.025 ke 4 barabar kadmon mein baanta hai. Boxes baayein se daayein a, c, b likhe hain, isliye a = 6.45, c = 6.525, b = 6.55.
4. (a) 11.10, 11.01, 1.101, 1.011, 1.01
   (b) 2.768, 2.698, 2.675, 2.567, 2.499
   (c) 4.678 g, 4.666 g, 4.656 g, 4.600 g, 4.595 g
   (d) 33.331 m, 33.313 m, 33.31 m, 33.133 m, 33.13 m
5. (a) 40.168. Tens mein 1 ho to sabse badi sankhya 18.640 hai, jo 30 se 11 se zyaada door hai, jabki 40.168 sirf 10.168 door hai. (b) 100 aur 1000 ke beech ki sankhya ko point se pehle teen digits chahiye; sabse chhoti 104.68 hai.
6. Nahi. 2.05 mein 2.5 se zyaada digits hain, phir bhi wo chhota hai, kyunki 2.5 mein 5 tenths hain aur 2.05 mein ek bhi nahi. Digits ke sthaan faisla karte hain, unki ginti nahi.
7. 0.25 + 0.3 + 0.5 + 0.2 + 0.05 = 1.3 kg.
8. Pehle teen din: 3.79 + 4.2 + 4.25 = 12.24 L. Aakhri teen din: 25 − 12.24 = 12.76 L.
9. 35.75 − 34.50 = 1.25, isliye Tinku ka vajan 1.25 kg kam hua.
10. Niyam hai 0.9 jodo, phir 0.01 ghataao, baar-baar. 6tha aur 7va term 8.18 aur 8.17 hone chahiye (kitaab mein 6.18 aur 6.17 chhapa hai), aur agle do 9.07, 9.06 hain.
11. 1 km = 1000 m = 1000 × 1000 mm = 10,00,000 mm.
12. 45 paise × 1,00,000 = 45,00,000 paise = ₹45,000.
13. (a) 1/10, kyunki 10/1000 = 1/100. (b) 90 thousandths, jo 9 hundredths hai. (c) 90 hundredths, jo 0.9 hai.
14. (b) 12 tens aur 12 tenths = 120 + 1.2 = 121.2 (c) 100 + 10 + 1 + 0.1 = 111.1 (d) 250 + 25 + 2.5 + 0.25 = 277.75.
15. 9.432 + 1.068 = 10.500, theek 10.5, isliye isse paas koi jod nahi ho sakta: poore hisse 9 + 1 = 10 dete hain aur decimal hisse 0.432 + 0.068 = 0.5. Kai aur tareekon se bhi theek 10.5 banta hai, jaise 5.603 + 4.897.
16. (a) 1/2 = 0.5 (b) 3/2 = 1.5 (c) 1/4 = 0.25 (d) 3/4 = 0.75 (e) 1/5 = 0.2 (f) 4/5 = 0.8.

Aur practice aur chhote notes topic pages mein hain: tenths aur hundredths, decimals ki tulna aur decimals jodna aur ghataana.