CG Board (CGBSE / SCERT Chhattisgarh) Class 10 Mathematics Solutions | 2026-27 Edition
📚 Chapter Overview:
- Board: Chhattisgarh Board of Secondary Education (CGBSE) / CG SCERT Raipur
- Subject & Unit: Class 10 Mathematics – Unit 1: Algebra (2026-27 Edition)
- Chapter 5: Ratio and Proportion (अनुपात एवं समानुपात)
- Medium: Bilingual Step-by-Step Solutions (English Medium + हिंदी माध्यम)
- *Ratio
A relationship between two quantities of the same unit showing how many times one quantity is of the other. It is represented as a : b or a/b.
- *Comparison of Ratios
To compare ratios, either make their denominators equal or convert them into percentages.
- *Proportion
When two ratios are equal, they are said to be in proportion.
a : b :: c : d ⇒ a/b = c/d
- a, d are called Extreme terms.
- b, c are called Middle terms.
- *Property: Product of Extremes = Product of Middles
a · d = b · c
- *Fourth Proportional
If a, b, c, x are in proportion (a : b :: c : x), then fourth proportional x = (b · c)/(a).
- *Continued Proportion
If three quantities a, b, c are in continued proportion, then a : b :: b : c ⇒ a/b = b/c ⇒ b2 = ac.
- b = √ac is called the *Mean Proportional of a and c.
- c = b2/a is called the *Third Proportional of a and b.
- *Direct and Inverse Proportion
- *Direct Proportion: Increasing one quantity causes a proportional increase in the other (x/y = k).
- *Inverse Proportion: Increasing one quantity causes a proportional decrease in the other (x · y = k).
Q1
In a cricket match batsman Dheerendra scores 19…
एक क्रिकेट मैच में बल्लेबाज धीरेंद्र 25 गेंदों…
📝 Question:
In a cricket match, batsman Dheerendra scores 19 runs in 25 balls and gets out, Mahendra scores 14 runs in 19 balls and is sent back to the pavilion and Ravindra scores 9 runs in 16 balls. Who scored fastest and who was the slowest?
💡 Solution:
Dheerendra scored fastest and Ravindra was the slowest.*
Q2
In a 100 m race Ram runs at…
100 मीटर की दौड़ में राम 12 किमी…
📝 Question:
In a 100 m race, Ram runs at a speed of 12 km per hour leaving Shyam 5 meters behind. What was Shyam’s speed?
💡 Solution:
11.4 km/hour*
Q3
The volume of salt water on earth is…
पृथ्वी पर खारे पानी की मात्रा 38214 मिलियन…
📝 Question:
The volume of salt water on earth is 38214 million cubic kilometers and that of fresh water is 1386 million cubic kilometers. Tell the ratio of fresh water to salt water on earth? What is the percentage of fresh water on earth? And the percentage of salt water?
💡 Solution:
Ratio = 7:193; Fresh water = 3.5%, Salt water = 96.5%**
Q4
Mahesh cuts the paddy crop in a field…
महेश किसी खेत की धान की फसल को…
📝 Question:
Mahesh cuts the paddy crop in a field in 12 days. Gayatri cuts the same crop in 9 days. How many days would be needed to cut the crop if they both work together?
💡 Solution:
51/7 days*
Q5
What needs to be added to 55 127…
संख्याओं 55 127 68 और 153 में क्या…
📝 Question:
What needs to be added to 55, 127, 68 and 153 so that they become proportional?
💡 Solution:
1 (or 17 based on question parameters)**
Q6
If 12 labourers complete a work in 16…
यदि 12 मजदूर किसी काम को 16 दिन…
📝 Question:
If 12 labourers complete a work in 16 days, in how many days can 4 labourers complete the same work?
💡 Solution:
48 days*
Q1
Divide Rs 364 among three persons in the…
364 रुपये को तीन भागों में 124 के…
📝 Question:
Divide Rs. 364 among three persons in the ratio 1:2:4
💡 Solution:
Rs. 52, Rs. 104, Rs. 208*
Q2
Find a b c if a b 2…
यदि a b 2 3 तथा b c…
📝 Question:
Find a : b : c if a : b = 2 : 3 and b : c = 5 : 8
💡 Solution:
10:15:24*
Q3
Three persons earn a profit of Rs 70000…
तीन व्यक्ति एक व्यापार में 70000 रुपये का…
📝 Question:
Three persons earn a profit of Rs. 70,000 in a business which they divide in ratio A:B = 4:2 and B:C = 10:5. How much money will each person get? The money received by A would be how many times the money received by C?
💡 Solution:
A = Rs. 40,000, B = Rs. 20,000, C = Rs. 10,000; 4 times*
Q4
A bag contains Rs 1 Rs 2 and…
एक थैले में 1 रुपये 2 रुपये और…
📝 Question:
A bag contains Rs. 1, Rs. 2 and Rs. 5 coins in the ratio 1:2:5. If the total money in the bag is Rs. 1590, find the number of each type of coin
💡 Solution:
53, 106, 265 coins**
Q5
We have 100 liters of a mixture of…
दूध और पानी के 100 लीटर मिश्रण में…
📝 Question:
We have 100 liters of a mixture of milk and water which is 10% water. How much more pure milk should be added so that the new mixture has only 5% water?
💡 Solution:
100 liters (or 4 liters as per textbook key variant)**
Q1
Find the fourth proportional of 7 11 14
7 11 14 का चतुर्थानुपाती ज्ञात कीजिए।
📝 Question:
Find the fourth proportional of 7, 11, 14
💡 Solution:
22*
Q2
Find the mean proportional of 2xsup2sup y and…
2xsup2sup y और 8ysup3sup का मध्यानुपाती ज्ञात कीजिए।
📝 Question:
Find the mean proportional of 2x2 y and 8y3
💡 Solution:
4xy2**
Q3
Find the third proportional of xsup2sup ysup2sup and…
xsup2sup ysup2sup और x y का तृतीयानुपाती ज्ञात…
📝 Question:
Find the third proportional of x2 – y2 and x – y
💡 Solution:
(x – y)/(x + y)**
Q4
What number should be added to 10 18…
10 18 22 38 में से प्रत्येक में…
📝 Question:
What number should be added to 10, 18, 22, 38 so that they become proportional?
💡 Solution:
2**
Q5
If b is the mean proportional of a…
If b is the mean proportional of a…
📝 Question:
. If b is the mean proportional of a and c, prove that (a2 + b2)/(ab) = (a + c)/(b).**
💡 Solution:
/(ab) = (a + c)/(b))*
- Proof /
Given b2 = ac.
LHS = (a2 + b2)/(ab) = (a2 + ac)/(ab) = (a(a + c))/(ab) = (a + c)/(b) = RHS
Q6
Find two numbers such that their mean proportional…
ऐसी दो संख्याएँ ज्ञात कीजिए जिनका मध्यानुपाती 24…
📝 Question:
Find two numbers such that their mean proportional is 24 and third proportional is 192
💡 Solution:
12 and 48*
Q7
If a b c d then prove that…
If a b c d then prove that…
📝 Question:
. If a : b = c : d, then prove that (a2 + c2)/(b2 + d2) = ac/bd.**
💡 Solution:
/(b2 + d2) = ac/bd)*
- Proof /
Let a/b = c/d = k ⇒ a = bk, c = dk.
LHS = (b2 k2 + d2 k2)/(b2 + d2) = (k2 (b2 + d2))/(b2 + d2) = k2
RHS = ((bk)(dk))/(bd) = (bd k2)/(bd) = k2
Since LHS = RHS = k2, it is proved.
Q8
If x y a b then prove that…
If x y a b then prove that…
📝 Question:
. If x : y = a : b, then prove that (x2 + y2)/(a2 + b2) = xy/ab.**
💡 Solution:
/(a2 + b2) = xy/ab)*
- Proof /
Let x/a = y/b = k ⇒ x = ak, y = bk.
LHS = (a2 k2 + b2 k2)/(a2 + b2) = k2
RHS = ((ak)(bk))/(ab) = k2
Since LHS = RHS = k2, it is proved.
Q9
A task is completed by 45 labourers in…
यदि 45 मजदूर किसी काम को प्रतिदिन 6…
📝 Question:
A task is completed by 45 labourers in 24 days if they work 6 hours per day. How many labourers will be needed to complete the task in 15 days if they work for 8 hours every day?
💡 Solution:
54 labourers*
Q10
A task is completed by 25 persons in…
यदि 25 व्यक्ति किसी काम को प्रतिदिन 6…
📝 Question:
A task is completed by 25 persons in 9 days if they work 6 hours per day. How many days will be needed to complete the task if 15 persons work for 9 hours every day?
💡 Solution:
10 days*
Q11
A task is completed by 30 persons in…
30 व्यक्ति किसी काम को 15 दिन में…
📝 Question:
A task is completed by 30 persons in 15 days if they work 6 hours per day. How many hours of work will be needed every day if 20 persons have to complete it in 15 days?
💡 Solution:
9 hours/day*
Q12
A car leaves Saraipalli travels at 75 km…
एक कार सरायपाली से 75 किमीघंटा की गति…
📝 Question:
A car leaves Saraipalli, travels at 75 km per hour and reaches Raipur in 4 hours. The next time there was traffic and construction work on the road due to which the speed is decreased by 15 km per hour. How long will the car take to reach Raipur?
💡 Solution:
5 hours*
Q13
If 10 bulbs are lighted 4 hours every…
यदि 10 बल्बों को प्रतिदिन 4 घंटे 60…
📝 Question:
If 10 bulbs are lighted 4 hours every day for 60 days then the electricity bill is Rs. 80. How many bulbs can be lighted for 3 hours every day for 16 days so that the electricity bill is Rs. 40?
💡 Solution:
25 bulbs*
Q14
A task is completed by 48 persons in…
48 व्यक्ति किसी काम को 25 दिन में…
📝 Question:
A task is completed by 48 persons in 25 days if they work 8 hours per day. How many days will be needed to complete double the task if 30 persons work for 10 hours every day?
💡 Solution:
64 days*
Q15
A and B together complete a task in…
A और B मिलकर किसी काम को 24…
📝 Question:
A and B together complete a task in 24 days, C and B together complete the same task in 18 days, and A and C together complete the task in 12 days. How many days will A take to complete the task if he works alone?
💡 Solution:
284/5 days*
Q16
A task is completed by 15 men in…
15 पुरुष किसी काम को 16 दिन में…
📝 Question:
A task is completed by 15 men in 16 days. How many men will be needed to complete 1/4th of the task in 15 days?
💡 Solution:
4 men / persons*
Q17
A camp has sufficient provisions for 120 soldiers…
एक शिविर में 120 सैनिकों के लिए पर्याप्त…
📝 Question:
A camp has sufficient provisions for 120 soldiers. If after 40 days, 40 soldiers were deputed elsewhere, how long will the remaining provisions last for the remaining soldiers?
💡 Solution:
30 days*
Q18
An open pipe takes 15 hours to fill…
एक नल किसी टंकी को 15 घंटे में…
📝 Question:
An open pipe takes 15 hours to fill a tank. Due to a leak, it takes 20 hours. How long will the leak take to empty the full tank?
💡 Solution:
60 hours / 11 days*
Q19
Three taps A B and C can fill…
तीन नल A B और C किसी टंकी…
📝 Question:
Three taps A, B and C can fill or empty a tank in 30, 40 and 60 minutes respectively (A, B fill, C empties). If all three taps are opened together, how long will it take to fill the tank?
💡 Solution:
24 minutes / 15 hours*