Ratio and Proportion: Concepts, Formulas, and Practice Questions

1. The ratio of the ages of A and B is 4:5. If B’s age is 25 years, find A’s age.

• Answer:
  The ratio of A to B is 4:5.
  Let A’s age = $4x$4x and B’s age = $5x$5x.
  Given that B’s age = 25 years, $$5x = 25 \implies x = 5$$5x=25⟹x=5 Therefore, A’s age = $4x = 4 \times 5 = 20$4x=4×5=20 years.

2. The ratio of the number of boys to girls in a class is 3:4. If there are 60 boys, how many girls are there?

• Answer:
  The ratio is 3:4, so the number of boys to girls is in the ratio 3:4.
  If there are 60 boys, let the number of girls be $x$x. $$\frac{60}{x} = \frac{3}{4} \implies 4 \times 60 = 3 \times x \implies x = \frac{240}{3} = 80$$x60​=43​⟹4×60=3×x⟹x=3240​=80 Therefore, there are 80 girls.

3. The ratio of the circumference of two circles is 2:3. If the radius of the first circle is 10 cm, find the radius of the second circle.

• Answer:
  The ratio of circumferences is given as 2:3, and the circumference of a circle is proportional to its radius.
  Let the radius of the second circle be $r_2$r2​. $$\frac{10}{r_2} = \frac{2}{3} \implies r_2 = \frac{10 \times 3}{2} = 15 \, \text{cm}$$r2​10​=32​⟹r2​=210×3​=15cm Therefore, the radius of the second circle is 15 cm.

4. A bag contains 30 red balls and 45 green balls. What is the ratio of red balls to green balls?

• Answer:
  The ratio of red balls to green balls is $30:45$30:45.
  Simplifying this ratio by dividing both numbers by 15: $$30:45 = 2:3$$30:45=2:3 Therefore, the ratio is $2:3$2:3.

5. If 3 men can complete a job in 5 days, how many days will it take 9 men to complete the same job?

• Answer:
  The work done is inversely proportional to the number of men.
  If 3 men take 5 days, the number of days taken by 9 men is: $$\text{Time} = \frac{3 \times 5}{9} = \frac{15}{9} = 5 \, \text{days}.$$Time=93×5​=915​=5days. Therefore, 9 men will take 5 days to complete the same job.

6. If 7x=1220\frac{7}{x} = \frac{12}{20}x7​=2012​, find xxx.

• Answer: $$\frac{7}{x} = \frac{12}{20} \implies x = \frac{7 \times 20}{12} = \frac{140}{12} = 11.67$$x7​=2012​⟹x=127×20​=12140​=11.67 Therefore, $x = 11.67$x=11.67.

7. If 34=x8\frac{3}{4} = \frac{x}{8}43​=8x​, find xxx.

• Answer: $$\frac{3}{4} = \frac{x}{8} \implies x = \frac{3 \times 8}{4} = 6$$43​=8x​⟹x=43×8​=6 Therefore, $x = 6$x=6.

8. The ratio of the heights of two trees is 5:7. If the height of the smaller tree is 10 meters, find the height of the larger tree.

• Answer:
  The ratio of heights is 5:7, and the height of the smaller tree is 10 meters.
  Let the height of the larger tree be $x$x. $$\frac{10}{x} = \frac{5}{7} \implies x = \frac{10 \times 7}{5} = 14 \, \text{meters}$$x10​=75​⟹x=510×7​=14meters Therefore, the height of the larger tree is 14 meters.

9. If 15x=912\frac{15}{x} = \frac{9}{12}x15​=129​, find xxx.

• Answer: $$\frac{15}{x} = \frac{9}{12} \implies x = \frac{15 \times 12}{9} = 20$$x15​=129​⟹x=915×12​=20 Therefore, $x = 20$x=20.

10. A car travels 200 km in 4 hours. How far will it travel in 10 hours at the same speed?

• Answer:
  The speed of the car is $\frac{200}{4} = 50 \, \text{km/h}$4200​=50km/h.
  In 10 hours, the car will travel: $$50 \times 10 = 500 \, \text{km}$$50×10=500km Therefore, the car will travel 500 km in 10 hours.

11. A person divides a sum of money in the ratio of 3:2 between A and B. If the total amount is ₹500, how much does each person receive?

• Answer:
  The ratio of A’s share to B’s share is 3:2.
  Let the total amount be divided into 5 parts (3 parts for A and 2 parts for B).
  The value of each part is: $$\frac{500}{5} = 100$$5500​=100 A’s share = $3 \times 100 = 300$3×100=300
  B’s share = $2 \times 100 = 200$2×100=200
  Therefore, A receives ₹300 and B receives ₹200.

12. In a school, the ratio of boys to girls is 5:7. If there are 280 girls, how many boys are there?

• Answer:
  The ratio of boys to girls is 5:7.
  Let the number of boys be $x$x. $$\frac{x}{280} = \frac{5}{7} \implies x = \frac{5 \times 280}{7} = 200$$280x​=75​⟹x=75×280​=200 Therefore, there are 200 boys.

13. The ratio of two numbers is 5:3. If the sum of the numbers is 40, find the numbers.

• Answer:
  Let the two numbers be $5x$5x and $3x$3x.
  The sum of the numbers is: $$5x + 3x = 40 \implies 8x = 40 \implies x = 5$$5x+3x=40⟹8x=40⟹x=5 Therefore, the numbers are $5 \times 5 = 25$5×5=25 and $3 \times 5 = 15$3×5=15.

14. A team of 12 people can finish a task in 10 days. How many days will it take for 18 people to finish the same task?

• Answer:
  The work done is inversely proportional to the number of people.
  Time taken by 18 people = $\frac{12 \times 10}{18} = \frac{120}{18} = 6.67$1812×10​=18120​=6.67 days.
  Therefore, it will take 18 people approximately 6.67 days.

15. The ratio of the present ages of A and B is 5:7. If 5 years ago, A’s age was 15 years, find the present age of B.

• Answer:
  Let A’s present age be $5x$5x and B’s present age be $7x$7x.
  5 years ago, A’s age was $5x – 5 = 15$5x−5=15. $$5x = 20 \implies x = 4$$5x=20⟹x=4 Therefore, B’s present age = $7x = 7 \times 4 = 28$7x=7×4=28.
  B’s present age is 28 years.

16. If the compound ratio of 4:3 and 5:2 is asked, what is the result?

• Answer:
  The compound ratio of $\frac{4}{3}$34​ and $\frac{5}{2}$25​ is: $$\frac{4}{3} : \frac{5}{2} = \frac{4 \times 5}{3 \times 2} = \frac{20}{6} = \frac{10}{3}$$34​:25​=3×24×5​=620​=310​ Therefore, the compound ratio is $10:3$10:3.

17. A person walks 3 km in 30 minutes. If another person walks 5 km in 50 minutes, what is the ratio of their speeds?

• Answer:
  First person’s speed = $\frac{3}{0.5} = 6 \, \text{km/h}$0.53​=6km/h.
  Second person’s speed = $\frac{5}{\frac{50}{60}} = \frac{5 \times 60}{50} = 6 \, \text{km/h}$6050​5​=505×60​=6km/h.
  The ratio of their speeds = $6:6 = 1:1$6:6=1:1.

18. A mixture contains milk and water in the ratio 3:2. How much water should be added to 30 liters of the mixture to make the ratio 1:2?

• Answer:
  Initially, the total mixture is 30 liters, with 18 liters of milk and 12 liters of water.
  Let the amount of water to be added be $x$x. After adding $x$x liters, the new ratio becomes $1:2$1:2. $$\frac{18}{12 + x} = \frac{1}{2} \implies 2 \times 18 = 12 + x \implies 36 = 12 + x \implies x = 24$$12+x18​=21​⟹2×18=12+x⟹36=12+x⟹x=24 Therefore, 24 liters of water should be added.

19. If a man’s salary is increased by 20%, find the new ratio of his salary to his previous salary.

• Answer:
  The new salary is 120% of the previous salary.
  Therefore, the ratio of the new salary to the previous salary is $\frac{120}{100} = 6:5$100120​=6:5.

20. In a mixture of 40 liters, the ratio of water to alcohol is 3:7. How much alcohol is there in the mixture?

• Answer:
  The total amount of alcohol = $\frac{7}{10} \times 40 = 28 \, \text{liters}$107​×40=28liters.

21. If 5 men can complete a job in 20 days, how many men are required to complete the same job in 10 days?

• Answer:
  The number of men is inversely proportional to the number of days.
  Number of men required = $\frac{5 \times 20}{10} = 10$105×20​=10 men.

22. The time taken to fill a tank is inversely proportional to the number of pipes working. If 4 pipes take 6 hours, how many pipes will be needed to fill the tank in 3 hours?

• Answer:
  Number of pipes needed = $\frac{4 \times 6}{3} = 8$34×6​=8 pipes.

23. The number of workers required to complete a task is inversely proportional to the time taken to complete it. If 8 workers can complete the task in 10 days, how many workers are needed to complete the same task in 5 days?

• Answer:
  Number of workers needed = $\frac{8 \times 10}{5} = 16$58×10​=16 workers.

24. The speed of a vehicle is inversely proportional to the time it takes to cover a fixed distance. If the speed is increased by 20%, how much time will be reduced in completing the same journey?

• Answer:
  The time taken is reduced by $\frac{1}{20}$201​ or 5%.

25. If the area of a rectangle is directly proportional to its length and inversely proportional to its width, find the change in area when the length is doubled and the width is halved.

Answer:
The area will increase by a factor of 4 (because $2 \times \frac{1}{2} = 4$2×21​=4).