CHAPTER-12 RATIO AND PROPORTION

1. There are 20 girls and 15 boys in a class.

(a) What is the ratio of the number of girls to the number of boys?

(b) What is the ratio of the number of girls to the total number of students in the class?

Explanation:

Given

Girls count = 20 girls

Boys count = 15 boys

Total student counts = 20 + 15 = 35

(a) Ratio of count of girls to the count of boys = count of girls / count of boys

= 20 / 15 (simplify)

= 4 / 3

(b) Ratio of count of girls to the total count of students = count of girls/ total student count

= 20 / 35 (simplify)

= 4 / 7


2. Out of 30 students in a class, 6 like football, 12 like cricket and the remaining like tennis. Find the ratio of






(a) The number of students liking football to the number of students liking tennis.

(b) The number of students liking cricket to the total number of students.

Explanation:

It has been given that

Count of students that like to play cricket = 12

Count of students that like to play football = 6

Count of students that like to play tennis = Total – students that like other sports

= 30 – 6 – 12 (simplify)

= 12

(a) Ratio of count of students that play football to the count of students that play tennis = Count of football liking players / count of tennis liking player

= 6 / 12 (simplify)

= 1 / 2

(b) Ratio of the count of players that play cricket to the total count of players = cricket players / total players

= 12 / 30 (simplify)

= 2 / 5


3. See the figure and find the ratio of










(a) Number of triangles to the number of circles inside the rectangle.

(b) Number of squares to all the figures inside the rectangle.

(c) Number of circles to all the figures inside the rectangle.

Explanation:

It has been given that

circles count = 2

triangles count = 3

squares count = 2

Total shapes count = 7

(a) Ratio of count of triangles to count of circles in the box = triangles count / circles count

= 3 / 2

(b) Ratio of count of squares to all shapes in the box = squares count / total shapes count

= 2 / 7

(c) Ratio of count of circles to all shapes in the box = circles count / total shapes count

= 2 / 7


4. The distance travelled by Hamid and Akhtar in an hour is 9 km and 12 km, respectively. Find the ratio of the speed of Hamid to the speed of Akhtar.

Explanation:

Speed is equal to the distance travelled in an hour of time.

Hamid travelled in 1 hour = 9 km

Akhtar travelled in 1 hour = 12 km

Hamid speed will be= 9 km/hr

Akhtar speed will be = 12 km/hr

Ratio of Hamid speed to Akhtar speed = Hamid’s speed / Akhtar’s speed

= 9 / 12 (simplify)

= 3 / 4


5. Fill in the following blanks:

15 / 18 = __ / 6 = 10 / __ = __ / 30 [Are these equivalent ratios?]

Explanation:

⇨ 15 / 18 = (3 × 5) / (3 × 6) (simplify)

= 5 / 6

⇨ 5 / 6 = (2 × 5) / (2 × 6) (simplify)

= 10 / 12

⇨ 10 / 12 = (10 × 5/2) / (12 × 5/2)

= 25 / 30

So, 5, 12 and 25 are the respective answers.

These are equivalent ratios.


6. Find the ratio of the following:

(a) 81 to 108

(b) 98 to 63

(c) 33 km to 121 km

(d) 30 minutes to 45 minutes

Explanation:

(a) 81 / 108 (prime factorise)

= (3 × 3 × 3 × 3) / (3 × 3 × 3 × 2 × 2) (simplify)

= 3 / 4

(b) 98 / 63 (prime factorise)

= (7 × 7 × 2) / (7 × 9) (simplify)

= 14 / 9

(c) 33 / 121 (prime factorise)

= (11 × 3) / (11 × 11) (simplify)

= 3 / 11

(d) 30 / 45 (prime factorise)

= (5 × 2 × 3) / (5 × 3 × 3) (simplify)

= 2 / 3


7. Find the ratio of the following:

(a) 30 minutes to 1.5 hours

(b) 40 cm to 1.5 m

(c) 55 paise to ₹ 1

(d) 500 ml to 2 litres

Explanation:

(a) 30 minutes to 1.5 hours

Convert 1.5 hours to minutes

⇨ 1.5 hours = 1.5 × 60

⇨ 1.5 hours = 90 minutes

Ratio = 30 / 90

= (30 × 1) / (30×3) (simplify)

= 1 / 3

(b) 40 cm to 1.5 m

Convert 1.5 m to cm

⇨ 1.5 m = 1.5×100 cm

⇨ 1.5 m = 150 cm

Ratio = 40 / 150

= (10×4) / (10×15) (simplify)

= 4 / 15

(c) 55 paise to ₹ 1

Convert ₹ to paise

⇨ ₹ 1 = 100 paise

Ratio = 55 / 100 

= (5 × 11) / (5 × 20) (simplify)

= 11 / 20

(d) 500 ml to 2 litres

Convert litres to ml

⇨ 1 litre = 1000 ml

⇨ 2 litre = 2×1000 ml

⇨ 2 litre = 2000 ml

Ratio = 500 / 2000 

= (500×1) / (500×4) (simplify)

= 1 / 4


8. In a year, Seema earns ₹ 1,50,000 and saves ₹ 50,000. Find the ratio of

(a) Money that Seema earns to the money she saves

(b) Money that she saves to the money she spends.

Explanation:

Earned money of Seema = ₹ 150000

Saved money of Seema = ₹ 50000

So, Spent money of Seema = ₹ 150000 – ₹ 50000 

= ₹ 100000

(a) Ratio = earned money / saved money 

= 150000 / 50000 (simplify)

= 15 / 5 (simplify)

= 3 / 1

(b) Ratio = saved money / spent money 

= 50000 / 100000 (simplify)

= 5/10 (simplify)

= 1/2

 

9. There are 102 teachers in a school of 3300 students. Find the ratio of the number of teachers to the number of students.

Explanation:

It has been given that

teachers count = 102

students count = 3300

Ratio = teachers count / students count = 102 / 3300

= (17 × 2 × 3) / (550 × 2 × 3) (simplify)

= 17/550


10. In a college, out of 4320 students, 2300 are girls. Find the ratio of

(a) Number of girls to the total number of students.

(b) Number of boys to the number of girls.

(c) Number of boys to the total number of students.

Explanation:

It has been given that

girls count = 2300

Total students count = 4320

boys count = Total count – girls count

= 4320 – 2300

= 2020

(a) Ratio = girls count / total count 

= 2300 / 4320

= (115 × 5 × 2 × 2) / (216 × 5 × 2 × 2) (simplify)

= 115/216

(b) Ratio = boys count / girls count 

= 2020 / 2300

= (101 × 2 × 5 × 2) / (115 × 5 × 2 × 2) (simplify)

= 101/115

(c) Ratio = boys count / total count 

= 2020 / 4320

= (101 × 2 × 2 × 5) / (216 × 5 × 2 × 2) (simplify)

= 101/216


11. Out of 1800 students in a school, 750 opted for basketball, 800 opted for cricket, and the remaining opted for table tennis. If a student can opt for only one game, find the ratio of

(a) The number of students who opted for basketball to the number of students who opted for table tennis.

(b) The number of students who opted for cricket to the number of students opting for basketball.

(c) The number of students who opted for basketball to the total number of students.

Explanation:

It has been given that

Count of total students = 1800

Count of students who took cricket = 800

Count of students who took basketball = 750

Count of students who took table tennis = Total students – cricket taking students – basketball taking students

= 1800 – 800 – 750 

= 250

(a) Ratio = basketball players / table tennis player 

= 750/250 

= (3×250) / (1×250) (simplify)

= 3/1

(b) Ratio = cricket players / basketball players

= 800/750

= (16×50) / (15×50) (simplify) 

= 16 / 15

(c) Ratio = basketball players / total players

= 750 / 1800

= 75 / 180 

= (5×15) / (12×15) (simplify)

= 5 / 12


12. Cost of a dozen pens is ₹ 180, and the cost of 8 ball pens is ₹ 56. Find the ratio of the cost of a pen to the cost of a ball pen.

Explanation:

One dozen pens price is = ₹ 180

One pen price = ₹ 180/12

= ₹ 15

8 ball pens price is = ₹ 56

One ball pen price is = ₹ 56/8

= ₹ 7

Ratio = 15/7


13. Consider the statement: The ratio of breadth and length of a hall is 2:5. Complete the following table that shows some possible breadths and lengths of the hall.

Breadth of the hall (in metres)

10


40

Length of the hall (in metres)

25

50


Explanation:

Hall length/ Hall Breadth = 5/2

(i) l = 50 m

⇨ 50 / b = 5/2 (simplify)

⇨ b = (50×2)/5

⇨ b = 20

Hall Breadth = 20 m

(ii) b = 40 m

⇨ l / 40 = 5/2 (simplify)

⇨ l = (40×5)/2

⇨ l = 100

Hall length = 100 m 


14. Divide 20 pens between Sheela and Sangeeta in a ratio of 3:2.

Explanation:

Ratio = 3/2

Sum of numerator and denominator = 3+2=5

So, Sangeeta gets 2/5 of pens available and Sheela gets 3/5 of pens available so that their ratio remains 3/2.

Count of pens that Sangeeta gets = (2/5) × 20

= 40/5 (simplify)

= 8

Count of pens that Sheela gets = (3/5) × 20

= 60/5 (simplify)

= 12

So, Sheela get 12 pens and Sangeeta gets 8 pens.


15. Mother wants to divide ₹ 36 between her daughters Shreya and Bhoomika in the ratio of their ages. If the age of Shreya is 15 years and the age of Bhoomika is 12 years, find how much Shreya and Bhoomika will get.


Explanation:

Age ratio of Shreya and Bhoomika = Shreya age / Bhoomika age

= 15/12 (simplify)

= 5/4

Divide ₹ 36 in ratio of 5 and 4

Sum of numerator and denominator = 5+4 = 9

So, Shreya gets 5/9 of total money and Bhoomika gets 4/9 of total money.

Amount that Shreya gets = (5/9)×36

= 180/9 (simplify)

= 20

Amount that Bhoomika gets = (4/9)×36

= 144/9 (simplify)

= 16

So, Shreya gets ₹ 20 and Bhoomika gets ₹ 16.


16. Present age of the father is 42 years, and that of his son is 14 years. Find the ratio of

(a) Present age of the father to the present age of the son

(b) Age of the father to the age of the son, when the son was 12 years old

(c) Age of father after 10 years to the age of son after 10 years

(d) Age of father to the age of son when father was 30 years old

Explanation:

(a) father’s age now = 42 years

son’s age now = 14 years

Ratio = (14×3) / (14) (simplify)

= 42 / 14 

= 3/1

(b) 

Father’s age 2 years ago= 42–2=40

Ratio = 40/12 

= (10 × 4) / (3 × 4) (simplify)

= 10/3

(c) Father’s age after ten years = 42 + 10 = 52 years

Son’s age after 10 years = 14+10 = 24 years

Ratio = 52/24 

= (13 × 4) / (6 × 4) (simplify)

= 13/6

(d) Father’s age 12 years ago = 42 – 12 = 30 years

Son’s age 12 years ago = 14–12 =2 years

Ratio = 30/2 

= (15 × 2) / (2×1)

= 15/1


17. Determine if the following are in proportion.

(a) 15, 45, 40, 120

(b) 33, 121, 9, 96

(c) 24, 28, 36, 48

(d) 32, 48, 70, 210

(e) 4, 6, 8, 12

(f) 33, 44, 75, 100

Explanation:

(a) 15, 45, 40, 120

⇨ 15 / 45 

⇨ (1×15) / (3×15)

⇨ 1/3

⇨ 40 / 120 

⇨ (1×40)/(3×40)

⇨ 1/3

So, 40:120 = 15:45 

Yes, in proportion.

(b) 33, 121, 9, 96

⇨ 33 / 121 

⇨ (11×3)/(11×11)

⇨ 3/11

⇨ 9 / 96 

⇨ (3×3)/(3×32)

⇨ 3/32

So, 9: 96 ≠ 33:121 

Not, in proportion

(c) 24, 28, 36, 48

⇨ 24 / 28 

⇨ (4×6)/(4×7)

⇨ 6/7

⇨ 36 / 48 

⇨ (12×3)/(12×4)

⇨ 3/4

So, 36:48 ≠ 24:28 

Not, in proportion.

(d) 32, 48, 70, 210

⇨ 32 / 48

⇨ (16×2)/(16×3)

⇨ 2/3

⇨ 70 / 210

⇨ (70×1)/(70×3) 

⇨ 1/3

So, 70:210 ≠ 32:48

Not, in proportion.

(e) 4, 6, 8, 12

⇨ 4 / 6 

⇨ (2×2)/(2×3)

⇨ 2/3

⇨ 8 / 12 

⇨ (4×2)/(4×3)

⇨ 2/3

So, 8:12 = 4:6

Yes, in proportion

(f) 33, 44, 75, 100

⇨ 33/ 44 

⇨ (11×3)/(11×4)

⇨ 3/4

⇨ 75 / 100 

⇨ (25×3)/(25×4)

⇨ 3 / 4

So, 75:100 = 33:44

Yes, in proportion.


18. Write True (T) or False ( F ) against each of the following statements :

(a) 16:24 :: 20:30

(b) 21:6 :: 35:10

(c) 12:18 :: 28:12

(d) 8:9 :: 24:27

(e) 5.2:3.9 :: 3:4

(f) 0.9:0.36 :: 10:4

Explanation:

(a) 16:24 :: 20:30

⇨ 20/30 = 2 / 3

⇨ 16/24 = 2 / 3

Both are equal. So, True.

(b) 21:6 :: 35:10

⇨ 35/10 = 7 / 2

⇨ 21/6 = 7 / 2

Both are equal. So, True.

(c) 12:18 :: 28:12

28/12 = 7 / 3

12/18 = 2 / 3

Both are unequal. So, False.

(d) 8:9:: 24:27

⇨ 8/9 = 8/9

⇨ 24/27 = 8/9

Both are equal. So, True.

(e) 5.2:3.9 :: 3:4

⇨ 5.2 / 3.9 = 4 / 3

⇨ 3/4 = 3/4

Both are unequal. So, False.

(f) 0.9:0.36 :: 10:4

⇨ 0.9/0.36 = 10 / 4

⇨ 10/4 = 10/4

Both are equal. So, True.


19. Are the following statements true?

(a) 40 persons: 200 persons = ₹ 15 : ₹ 75

(b) 7.5 litres: 15 litres = 5 kg : 10 kg

(c) 99 kg: 45 kg = ₹ 44 : ₹ 20

(d) 32 m: 64 m = 6 sec : 12 sec

(e) 45 km : 60 km = 12 hours : 15 hours

Explanation:

(a) 40 persons : 200 persons = ₹ 15 : ₹ 75

⇨ 15/75 = 1/5

⇨ 40/200 = 1/5

Both are equal. So, True.

(b) 7.5 litres : 15 litres = 5 kg : 10 kg

⇨ 5/10 = 1/2

⇨ 7.5/15 = 1/2

Both are equal. So, True.

(c) 99 kg : 45 kg = ₹ 44 : ₹ 20

⇨ 44/20 = 11/5

⇨ 99/45 = 11/5

Both are equal. So, True.

(d) 32 m : 64 m = 6 sec : 12 sec

⇨ 6/12 = 1/2

⇨ 32/64 = 1/2

Both are equal. So, True.

(e) 45 km : 60 km = 12 hours : 15 hours

⇨ 12/15 = 4/5

⇨ 45/60 = 3/4

Both are unequal. So, False.


20. Determine if the following ratios form a proportion. Also, write the middle terms and extreme terms where the ratios form a proportion.

(a) 25 cm : 1 m and ₹ 40 : ₹ 160

(b) 39 litres : 65 litres and 6 bottles : 10 bottles

(c) 2 kg : 80 kg and 25 g : 625 g

(d) 200 mL : 2.5 litre and ₹ 4 : ₹ 50

Explanation:

(a) 25 cm : 1 m and ₹ 40 : ₹ 160

1 m = 100 cm

⇨ 40/160 = 1/4

⇨ 25/100 = 1/4

Both are equal. So, True.

Extreme terms- 25 & 160, Middle terms- 1 & 40

(b) 39 litres : 65 litres and 6 bottles : 10 bottles

⇨ 6/10 = 3/5

⇨ 39/65 = 3/5

Both are equal. So, True.

Extreme terms- 39 & 10, Middle terms- 65 & 6

(c) 2 kg : 80 kg and 25 g : 625 g

⇨ 25/625 = 1/25

⇨ 2/80 = 1/40

Both are unequal. So, False.

(d) 200 mL : 2.5 litre and ₹ 4 : ₹ 50

2.5 litre = 2500 ml

⇨ 4/50 = 2/25

⇨ 200/2500 = 2/25

Both are equal. So, True.

Extreme terms- 200 & 50, Middle terms- 2.5 & 4


21. If the cost of 7 m of cloth is ₹ 1470, find the cost of 5 m of cloth.

Explanation:

It has been given that

7 m of cloth has a price of = ₹ 1470

1 m cloth has a price of = (₹ 1470) / 7 = ₹ 210 (unitary method)

Then, 5 m cloth has a cost of = ₹ 210 × 5 = ₹ 1050 (unitary method)

So, 5 m of cloth has a price of ₹ 1050.


22. Ekta earns ₹ 3000 in 10 days. How much will she earn in 30 days?

Explanation:

It has been given that

In 10 days of time money that Ekta earned = ₹ 3000

So, By Unitary method

In one day she earned = ₹ 3000 / 10 = ₹ 300

Again, use unitary method

In 30 days of time she will get = ₹ 300 × 30 = ₹ 9000.

So, ₹ 9000 will be earned by Ekta in 30 days.


23. If it has rained 276 mm in the last 3 days, how many cm of rain will fall in one full week (7 days)? Assume that the rain continues to fall at the same rate.

Explanation:

In last 3 days the rain fall is = 276 mm

By unitary method

In one day rain fall = 276/3 = 92 mm

In a span of a week means 7 days the rain fall is = rain fall in one day × number of days

= 92×7

= 644 mm

= 64.4 cm (1 cm = 10 mm)


24. Cost of 5 kg of wheat is ₹ 91.50.

(a) What will be the cost of 8 kg of wheat?

(b) What quantity of wheat can be purchased for ₹ 183?

Explanation:

(a) A wheat bag of 5kg has a price = ₹ 91.50

By unitary method

A wheat bag of 1kg will have a price = ₹ 91.50/5 = ₹ 18.3

A wheat bag of 8kg will have a price = ₹ 18.3×8 = ₹ 146.40

(b) For a price of ₹91.50 quantity of wheat got = 5 kg

By unitary method

For a price of ₹ 1 quantity of wheat got = 5 / 91.50 (kg/₹)

So, for a price of ₹ 183 quantity of wheat got = 183×(5 / 91.50)

= 2×5

= 10 kg


25. The temperature dropped 150 C in the last 30 days. If the rate of temperature drop remains the same, how many degrees will the temperature drop in the next ten days?

Explanation:

In a span of 30 days temperature dropped = 150 C

By unitary method

In 1 day temperature drop = 15/30

= 15 / 30 (simplify)

= (1 / 2)0 C

= 0.5 0C

So, In a span of next 10 days temperature dropped = (0.5) × 10 = 50 C


26. Shaina pays ₹ 15000 as rent for 3 months. How much does she have to pay for a whole year if the rent per month remains the same?

Explanation:

In a span of 3 month rent given = ₹ 15000

By unitary method

In a span of 1 month rent given = ₹ 15000 / 3 = ₹ 5000

So, In a span of 1 year or 12 month rent given = (₹ 5000) × 12

= ₹ 60,000

So, in 1 year rent given is ₹ 60000.


27. Cost of 4 dozen bananas is ₹ 180. How many bananas can be purchased for ₹ 90?

Explanation:

At a price of ₹ 180 quantity of bananas = 4 dozens

Number of bananas = 12×4 = 48

By unitary method

At a price of ₹ 1 banana got = 48 / 180

So, At a price of ₹ 90 number of bananas got = 90×(48/180)

= 48/2

= 24 bananas

So, At a price of ₹ 90, number of bananas got is 24.


28. The weight of 72 books is 9 kg. What is the weight of 40 such books?

Explanation:

72 books have a weight of = 9 kg

By unitary method

One book weigh = 9 / 72 (simplify)

= 1 / 8 kg

So, 40 books will weigh = 40×(1/8) = 5 kg

So,40 books weigh 5 kg.


29. A truck requires 108 litres of diesel to cover a distance of 594 km. How much diesel will be required by the truck to cover a distance of 1650 km?

Explanation:

For a distance of 594 km, fuel required = 108 litres

By unitary method

For a distance of 1 km, fuel required = 108 / 594

= 2 / 11 litre

By unitary method

For a distance of 1650 km, fuel required = (2 / 11) × 1650

= 2×150

= 300 litres

 So, For a distance of 1650 km, diesel needed is 300 litres.


30. Raju purchases 10 pens for ₹ 150, and Manish buys 7 pens for ₹ 84. Can you say who got the pens cheaper?

Explanation:

For a price of ₹ 150, pens received by Raju = 10 pens

By unitary method

For Raju, one pen will cost = 150/10 = ₹ 15

For a price of ₹ 84, pens received Manish = 7 pens

By unitary method

For Manish, one pen will cost = 84 / 7 = ₹ 12

So, Manish bought pens at cheaper price than compare to Raju


31. Anish made 42 runs in 6 overs, and Anup made 63 runs in 7 overs. Who made more runs per over?

Explanation:

In 6 overs, Anish scored runs = 42

By unitary method

In 1 over, Anish scored runs = 42 / 6 = 7

In 7 overs, Anup scored runs = 63

By unitary method

In 1 over, Anup scored runs = 63/7 = 9

So, runs per over is more for Anup than compare to Anish.

Also Read: Chapter 13: Symmetry​