#### Ratio and Proportion

Exercise 12.1

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

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

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

Solution

- a) Number of girls = 20

Number of boys = 15

Total number of students in the class = 35

Number of girls : number of boys = 20 : 15

= 4 : 3

- b) Number of girls : total number of students in the class = 20 : 35 == 4 : 7

- Out of 30 students in a class, 6 like football, 12 like cricket and remaining like tennis.

Find the ratio of

(a) Number of students liking football to number of students liking tennis.

(b) Number of students liking cricket to total number of students.

Solution

Total number of students in the class = 30

Number of students who like football = 6

Number of students who like cricket = 12

Number of students who like tennis = 30 – (6 + 12) = 12

- a) Number of students who like football : number of students who like tennis = 6 : 12 = 1 : 2

- b) Number of students who like cricket : total number of students = 12 : 30 = 2 : 5

- 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.

Solution

Number of triangles = 3

Number of circles = 2

Number of squares = 2

Total number of figures inside the rectangle = 7

- a) Number of triangles to the number of circles inside the rectangle = 3 : 2

(b) Number of squares to all the figures inside the rectangle = 2 : 7

(c) Number of circles to all the figures inside the rectangle = 2 : 7

- Distances travelled by Hamid and Akhtar in an hour are 9 km and 12 km. Find the ratio of speed of Hamid to the speed of Akhtar.

Solution

Distance travelled by Hamid in 1 hour = 9 km

Speed of Hamid = distance/time = 9 km/hr

Distance travelled by Akhtar in 1 hour = 12 km

Speed of Hamid = distance/time = 12 km/hr

Ratio of speed of Hamid : Akhtar = 9 : 12 = 3 : 4

- Fill in the following blanks:

[Are these equivalent ratios?]

Solution

So, we have

Yes, the ratios are equivalent.

- 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

Solution

- a) 81 : 108 == 3 : 4

- b) 98 : 63 == 14 : 9

- c) 33 km : 121 km == 3 : 11

- d) 30 min : 45 min == 2 : 3

- Find the ratio of the following:

(a) 30 minutes to 1.5 hours (b) 40 cm to 1.5 m (c) 55 paise to Re 1 (d) 500 ml to 2 litres

Solution

The two quantities are not in the same unit. So, we convert to the same unit.

- a) 30 min == 0.5 hours

Therefore, the required ratio is 0.5 : 1.5

=

- b) 40 cm to 1.5 m

Therefore, the ratio is 0.4 : 1.5

- c) 55 paise to Re. 1

Re 1 = 100 paise

Therefore, the ratio is 55 : 100

- d) 500 ml to 2 l

1 litre = 1000 ml

2 litre = 2000 ml

So, the ratio is 500 : 2000

- In a year, Seema earns Rs 1,50,000 and saves Rs 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.

Solution

Amount earned by Seema = Rs 1,50,000

Amount saved = Rs. 50,000

Amount spent = 1,50,000 – 50,000 = 1,00,000

- a) Money Seema earns : money she saves =1,50,000 : 50,000

- b) Money saved : money spent = 50,000 : 1,00,000

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

Solution

Number of teachers in the school = 102

Number of students in the school = 3300

Number of teachers : number of students = 102 : 3300

- 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.

Solution

Total number of students in the school = 4320

Number of girls = 2300

Number of boys = 4320 – 2300 = 2020

- a) number of girls : total number of students = 2300 : 4320

- b) number of boys : number of girls = 2020 : 2300

- c) number of boys : total number of students = 2020 : 4320

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

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

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

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

Solution

Total number of students in a school = 1800

Students who opted for basketball = 750

Students who opted for cricket = 800

Students who opted for table tennis = 1800 – (750 + 800) = 1800 – 1550 = 250

- a) Students who opted basketball : Students who opted for table tennis = 750 : 250

- b) students who opted cricket : basketball = 800 : 750

- c) students who opted basketball : total number of students = 750 : 1800

- Cost of a dozen pens is Rs 180 and cost of 8 ball pens is Rs 56. Find the ratio of the cost of a pen to the cost of a ball pen.

Solution

Cost of 12 pens(a dozen) is = Rs. 180

Cost of 1 pen = 180 ÷ 12 = 15 Rs

Cost of 8 ball pens = Rs. 56

Cost of 1 ball pen = 56 ÷ 8 = Rs. 7

Cost of a pen : Cost of a ball pen = 15 : 7

- Consider the statement: 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.

Solution

Breadth of a hall : length of the hall = 2 : 5

To fill the missing numbers, we find the equivalent ratios.

Since 5 × 10 = 50, we find 2 × 10 = 20.

That is,

20 : 50 is the second ratio.

40 : 100 is the third ratio.

- Divide 20 pens between Sheela and Sangeeta in the ratio of 3 : 2.

Solution

The two parts are 3 and 2. Sum of the parts is 5.

So, Sheela gets 3 parts and Sangeetha gets 2 parts out of every 5 parts.

OR

Sheela gets 3/5 of the total pens and Sangeeta gets 2/5 of the total pens.

Number of pens Sheela gets =

Number of pens Sangeeta gets =

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

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

Solution

Ratio of their ages = 15 : 12 = 5 : 4

Mother divides Rs. 36 in the ratio of their ages 5 : 4

So, Shreya gets 5/9 of the total amount to be divided and Bhoomika gets 4/9 of the total amount to be divided.

Amount Shreya gets =

Amount Bhoomika gets =

Shreya gets Rs. 20 and Bhoomika gets Rs. 16.

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

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

(b) Age of the father to the age of son, when 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.

Solution

Present age of father = 42 years

Present age of his son = 14 years

- a) Age of father : age of son = 42 : 14

- b) When the son was 12 = (14 – 2) years old, the father would have been 40 = (42 – 2) years old.

Age of father : son = 40 : 12

- c) Age of father after 10 years = Present age + 10 = 42 + 10 = 52 years

Age of son after 10 years = 14 + 10 = 24 years

Age of father : son = 52 : 24

- d) When was the father 30 years old?

Since, 42 – 30 = 12, we know that 12 years back, the father was 30 years old.

What was the son’s age 12 years back?

His age 12 years back was = present age – 12 = 2 years

Ratio of father : son = 30 : 2 = 15 : 1

Exercise 12.2

- 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

*If two ratios are equal, we say that they are in proportion.*

- a) Ratio of 15 and 45 =

Ratio of 40 and 120 =

Since 15 : 45 and 40 : 120 are equal, we say that they are in proportion.

15 : 45 :: 40 : 120

- b) Ratio of 33 and 121 =

Ratio of 9 and 96 =

Since 33 : 121 and 9 : 96 are unequal, we say that they are not in proportion.

- c)Ratio of 24 and 28 =

Ratio of 36 and 48 =

Since 24 : 28 and 36 : 48 are unequal, we say that they are not in proportion.

- d)Ratio of 32 and 48 =

Ratio of 70 and 210 =

Since 32 : 48 and 70 : 210 are unequal, we say that they are not in proportion.

- e)Ratio of 4 and 6 =

Ratio of 8 and 12 =

Since 4 : 6 and 8 : 12 are equal, we say that they are in proportion.

4 : 6 :: 8 : 12

- f)Ratio of 33 and 44 =

Ratio of 75 and 100 =

Since 33 : 44 and 75 : 100 are equal, we say that they are in proportion.

33 : 44 :: 75 : 100

- 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

Solution

(a) 16 : 24 :: 20 : 30

The ratios are equal and hence True.

(b) 21: 6 :: 35 : 10

The ratios are equal and hence True.

(c) 12 : 18 :: 28 : 12

The ratios are unequal. Hence False

(d) 8 : 9 :: 24 : 27

The ratios are equal, hence True.

(e) 5.2 : 3.9 :: 3 : 4

The ratios are unequal, hence False

(f) 0.9 : 0.36 :: 10 : 4

The ratios are equal, hence True

- Are the following statements true?

- a) 40 persons : 200 persons = Rs 15 : Rs 75

True

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

True

(c) 99 kg : 45 kg = Rs 44 : Rs 20

True

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

True

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

False

- 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 Rs 40 : Rs 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 Rs 4 : Rs 50

Solution

(a) 25 cm : 1 m and Rs 40 : Rs 160

Converting to the same unit, 1 m = 100 cm.

So, the ratios are equal and hence are in proportion.

25 cm : 100 cm :: Rs. 40 : Rs. 160

Middle terms are 100 cm and Rs. 40.

Extremes are 25 cm and Rs. 160

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

So, the ratios are equal and hence 39 *l* : 65 *l* :: 6 bottles : 10 bottles.

Middle terms are 65 litres and 6 bottles, Extremes are 39 litres and 10 bottles.

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

The ratios are unequal and hence they are not in proportion.

(d) 200 ml : 2.5 litre and Rs 4 : Rs 50

The ratios are equal and hence they are in proportion.

200 : 2.5 :: 4 : 50

Middle terms are 2.5 litre and Rs. 4

Extremes are 200 ml and Rs. 50

Exercise 12.3

- If the cost of 7 m of cloth is Rs 294, find the cost of 5 m of cloth.

Solution

Cost of 7 m of cloth = Rs 294

So, the cost of 1 m of cloth = 294 ÷ 7 = Rs 42

Cost of 5 m of cloth = 42 × 5 = Rs 210

- Ekta earns Rs 1500 in 10 days. How much will she earn in 30 days?

Solution

Amount Ekta earns in 10 days = Rs 1500

So, amount earned in 1 day = Rs 1500 ÷ 10 = Rs. 150

Amount she will earn in 30 days = Rs. 150 × 30 = Rs. 4500

- 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.

Solution

Measure of rain in 3 days = 276 mm = 27.6 cm

So, measure of rain in 1 day = 27.6 ÷ 3 = 9.2 cm

We know that one week has 7 days.

So, measure of rain in 7 days = 9.2 × 7 = 64.4 cm

- Cost of 5 kg of wheat is Rs 30.50.

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

(b) What quantity of wheat can be purchased in Rs 61?

Solution

Cost of 5 kg of wheat = Rs. 30.50

Cost of 1 kg = Rs. 30.50 ÷ 5 = Rs. 6.10

- a) Cost of 8 kg of wheat = 8 × Rs. 6.10 = Rs. 48.80

- b) Amount of wheat purchased for Rs. 6.10 = 1 kg

Amount of wheat purchased for Rs 61 = 61 ÷ 6.10 = 6100 ÷ 610 = 10 kg

- The temperature dropped 15 degree celsius 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?

Solution

Temperature drop in 30 days = 15°C

Temperature drop in 1 day = 15 ÷ 30 = (½)°C

Temperature drop in 10 days = (½)°C × 10 = 5°C

- Shaina pays Rs 7500 as rent for 3 months. How much does she has to pay for a whole year, if the rent per month remains same?

Solution

Rent paid for 3 months = Rs 7500

Rent paid per month = 7500 ÷ 3 = Rs 2500

In a year, there are 12 months.

So, the rent paid for a year = Rs 2500 × 12 = Rs 30,000

- Cost of 4 dozens bananas is Rs 60. How many bananas can be purchased for Rs 12.50?

Solution

1 dozen = 12

So, 4 dozen = 48

The cost of 48 bananas = Rs. 60

Cost of 1 banana= 60 ÷ 48 = 5 ÷ 4 = Rs 1.25

For Rs. 1.25, number of bananas that can be purchased = 1

For Rs. 12.50, number of bananas that can be purchased = 12.50 × 1 ÷ 1.25 = 12.50 ÷ 1.25

= 1250 ÷ 125 = 10

So, 10 bananas can be purchased.

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

Solution

Weight og 72 books = 9 kg

Weight of 1 book = 9 ÷ 72 = 1/8 Kg

Weight of 40 books = 40 × 1/8 = 5 kg

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

Solution

Amount of Diesel required for 594 km = 108 *l*

Amount of Diesel required for 1 km = 108 ÷ 594

Amount of diesel required for 1650 km =

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

Solution

Amount paid by Raju for 10 pens = Rs. 150

Amount paid for 1 pen = 150 ÷ 10 = Rs 15

Amount paid by Manish for 7 pens = Rs. 84

Amount paid for 1 pen = 84 ÷ 7 = Rs 12

Raju paid Rs 15 for 1 pen but Manish paid Rs. 12

So, Manish got the pens cheaper.

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

Solution

Runs made by Anish in 6 overs = 42

Runs made in 1 over = 42 ÷ 6 = 7

Runs made by Anup in 7 overs = 63

Runs made in 1 over = 63 ÷ 7 = 9

In one over, Anup made 9 runs but Anish made 7 runs. So, Anup made more runs in an over than Anish.