Exercise 3.5

Exercise 3.5

Miscellaneous Practice Problems

1. The maximum speed of some of the animals are given below : the Elephant=20 km/h; the Lion=80 km/h; the Cheetah=100 km/h

Find the following ratios of their speeds in simplified form and find which ratio is the least?.

(i) the Elephant and the Lion

(ii) the Lion and the Cheetah

(iii) the Elephant and the Cheetah

Solution

(i) Elephant: Lion = 20 : 80

Fraction form =  20/80 ÷ 20/20 = 1/4

Elephant : Lion =  1 : 4

(ii) Lion : Cheetah = 80 : 100

Fraction form =  80/100 ÷ 20/20 = 4/5

Lion : Cheetah = 4 : 5

(iii) Elephant : Cheetah = 20 : 100

Fraction form =  20/100 ÷ 20/20 = 1/5

Elephant : Cheetah = 4 : 5

Comparing 1/4 , 4/5, 1/5 ⇒ 1/5, < 1/4, < 4/5

Ratio of elephant to cheetah is least.

2. A particular high school has 1500 students 50 teachers and 5 administrators. If the school grows to 1800 students and the ratios are maintained, then find the number of teachers and administrators.

Solution:

The ratios are in proportion

Teachers ratio in proportion is

1500 : 50 : : 1800 : x

Extremes = 1500 x

Means = 50 × 1800

Means =  Extrems

50 × 1800 = 1500x

x = [50 × 1800] /1500 = 60

No.of teachers = 60

Admistratives ratio is

1500 : 5 ::1800 : x

Extremes = 1500x

Means = 5 × 1800

1500x = 5 × 1800

x = [5 × 1800] /1500 = 6

No. of Admistratives = 6

So 60 teachers and 6 administratives.

3. I have a box which has 3 green, 9 blue, 4 yellow, 8 orange coloured cubes in it.

(a) What is the ratio of orange to yellow cubes?

(b) What is the ratio of green to blue cubes?

(c) How many different ratios can be formed, when you compare each colour to anyone of the other colours?

Solution:

A box contains cubes in which = 3 green

9 blue

4 yellow

8 orange

(a) orange : yellow = 8 : 4

simplified form = 8/4 ÷ 4/4 = 2/1

orange : yellow = 2:1

(b) green : blue = 3 : 9

simplified form = 3/9 ÷ 3/3 = 1/3

green : blue = 1 : 3

(c) No. of ratios (np_r) = n! / ( n – r ) !

n = 4 r = 2

4p₂ = [ 1 × 2 × 3 × 4 ] / [ ( 4 – 2 ) ! ] = [ 1 × 2 × 3 × 4 ] / [1 × 2] = 12

No. of ratios = 12.

4. A gets double of what B gets and B gets double of what C gets. Find A : B and B : C and verify whether the result is in proportion or not.

Solution: If B = 1     A = 2

A : B = 2 : 1

If  C = 1       B = 2

B : C = 2 : 1

A : B : : B : C

i.e., 2 : 1 : : 2 : 1

2 = 2

They are in proportion.

5. The ingredients required for the preparation of Ragi Kali, a healthy dish of Tamilnadu is given below.

(a) If one cup of ragi flour is used then, what would be the amount of raw rice required?

(b) If 16 cups of water is used, then how much of ragi flour should be used?

(c) Which of these ingredients cannot be expressed as a ratio? Why?

Solution:

(a) Ragi flour : raw rice = 4:1

If ragi flour = 1 cup

The ratio becomes  4/1 × 4/4 = 1 / 1/4

Ragi flour : raw rice = 1 : 1/4

Raw rice = 1/4 cup

(b) Water : ragi flour = 8 : 4

for 16 cups of water

8/4 × 2 /2 = 16/8  

water : ragi flour = 16 : 8

for 16 cups we need 8 cup Ragi flour.

(c) sesame oil and oil cannot be expressed as a ratio because both of them as different units.

Challenging Problems

6. Antony brushes his teeth in the morning and night on all days in a week. Shabeen brushes her teeth only in the morning. What is the ratio of the number of times they brush their teeth in a week?

Solution:

No. of times for Antony = 2

No. of times for Shabeen =1

for one day    Antony : Shabeen = 2 : 1

for a week Antony : Shabeen the smallest ratio is same.

Ratio is 2 : 1

7. Thirumagal's mother wears a bracelet made of 35 red beads and 30 blue beads. Thirumalgal wants to make smaller bracelets using the same two coloured beads in the same ratio. In how many different ways can she make the bracelets?

Solution: Thirumagal’s mother bracelet ratio = 35 : 30 ÷ 5 gives smallest ratio = 7 : 6

Thirumagal make smaller bracelets in different ways. The ratios are:

7/6 × 1/1  = 7/6 ; 7/6 × 2/2  = 14/12 ; 7/6 × 3/3  = 21/18 ; 7/6 × 4/4  = 28/24

The ratios are 7 : 6 ; 14 : 12 ; 21 : 18 ; 28 : 24

8. Team A wins 26 matches out of 52 matches. Team B wins three-fourth of 52 matches played. Which team has a better winning record?

Solution: Team A wins = 26 out of 52 = 26 : 52

Team B wins = 3/4 × 52 = 39 out of 52

Fractional form = 26/52 < 39/52

Team  B has better record.

9. In a school excursion, 6 teachers and 12 students from 6th standard and 9 teachers and 27 students from 7th standard, 4 teachers and 16 students from 8th standard took part. Which class has the least teacher to student ratio?

Solution: Teacher : Student

Class 6 = 6 : 12 ; Class 7 = 9 : 27 ; Class 8 = 4 : 16

Class 6 = 6/12 ÷ 6/6 = 1/2 ; Class 7 = 9/27 ÷ 9/9 = 1/3 ; Class 8 = 4/16 ÷ 4/4 = 1/4 ;

Teacher : Student

1/2 > 1/3 > 1/4 

Class 6 ⇒ 1:2 ; Class 7 ⇒ 1:3 ; Class 8 ⇒ 1:4

The standard 8 is the least ratio. 

10. Fill the boxes using any set of suitable numbers 6 :  : :  : 15.

Answer: 6:2 :: 45:15

11. From your school diary, write the ratio of the number of holidays to the number of working days in the current academic year.

Answer: = 29 : 44 

12. If the ratio of Green, Yellow and Black balls in a bag is 4 : 3 : 5, then

(a) Which is the most likely ball that you can choose from the bag?

Answer: Black balls 

(b) How many balls in total are there in the bag if you have 40 black balls in it?

Answer: 96 balls 

(c) Find the number of green and yellow balls in the bag.

Answer: green=32 balls, yellow = 24 balls

Answers:

Exercise 3.5

1) (i) 1 : 4 (ii) 4 : 5 (iii) 1 : 5 (iv) Ratio of elephant to cheetah is least

2) 60 teachers and 6 administrators

3) (i) 2 : 1 (ii) 1 : 3 (iii) 12 ratios

4) A : B = 2 : 1, B : C = 2 : 1; They are in proportion.

5) (a) 1/4 cup (b) 8 cups

(c) Ragi flour, Raw rice and water are in one unit, Sesame oil and salt are in different units. these different units cannot be compared and cannot be expressed as a ratio.

6) 2 : 1

7) There are four different ways.

8) Team B has better record

9) The standard 8 is the least ratio

10) The four different answers are : 1 and 90; 2 and 45; 30 and 3; 5 and 18; 6 and 15

11) 29:44

12) (a) Black balls (b) 96 balls (c) 32 balls, 24 balls