Class 8 Maths Solution | Lesson 4 Ratio and Proportion | Curriculum Development Centre (CDC)

Lesson 4 Ratio and Proportion

Exercise 4.1

(a) 4 hours and 6 hours

Ratio = 4/6 = 2/3

= 2 : 3

(b) 12 feet and 9 feet

Ratio = 12/9 = 4/3

= 4 : 3

(c) 250 grams and 2 kg

Converting 2 kg to grams: 2 kg = 2000 grams

Ratio = 250/2000 = 1/8

= 1 : 8

(d) 3 kg and 850 grams

Converting 3 kg to grams: 3 kg = 3000 grams

Ratio = 3000/850 = 600/170 = 60/17

= 60 : 17

(e) 2 hours and 45 minutes

Converting 2 hours to minutes: 2 hours = 120 minutes

Ratio = 120/45 = 8/3

= 8 : 3

(f) 40 minutes and 1 hour

Converting 1 hour to minutes: 1 hour = 60 minutes

Ratio = 40/60 = 2/3

= 2 : 3

(g) 2 liters and 850 ml

Converting 2 liters to milliliters: 2 liters = 2000 ml

Ratio = 2000/850 = 400/170 = 40/17

= 40 : 17

(h) Rs 5 and 90 paisa

Converting Rs 5 to paisa: 5 Rs = 500 paisa

Ratio = 500/90 = 50/9

= 50 : 9

2. (a) The ratio of the boys students and girls students in Kalika Secondary School is 5 : 8. If the number of girls in that school is 480, what is the number of boys?

Solution:

Here, the ratio of boys to girls = 5 : 8

Number of girls = 480

Let the number of boys be x.

Then,

x / 480 = 5 / 8

or, x = (480 × 5) / 8

= 2400 / 8

= 300

⸫ The number of boys is 300.

(b) The ratio of the number of teachers and students in Nepal Secondary School is 1:32. If there are 25 teachers, what is the number of students?

Solution:

Let the number of students be y.

The ratio of teachers to students is 1:32. So,

1 / 32 = 25 / y

Cross-multiply:

or, 1 x y = 25 x 32

or, y = 25 x 32 = 800

⸫ The number of students is 800.

3. (a) Divide Rs. 840 to John and Joseph in the ratio of 3:4. How much will each get?

Solution:

Let the shares of John and Joseph be 3x and 4x, respectively.

Now, we know that the total amount is Rs. 840:

3x + 4x = 840

or, 7x = 840

or, x = 840 / 7 = 120

John’s share = 3x = 3 x 120 = 360

Joseph’s share = 4x = 4 x 120 = 480

⸫ John will get Rs. 360, and Joseph will get Rs. 480.

(b) Two booksellers buy the same type of 50 dozen exercise books from the wholesale shop. If they divided the exercise books in the ratio of 5:7, how many will each receive?

Solution:

Let the shares of the two booksellers be 5x and 7x, respectively.

The total number of exercise books = 50 dozen = 50 x 12 = 600 books.

Now, the total shares = 5x + 7x = 600, so:

12x = 600

or, x = 600 / 12 = 50

The first bookseller will receive 5x = 5 x 50 = 250 books.

The second bookseller will receive 7x = 7 x 50 = 350 books.

⸫ The first bookseller will receive 250 books, and the second bookseller will receive 350 books.

4. (a) A map is drawn in a scale of 1 : 2,000. The distance between two places is 8 cm. What is the real distance between the two places?

Solution:

Scale of the map = 1 : 2,000

Distance on the map = 8 cm

The real distance = 8 × 2,000 = 16,000 cm

Converting cm to meters: 16,000 cm = 160 meters

⸫ The real distance is 160 meters.

4. (b) A map is drawn in a scale of 1 : 4,000. The distance between two places is 5 cm. What is the real distance between the two places?

Solution:

Scale of the map = 1 : 4,000

Distance on the map = 5 cm

The real distance = 5 × 4,000 = 20,000 cm

Converting cm to meters: 20,000 cm = 200 meters

⸫ The real distance is 200 meters.

5. (a) The ratio of the present age of Rita and Nagma is 4:5. If the ratio of their ages was 3:4 two years ago, what are their present ages?

Solution:

Let the present ages of Rita and Nagma be 4x and 5x, respectively.

Two years ago, their ages were (4x - 2) and (5x - 2), and the ratio was 3:4.

So, (4x - 2) / (5x - 2) = 3 / 4

Cross-multiply:

4(4x - 2) = 3(5x - 2)

or, 16x - 8 = 15x - 6

or, x = 2

Rita's age = 4x = 4 × 2 = 8 years

Nagma's age = 5x = 5 × 2 = 10 years

⸫ Their present ages are 8 years and 10 years.

5. (b) Two brothers spent in the ratio of 4:5 on the occasion of the birthday of a friend. If they had added Rs. 10 to each expense, the ratio of their expenses becomes 5:6. Find each of their expenses.

Solution:

Let the expenses of the two brothers be 4x and 5x.

After adding Rs. 10 to each expense, the new expenses are (4x + 10) and (5x + 10), and the ratio is 5:6.

(4x + 10) / (5x + 10) = 5 / 6

Cross-multiply:

6(4x + 10) = 5(5x + 10)

or, 24x + 60 = 25x + 50

or, x = 10

First brother's expense = 4x = 4 × 10 = Rs. 40

Second brother's expense = 5x = 5 × 10 = Rs. 50

⸫ Their expenses are Rs. 40 and Rs. 50.

5. (c) Two numbers are in the ratio of 1:3. If 5 is added to both numbers, the ratio becomes 1:2. Find the numbers.

Solution:

Let the numbers be x and 3x.

After adding 5 to both, the new ratio becomes 1:2.

(x + 5) / (3x + 5) = 1 / 2

Cross-multiply:

2(x + 5) = 1(3x + 5)

or, 2x + 10 = 3x + 5

or, x = 5

The numbers are x = 5 and 3x = 15.

⸫ The numbers are 5 and 15.

5. (d) The ratio of the present age of Ali and Amir is 3:4. If their age ratio 3 years ago was 2:3, find their present ages.

Solution:

Let the present ages of Ali and Amir be 3x and 4x, respectively.

Three years ago, their ages were (3x - 3) and (4x - 3), and the ratio was 2:3.

(3x - 3) / (4x - 3) = 2 / 3

Cross-multiply:

3(3x - 3) = 2(4x - 3)

or, 9x - 9 = 8x - 6

or, x = 3

Ali's age = 3x = 3 × 3 = 9 years

Amir's age = 4x = 4 × 3 = 12 years

⸫ Their present ages are 9 years and 12 years.

7. (a) If B collects twice as much as A and C collects three times as much as B, Rs 98,460 is collected in total. How much amount has each of them collected?

Solution:

Let the amount collected by A be x. Then, the amount collected by B is 2x, and the amount collected by C is 3(2x) = 6x.

The total amount collected = Rs 98,460, so:

x + 2x + 6x = 98,460

or, 9x = 98,460

or, x = 98,460 / 9 = 10,940

Now, A’s share = x = Rs 10,940.

B’s share = 2x = 2 × 10,940 = Rs 21,880.

C’s share = 6x = 6 × 10,940 = Rs 65,640.

⸫ A will collect Rs 10,940, B will collect Rs 21,880, and C will collect Rs 65,640.

7. (b) If Krishna collects twice as much as Ram and Hari collects three times as much as Krishna so that Rs 16,200 is collected in total, how much has each of them collected?

Solution:

Let the amount collected by Ram be x. Then, Krishna collects 2x, and Hari collects 3(2x) = 6x.

The total amount collected = Rs 16,200, so:

x + 2x + 6x = 16,200

or, 9x = 16,200

or, x = 16,200 / 9 = 1,800

Now, Ram’s share = x = Rs 1,800.

Krishna’s share = 2x = 2 × 1,800 = Rs 3,600.

Hari’s share = 6x = 6 × 1,800 = Rs 10,800.

⸫ Ram will collect Rs 1,800, Krishna will collect Rs 3,600, and Hari will collect Rs 10,800.

8. (a) The ratio of the angles of a triangle is 1:1:2. Find the angles of the triangle.

Solution:

Let the angles of the triangle be x, x, and 2x.

The sum of the angles in a triangle is always 180°.

x + x + 2x = 180

or, 4x = 180

or, x = 180 / 4 = 45°

So, the angles of the triangle are 45°, 45°, and 90°.

⸫ The angles of the triangle are 45°, 45°, and 90°.

8. (b) The ratio of the angles of a triangle is 2:3:4. Find the angles of the triangle.

Solution:

Let the angles of the triangle be 2x, 3x, and 4x.

The sum of the angles in a triangle is always 180°.

2x + 3x + 4x = 180

or, 9x = 180

or, x = 180 / 9 = 20°

So, the angles of the triangle are:

2x = 2 × 20 = 40°

3x = 3 × 20 = 60°

4x = 4 × 20 = 80°

⸫ The angles of the triangle are 40°, 60°, and 80°.