Subject: Compulsory Maths

A ratio is a comparison of two or more numbers that are usually of the same type or measurement. If the numbers have different units, it is important to convert the units to be the same before doing any calculations.

The ratio is the method to show the relationship between two numbers or two quantities which indicate the number of times the first number contains the second. The ratio between two quantities is obtained by dividing the first quantity by the second. For example: the ratio between Rs 15 and Rs 30 = \(\frac{15}{30}\) = \(\frac{1}{2}\) = 1:2

There are three ways to write a ratio:

- As fraction = \(\frac{1}{2}\) ( 1 upon 2 )
- With a colon ( : ), 2 : 5 ( 2 is to 5)
- With the word " to", ( 2 to 5)

- A ratio doesn't contain any unit as it is pure number.
- While finding the ratio between two quantities, both quantities should be of the same unit. For example, the ratio between 20 cm and 3 m = ratio between 20 cm and 300 cm = \(\frac{20}{300}\) = \(\frac{1}{15}\) = 1:15.
- A ratio remains unchanged if both of its terms be multiplied or divided by the same number.

For example:

\(\frac{1}{3}\) = \(\frac{1}{3}\) x \(\frac{4}{4}\) = \(\frac{4}{12}\) = \(\frac{1}{3}\) - A ratio should always be expressed in its lowest terms.

For example;

\(\frac{20}{32}\) = \(\frac{20}{32}\) ÷ \(\frac{4}{4}\) = \(\frac{5}{8}\)

Let's divide Rs 450 among three persons in the ratio 2 : 3: 4

Since 2 + 3 + 4 =9

- Renu's share = \(\frac{2}{9}\) x Rs 450 = Rs 100
- Barsha's share = \(\frac{3}{9}\) x Rs 450 =Rs 150
- Sabina's share = \(\frac{4}{9}\) x Rs 450 =Rs 200

- We can use ratios to scale drawings up or down (by multiplying or dividing).
- The trick with ratios is to always multiply or divide the numbers by the same value.
- A ratio says how much of one thing there is compared to another thing.

- It includes every relationship which established among the people.
- There can be more than one community in a society. Community smaller than society.
- It is a network of social relationships which cannot see or touched.
- common interests and common objectives are not necessary for society.

Divide Rs 240 between Rita and Renu in the ratio of 3 : 5

Solution:

Since 3 : 5 = 8

Rita's share = \(\frac{3}{8}\) \(\times\)Rs 240 = Rs 90

Renu"s share = \(\frac{5}{8}\)\(\times\) Rs 240 = Rs 150

What is the ratio of 75 paisa to Rs 3.

Solution:

Here, Rs 3 = 300 paisa

Ratio of 75 paisa to Rs 3 = \(\frac{75}{300}\) = \(\frac{1}{4}\) = 1 : 4

Divide Rs.315 among Sushma, Renu and Anisha in the ratio of 2: 3: 4

Solution:

Since, 2 + 3 + 4 = 9

Sushma's share = \(\frac{2}{9}\)\(\times\) Rs 315 = Rs 70

Renu's share = \(\frac{3}{9}\)\(\times\) Rs 315 = Rs 105

Anisha's share = \(\frac{4}{9}\)\(\times\) Rs 315 = Rs 140

Hence, the required shares are Rs 70, Rs 105 and Rs 140

Two numbers are in the ratio 2 : 3, if 3 is added to both, the ratio will be 5 : 7. Find the numbers.

Solution:

Let the numbers be 2x and 3x.

According to the question,

\(\frac{2x + 3}{3x + 3}\) = \(\frac{5}{7}\)

or, 15x + 15 = 14x + 21

or, 15x - 14x = 21 - 15

or, x = 6

Now,

2x = 2\(\times\)6 = 12 and

3x =3\(\times\) 6 = 18

Hence, the required numbers are 12 and 18.

Two numbers are in the ratio 4 : 5, if 5 is subtracted from both, the ratio will be 3 : 4. Find the numbers.

Solution:

Let the numbers be 4x and 5x.

By question,

\(\frac{4x - 5}{5x - 5}\) = \(\frac{3}{4}\)

or, 16x - 20 = 15x -15

or, 16x - 15x = -15 + 20

or, x = 5

Now,

4x = 4\(\times\)5 = 20 and

5x = 5\(\times\)5 = 25

Hence, the required numbers are 20 and 25.

Find the ratio of 5 hrs and 10 hrs.

Solution:

Given information,

5 hrs and 10 hrs.

First quantity = 5 hrs

Second quantity = 10 hrs

Ratio = \(\frac{5hrs}{10hrs}\)

= \(\frac{ 1}{2}\)

= 1 : 2

Find the ratio of 20 cm and 25 cm.

Solution:

Given information:

20 cm and 25 cm.

First quantity = 20 cm

Second quantity = 25 cm

Two quantities are of same units.

Ratio = \(\frac{20cm}{25cm}\)

= \(\frac{4}{5}\)

= 4 : 5

Find the ratio of Rs 75 and 750 paisa.

Solution:

Given information:

Rs 75 and 750 paisa.

First quantity = Rs 75

Second quantity = 750 paisa

Since two quantities are not of same unit. So, convert Rs 75 into paisa. = Rs 75\(\times\)100

= 7500 paisa

Now, both quantities are of same units.

Ratio = \(\frac{7500 paisa}{750 paisa}\)

= \(\frac{10}{1}\)

= 10 : 1

Find the ratio of 750 grams and 1.5 kilograms.

Solution:

First quantity = 750 grams

Second quantity = 1.5 kg

Sine both quantities are not of same units. So, 1.5 kg = 1.5\(\times\) 1000 = 1500 grams

Now,

Ratio = \(\frac{750 grams}{1500 grams}\)

= \(\frac{1}{2}\)

= 1 : 2

Find the ratio of 3ft and 9ft.

Solution:

Given information:

3 ft and 9 ft.

First quantity = 3 ft

Second quantity = 9 ft

Since, both quantities are of same units.

Ratio = \(\frac{3 ft}{9ft}\)

= \(\frac{1}{3}\)

= 1 : 3

Divide Rs 500 between Saroj and Renu in the ratio of 3 : 5

Solution:

Since, 5 + 3 = 8

Saroj's share = \(\frac{3}{8}\)\(\times\) Rs 500 = Rs 187.5

Renu's share = \(\frac{5}{8}\)\(\times\)Rs 500 = Rs 312.5

Divide Rs 296 between two persons in the ratio of 3 : 5

Solution:

Since, 3 + 5 = 8

First's person = \(\frac{3}{8}\)\(\times\) Rs 296 = Rs 111

Second's person = \(\frac{5}{8}\)\(\times\) Rs 296 = Rs 185

What is the ratio of 90 paisa to Rs 9.

Solution:

Here, Rs 9 = 900 paisa

Ratio of 90 paisa to Rs 9

Rs 9 = 9 \(\times\) 100 paisa = 900

= \(\frac{90}{900}\)

= \(\frac{1}{10}\) = 1 : 10

The Ratio of the number of Secondary schools and the students are 1 : 32 . If there are 25 schools then how many are students ?

Solution:

The Ratio of Secondary school and students = 1 : 32

Number of schools = 25

Number of Students = ?

Now,

Ratio = 1 : 32

or, \(\frac{Teacher}{Student}\) = \(\frac{1}{32}\)

or, \(\frac{25}{Student}\) = \(\frac{1}{32}\)

or, Student = 25\(\times\) 32

\(\therefore\) Student = 800

The Ratio of the Food and Education of a family is 4 : 5. If they spend Rs 6, 750 in Education, find the amount in Food?

Solution:

The Ratio of Food and Education of a family = 4 : 5

Monthly expenditure in Education = Rd 6750

Expenditure in Food = ?

Here, \(\frac{expenditure in food}{expenditure in education}\) = \(\frac{4}{5}\)

or, \(\frac{expenditure in food}{6750}\) = \(\frac{4}{5}\)

or, expenditure in food = \(\frac{4}{5}\) x 6750

\(\therefore\) expenditure in food = Rs 5400

The Ratio of the number of boys and the number of girls in a school of 450 students is 4 : 5. When some new boys and girls are admitted, the number of boys increase by 25 and the ratio of boys to girls changes to 9 : 13. Calculate, how many new girls are admitted?

Solution:

Since, 4 + 5 = 9,

Number of boys in the School = \(\frac{4}{9}\) x 450 = 200

Number of girl in the School = \(\frac{5}{9}\) x 450 = 250

Since the num,ber of boys increases by 25, the new number of boys = 200 + 25 = 225.

Let the number of new girls admitted be x.

Then, new number of girls = 250 + x.

By question,

\(\frac{225}{250 + x }\) = \(\frac{9}{13}\)

or, 2250 + 9x = 2925

or, 9x = 2925 - 2250

or, x = \(\frac{675}{9}\)

\(\therefore\) x = 75

Two numbers are in the Ratio 5 : 7. If 3 is added to both of them, the ratio will be 4 : 5. What are the numbers?

Solution:

Let the numbers be 5x and 7x.

According to the question, \(\frac{ 5x + 3}{7x + 3}\) = \(\frac{4}{5}\)

or, 28x + 12 = 25x + 15

or, 28x - 25x = 15 - 12

or, 3x = 3

or, x =\(\frac{3}{3}\)

\(\therefore\) x = 1

Now,

5x = 5X 1 = 5 and

7x = 7X1 =7

Since the required numbers are 5 and 7.

The Ratio of a height of Sony and Jenny is 4 : 5. If the height of Sony is 40 inches, find the height of Jenny?

Solution:

The Ratio of height of Sony and Jenny = 4 : 5

or, \(\frac{height of Sony}{height of Jenny}\) = \(\frac{4}{5}\)

or, \(\frac{40 inches}{Jenny's height}\) = \(\frac{4}{5}\)

or, Jenny's height = \(\frac{40 \times 5}{4}\)

\(\therefore\) Jenny's height = 50 inches

The Ratio of the Investment between Sunil and Saroj is 10 : 13. If Sunil has invested Rs 5000, How much did Saroj invest?

Solution:

The Ratio of the investment of Saroj and Sunil = 10 : 13

Sunil's investment = Rs 5000

Saroj's investment =?

Here,

\(\frac{Investment of Sunil}{Investment of Saroj}\) = \(\frac{10}{13}\)

or, \(\frac{Rs 5000}{Investment of Saroj}\) = \(\frac{10}{13}\)

or, Saroj's investment = \(\frac{ 13 X 5000}{10}\)

or, Saroj's investment = Rs 6500

Investment of Saroj is Rs 65,00

Salaka, Mausami and Smriti invested in the ratio of 3 : 4 : 5. If the total amount of their investment is Rs 36,000,000. Find how much did each invest?

Solution:

Total amount = Rs 36,000,000

Suppose x be the Ratio, So 3x, 4x and 5x.

According to the question,

3x + 4x + 5x = Rs 36,000,000

Or, 12x = 36,000,000

Or, x = \(\frac{36,000,000}{12}\)

= Rs 3,000,000

Now,

Investment of Salaka = 3x = 3X Rs 3,000,000 = Rs9,000,000

Investment of Mausami = 4x = 4 X Rs 3,000,000 = Rs 12,000,000

Investment of Smriti = 5x = 5 X Rs 3,000,000 = Rs 15,000,000

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