Ratio

Subject: Compulsory Maths

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Overview

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.

Ratio
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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)

 

Things to remember

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

To divide a given quantity in a given ratio

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

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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
Things to remember
  • 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.
Questions and Answers

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

Solution:

Here, Rs 3 = 300 paisa

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

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.

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.

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

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

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

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

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

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

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

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

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

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

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

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.

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

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

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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