Note on Proportion

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When two ratios are equal, then it is called as a proportion. It is an equation that can be solved. It is a case where two fractions are equal. It can be written in two ways:

  1. Two equal fractions, \(\frac{a}{b}\) = \(\frac{c}{d}\)
  2. Using a colon, a : b = c : d


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There are four quantities in proportions where the first and fourth terms are known as extremes and the second and third terms are called means. It shows or tells two fraction or ratios are equal.

Terms are the four different quantities in the proportion. In proportion a: b: :c: d, a, b, c, and d are its first term, second term, third term& fourth terms. The fourth term is called the fourth proportional to the numbers a, b, and c.

In a proportion, the first and fourth terms are called extremes and the second and third terms are called means.

Examples

  1. Find the fourth proportional of 6, 8, 9.
    Solution:
    Let the fourth proportional to 6, 8, 9
    Then, \(\frac{6}{8}\) = \(\frac{9}{x}\)
    or, 6x = 72
    or, x = \(\frac{72}{6}\) = 12

  2. Find the value of x in 16 : 8 = x : 5
    Solution:
    16 : 8 = x : 5
    or, \(\frac{16}{8}\) = \(\frac{x}{5}\)
    or, 8x =16 \(\times\) 5
    or, x = \(\frac{80}{8}\)
    \(\therefore\) x = 10

  • When two ratios are equal, a name is given to a statement which is called a proportion.
  • Proportion is an equation where two ratios are equal to each other.
  • In a proportion, the first and fourth terms are called extremes and the second and third terms are called means.
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Very Short Questions

Solution:

Let the required third proportion be x.

2 : 22 = 22 : x

or, \(\frac{2}{22}\) = \(\frac{22}{x}\)

or, 2x = 22\(\times\) 22

or, x = \(\frac{22 \times 22}{2}\)

\(\therefore\) x = 242

Solution:

Let the required mean proportion be x. Then,

2 : x = x : 32

or, \(\frac{2}{x}\) = \(\frac{x}{32}\)

or, x2 = 64

or, x = \(\sqrt{64}\)

\(\therefore\) x = 8

Solution:

Let x be the proportional to 5, 7 and 8

Then, \(\frac{5}{7}\) = \(\frac{8}{x}\)

or, 5x =7 \(\times\) 8

or, x = \(\frac{56}{5}\)

\(\therefore\) x = 11.2

Solution:

16 : 8 = x : 5

or, \(\frac{16}{8}\) = \(\frac{x}{5}\)

or, 8x =16 \(\times\) 5

or, x = \(\frac{80}{8}\)

\(\therefore\) x = 10

Solution:

The ratio of first two numbers = \(\frac{7}{9}\)

The ratio of second two numbers = \(\frac{z}{18}\)

Now,

\(\frac{7}{9}\) = \(\frac{z}{18}\)

or, z = \(\frac{7\times18}{9}\)

or, z = 7\(\times\)2

\(\therefore\) z = 14

Solution:

Given, x:5 = 10:25

or, \(\frac{x}{5}\) = \(\frac{10}{25}\)

or, 25x = 5 \(\times\) 10

or, x = \(\frac{50}{25}\)

\(\therefore\) x = 2

Solution:

Given, 3:7 = 21:x

or, \(\frac{3}{7}\) = \(\frac{21}{x}\)

or, x = \(\frac{21\times7}{3}\)

\(\therefore\) x = 49

Solution:

The first two number of ratio = \(\frac{3}{a}\)

Second two number of ratio = \(\frac{9}{21}\)

given numbers are in proportion, so

\(\frac{3}{a}\) = \(\frac{9}{21}\)

or, a = \(\frac {3\times21}{9}\)

\(\therefore\) a = 7

Solution:

Given,

25 : 15 = x : 3

or, \(\frac{25}{15}\) = \(\frac{x}{3}\)

or, x = \(\frac{25\times3}{15}\)

or, x = \(\frac{75}{15}\)

\(\therefore\) x = 5

Solution:

The ratio of first and second numbers = \(\frac{7}{8}\) = 7:8

The ratio of third and fourth numbers = \(\frac{14}{20}\) = 7:10

Hence, the first ratio 7:8 and the second ratio 14:20 = 7:10 are not not in the proportional form.

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  • Find the  value of x?

    x, 2, 6, 4

     

    7
    5
    4
    3
  • Find the value of x?

    x:5 = 10:25

    5
    2
    7
    3
  • Find the value of x?

    3 : 7 = 21 : x

    45
    47
    49
    50
  • Find the value of x?

    10 : x = 2 : 11

    55
    58
    60
    57
  • Find the value of y?

    16, 4, 4, y

    2
    1
    5
    3
  • Find the value of a?

    3, a, 9, 21

    9
    5
    11
    7
  • Find the value of x in 16 : 8 = x : 5

    14
    12
    10
    16
  • Find the fourth proportional to :

    5, 7 ,8

    ;i:4;s:3:
    ;i:3;s:4:
    11.2
    12.3
  • If 5 pens cost Rs 150, what is the cost of 12 pens ?

    Rs 350 
    Rs 300
    Rs 360
    Rs 250 
  • Find the third proportional to 2 and 32.

    512
    445
    500
    568
  • Find the mean proportion between :

    0.9 and 2.5

    1.5
    1.8
    1.4
    2.2
  • Find the mean proportional between:

    0.6 and 9.6

     

    2.4
    2.2
    3.2
    3.4
  • If x : y = 2 : 3 and y : z = 5 : 7, find x : y : z.

    11 : 14 : 20
    16 : 11 : 22
    15 : 12 : 23
    10 : 15 : 21
  • If m : n = 4 : 9 and n : s = 3 : 7, find m : s.

    5 : 20
    4 : 23
    5 : 5
    4 : 21
  • Find x in the following proportions:

    4 : 8 : : x : 16

    8
    2
    4
    10
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DISCUSSIONS ABOUT THIS NOTE

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Sanjiv

If the distance beteen two places is 18000 and thw ratio of two places is1:200000.find the distance


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