Note on Ratio and Proportion

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 1

Ratio

The ratio of two quantities of the same type ( let a and b) is used to express how many times bigger or smaller, one quantity is compared to other. For example , if a = 3 and b = 4, then we can write \(\frac{a}{b}\) = \(\frac{3}{4}\). Also, we can write a = \(\frac{3}{4}\) b or a is three-fourth of b. The ratios 2:5 and 8:20.

In a ratio a : b or \(\frac{a}{b}\), a is called the antecedent and b is called the consequent. The ratio b : a is the inverse ratio of a:b and vice-versa.

Compound ratio
 If a : b and c : d be any two ratios. then a : b \(\times\) c : d = \(\frac{a}{b}\) \(\times\) \(\frac{c}{d}\) = \(\frac{ac}{bd}\) = ac : bd is called compound ratio.

Duplicate and sub -duplicate ratio
If a : b be a ratio, then the duplicate ratio of \(\frac{a}{b}\) = (\(\frac{a}{b}\))2 = \(\frac{a^2}{b^2}\) = a2: b2
And, the sup-duplicate ratio of \(\frac{a}{b}\) = √ \(\frac{a}{b}\)

Triplicate and sub - triplicate ratio
If a : b be a ratio, then the triplicate ratio of \(\frac{a}{b}\) = (\(\frac{a}{b}\))3 and sub-triplicate vratio of \(\frac{a}{b}\) =3√ \(\frac{a}{b}\)

Proportion

Similarly, if two or more than two ratios are equal, those quantities which make ratios are proportional.
Two ratios a : b and c : d equal or \(\frac{a}{b}\) = \(\frac{c}{d}\), then a, b, c and d are in proportion.
Now, let us study some related examples of proportion.
Example: \(\frac{8}{20}\) = \(\frac{2}{5}\)
Or, \(\frac{20}{8}\) = \(\frac{5 \times 4}{2 \times 4}\) = \(\frac{5}{2}\)
\(\therefore\) \(\frac{20}{8}\) = \(\frac{5}{2}\)

Continued proportion
If a , b and c be any three number such that the ratio of the a and b is equal to the ratio of b and c, then such ratio is known as a compound proportion.

\(\therefore\) \(\frac{a}{b}\) = \(\frac{b}{c}\) is said to be continued proportion. Then, ac= b2

a : b = b : c

Here, a is 1st proportion

b is mean proportion

c is 3rd proportion

Mean proportion (b) =√ac

Properties of proportion

If a, b, c and d are in proportion, then we can verify the following six properties of proportion.

  1. Invertendo
  2. Alternendo
  3. Componendo
  4. Dividendo
  5. Componendo and Dividendo
  6. Addendo

a) Invertendo
If \(\frac{a}{b}\) =\(\frac{c}{d}\), then \(\frac{b}{a}\) =\(\frac{d}{c}\) is known as invertendo properties of proportion.

Proof:

Here, \(\frac{a}{b}\) = \(\frac{c}{d}\)

Then, 1 \(\div\)\(\frac{a}{b}\) = 1\(\div\)\(\frac{c}{d}\) (1 is divided by both ratio)

1 \(\times\)\(\frac{b}{a}\) = 1\(\times\)\(\frac{d}{c}\)

\(\therefore\) \(\frac{b}{a}\) =\(\frac{d}{c}\)

Hence, if \(\frac{a}{b}\) = \(\frac{c}{d}\), then \(\frac{b}{a}\) = \(\frac{d}{c}\)

b) Alternendo
If \(\frac{a}{b}\) = \(\frac{c}{d}\), then \(\frac{a}{c}\) = \(\frac{b}{d}\) is known as alternendo property of proportion.

Proof:

Here, \(\frac{a}{b}\) = \(\frac{c}{d}\)

Multiplying both by \(\frac{b}{c}\), we get \(\frac{a}{b}\) \(\times\) \(\frac{b}{c}\) = \(\frac{c}{d}\) \(\times\) \(\frac{b}{c}\)

or, \(\frac{a}{c}\) = \(\frac{b}{d}\)

\(\therefore\) \(\frac{a}{c}\) = \(\frac{b}{d}\)

c) Componendo
If \(\frac{a}{b}\) = \(\frac{c}{d}\), then \(\frac{a + b}{b}\) = \(\frac{c + d}{d}\) is known as componendo property of proportion.

Proof:

Here, \(\frac{a}{b}\) = \(\frac{c}{d}\)

Then, adding one on both side, we get

\(\frac{a}{b}\) + 1 = \(\frac{c}{d}\) + 1

\(\frac{a + b}{b}\) = \(\frac{c + d}{d}\)

Hence, if \(\frac{a}{b}\) = \(\frac{c}{d}\), then \(\frac{a + b}{b}\) = \(\frac{c + d}{d}\)

d) Dividendo
If \(\frac{a}{b}\) = \(\frac{c}{d}\), then \(\frac{a - b}{b}\) = \(\frac{c - d}{d}\) is known as dividendo property of proportion.

Proof:

Here, \(\frac{a}{b}\) = \(\frac{c}{d}\)

subtracting 1 from both sides, we get

\(\frac{a}{b}\) - 1 = \(\frac{c}{d}\) - 1

or, \(\frac{a - b}{b}\) = \(\frac{c - d}{d}\)

\(\therefore\) \(\frac{a - b}{b}\) = \(\frac{c - d}{d}\)

e) Componendo and dividendo
If \(\frac{a}{b}\) = \(\frac{c}{d}\), then \(\frac{a + b}{a - b}\) = \(\frac{c + d}{c - d}\) is known as componendo and dividendo property of proportion.

Proof:

Here, \(\frac{a}{b}\) = \(\frac{c}{d}\)

By compendendo we have,

\(\frac{a + b}{b}\) = \(\frac{c + d}{d}\)................. (1)

Again, by dividendo, we have

\(\frac{a - b}{b}\) = \(\frac{c - d}{d}\) ..................... (2)

Now, dividing equation (1) by (2), we get

\(\frac {\frac {a+b}b}{\frac {a-b}b}\) = \(\frac {\frac {c+d}d}{\frac {c-d}d}\)

or, \(\frac{a + b}{b}\) \(\times\) \(\frac{b}{a - b}\) = \(\frac{c + d}{d}\) \(\times\) \(\frac{d}{c - d}\)

or, \(\frac{a + b}{a - b}\) = \(\frac{c + d}{c - d}\)

\(\therefore\) \(\frac{a + b}{a - b}\) = \(\frac{c + d}{c - d}\)

f) Addendo
If \(\frac{a}{b}\) = \(\frac{c}{d}\), then, \(\frac{a}{b}\) = \(\frac{c}{d}\) = \(\frac{a + c}{b + d}\) is known as addendo property of proportion.

Proof:

Here, \(\frac{a}{b}\) = \(\frac{c}{d}\)

By alternendo, we get,

\(\frac{a}{c}\) = \(\frac{b}{d}\)

By alternendo we get,

\(\frac{a + c}{c}\) = \(\frac{b + d}{d}\)

Again, by alternendo we get,

\(\frac{a + c}{b + d}\) = \(\frac{c}{d}\)

\(\therefore\)\(\frac{a}{b}\) = \(\frac{c}{d}\) = \(\frac{a + c}{b + d}\)

Hence, if \(\frac{a}{b}\) = \(\frac{c}{d}\) then, \(\frac{a}{b}\) = \(\frac{c}{d}\) = \(\frac{a + c}{b + d}\)

Similarly, if \(\frac{a}{b}\) = \(\frac{c}{d}\) = \(\frac{e}{f}\) then, \(\frac{a}{b}\) = \(\frac{c}{d}\) = \(\frac{e}{f}\) = \(\frac{a + c + e}{b + d + f}\) and so on.

Solution of discontinued and continued proportion (k-method):

Discontinued proportion

If a, b, c and d are in discontinued proportion,

Let, \(\frac ab\) = \(\frac cd\) = k 

Then,

\(\frac ab\) = k,

∴ a = bk....................(i)

\(\frac cd\) = k,

∴ c = dk.....................(ii)

In terms of the denominator with 'k' is a constant number, express the two numerators. We solve the problems related to proportion.

Continued proportion

If a, b, c and d are in a continued proportion, 

Let, \(\frac ab\) = \(\frac bc\) = \(\frac cd\)= k 

or, \(\frac cd\) = k

∴ c = dk...................(i)

or, \(\frac bc\) = k

∴ b = ck = d.k.k = dk2..............(II)

or, \(\frac ab\) = k,

∴ a = bk = dk2.k = dk3.............(iii)

∴ a = dk3, b = dk2 and c = dk

So, if a, b, c and d are in contiuned proportion, we express a, b, c in terms of d with 'k' constant and solve the problem.

A proportion is a name we give to a statement that two ratios are equal. It can be written in following way:

  • using a colon,    a:b = c:d

When two ratios are equal, then the cross products of the ratios are equal.

That is, for the proportion, a:b = c:d ,  a x d = b x c 

.

Very Short Questions

Here, by chain rule
The work done by Ram in 3 days = the work done by Shyam in 4 days.
The work done by Shyam in 5 days = the work done by Hari in 6 days.
Let, the work done by Hari 16 days = the work done by Ram in x days

Now , 3 \(\times\) 5 \(\times\)16 = 4 \(\times\) 6 \(\times\) x

\(\therefore\) x= \(\frac{3 \times 5 \times 16}{4 \times 6}\) = 10.

Hence, the work done by Hari in 16 days can be done by Ram in 10 days. Ans.

Here , solving the given problem by chain rule ,
The price of 3 ducks = Price of 4 hens
Price of 2 hens= Rs. Rs.750

Let , Rs.x= price of 1 peasants
price of 4 peasants= Price of 7 ducks
Now , 3 \(\times\) 2 \(\times\) x \(\times\) 4 = 4 \(\times\) 750 \(\times\) 1 \(\times\) 7

or , x= \(\frac{4 \times 750\times 1 \times 7}{3 \times 2 \times 4}\) = 875.

The price of 1 peasant = Rs. 875.

Here , solving the given problem by chain rule ,
Food for x horses =food for 153 oxen
Food for 12 oxen = food for 24 sheep
Food for 15 sheep = food for 25 goats
Food for 17 goats= food for 3 baby elephants
Food for 8 baby ele

Let, x eggs of Swan's can be exchanges with two eggs of the hen.
Here, solving the given problem by chain rule,
4 eggs of hen = 3 eggs of duck.
7 eggs of duck= 4 eggs of a swan.

Hence , x × 4 × 7 = 2 × 3 × 4

or , 28x = 24
x = \(\frac{24}{28}\) = \(\frac{6}{7}\)

Now, using unitary method
\(\frac{6}{7}\) eggs of swan = 2 eggs of hen

\(\therefore\) cost of \(\frac{6}{7}\) eggs of swan = Rs. 7.50

or , cost of 1 eggs of swan = Rs. 7.5 \(\times\) \(\frac{7}{6}\) = Rs. 8.75 Ans.

Let , the cost of mixture after 2x kg of rice costing Rs. 15 and 3x kg of rice costing Rs. 20 be Rs. y per kg.

Now , Rs. 15 × 2x + Rs. 20 × 3x = Rs. y (2x + 3x)
or , (30 x + 60x) = 5xy
or , y= \(\frac{90x}{5x}\) = 18

\(\therefore\) The cost of mixture is Rs. 18 per kg.

Here , the sum of the propertional parts = 56l
Quantity of milk = \(\frac{4}{7}\) \(\times\) 56 l = 4 \(\times\) 8 l= 32l
Quantity of water = \(\frac{3}{7}\) \(\times\) 56 l = 3 \(\times\) 8 l = 24l Ans.

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  • If (5a+2b) : (7a+3b) = 9:13, find the value of a:b

    1:3
    1:5
    1:4
    1:2
  • When rs.105 is divided into two parts in the ratio of 2:5, how much money will be in the first part?

    Rs.10


    Rs.30


    Rs.20


    Rs.15


  • What number must be subtracted from each term of ratio 15:18 so that it may become equal to the term of the ratio 3:4?

    4


    8


    6


    7


  • When Rs.1050 was divided between Ram and Hari in the  ratio of 3:4, How much will Hari get?

    Rs.250


    Rs.300


    Rs.500


    Rs.600


  • What number must be subtracted from each term of ratio 4:9 so that it may become equal to 9:4?

    13


    10


    12


    11


  • If 1,a,5 and 15 are in proportional , Find the value of a.

    6


    3


    2


    1


  • If 2,x,18 and 54 are in proportion then what is the value of x?

    2


    4


    3


    6


  •  If(frac{3m-5n}{3m+5n}) =(frac{1}{4}),what is the value of (frac{m}{n})?

    (frac{24}{9})


    (frac{25}{9})


    (frac{21}{9})


    (frac{15}{9})


  • If (frac{p}{q})=(frac{3}{4}), What is value of (frac{2p-3q}{2p+3q})?

    -(frac{1}{8})


    -(frac{1}{3})


    -(frac{1}{4})


    -(frac{1}{7})


  • If (frac{a}{b})=(frac{3}{4}), what is the value of (frac{2a-3b}{2a+3b})?

    (frac{-1}{3})


    (frac{-1}{6})


    (frac{-1}{5})


    (frac{-1}{3})


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DISCUSSIONS ABOUT THIS NOTE

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(n p) 3÷(n p)3

(nk pk)³÷(n p)3


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Aarjan

When rs 105 is divided in the ratio of 2:5,how much money will be?


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Sworup

When rs 105 is divided in the ratio of 2:5,how much money will be in the first part


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Sadikshya

if a/b=b/c=c/d,then prove a/d =a^3 b^3 c^3/b^3 c^3 d^3


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