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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} = \(\frac{a^2}{b^2}\) = a^{2}: b^{2}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}√ \(\frac{a}{b}\)
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= b^{2}
a : b = b : c
Here, a is 1^{st }proportion
b is mean proportion
c is 3^{rd }proportion
Mean proportion (b) =√ac
If a, b, c and d are in proportion, then we can verify the following six properties of proportion.
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.
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.
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 = dk^{2..............(II)}
or, \(\frac ab\) = k,
∴ a = bk = dk^{2}.k = dk^{3}.............(iii)
∴ a = dk^{3}, b = dk^{2} 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:
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
.
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.
If (5a+2b) : (7a+3b) = 9:13, find the value of a:b
When rs.105 is divided into two parts in the ratio of 2:5, how much money will be in the first part?
Rs.30
Rs.10
Rs.15
Rs.20
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?
8
6
7
4
When Rs.1050 was divided between Ram and Hari in the ratio of 3:4, How much will Hari get?
Rs.600
Rs.300
Rs.250
Rs.500
What number must be subtracted from each term of ratio 4:9 so that it may become equal to 9:4?
10
13
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?
6
4
3
2
If(frac{3m-5n}{3m+5n}) =(frac{1}{4}),what is the value of (frac{m}{n})?
(frac{25}{9})
(frac{21}{9})
(frac{15}{9})
(frac{24}{9})
If (frac{p}{q})=(frac{3}{4}), What is value of (frac{2p-3q}{2p+3q})?
-(frac{1}{3})
-(frac{1}{8})
-(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}{5})
(frac{-1}{3})
(frac{-1}{6})
(frac{-1}{3})
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(n p) 3÷(n p)3
(nk pk)³÷(n p)3
Feb 06, 2017
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Aarjan
When rs 105 is divided in the ratio of 2:5,how much money will be?
Feb 06, 2017
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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
Dec 31, 2016
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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
Dec 31, 2016
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