Ratio & Proportion
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Introduction
RATIO: When
we say that the length of a line AB is 5 centimeters, we mean that a unit of
length called 1 centimeter is contained in AB five times. If we have two lines
AB and CD and their lengths are 2 and 3 centimeters respectively, we say that
the length of AB is 2/3 of the length of CD.
DEFINITION: The
number of times one quantity contains another quantity of the same kind is
called as the ratio of the two quantities. Clearly, the ratio of two quantities
is equivalent to the fraction that one quantity is of the other. Observe
carefully that the two quantities must be of the same kind. There can be a
ratio between Rs.20 and Rs.30, but there can be no ratio between Rs.20 and 30
mangoes.
In other words, if the values of two quantities A and B are 4 and 6
respectively, then we say that they are in the ratio 4 : 6. Ratio is the
relation which one quantity bears to another of the same kind, the comparison
being made by considering what multiple, part or parts, one quantity is of the
other.
Since the quotient obtained on dividing one concrete quantity by another of the
same kind is an abstract number, the ratio between two concrete quantities of
the same kind is an abstract number. It may be an integer or fraction. Thus the
ratio between Rs.5 and Rs.7 is 5 : 7.
REPRESENTATION: The
ratio of two quantities “a” and “b” is represented as a : b and read as “a is
to b”. A ratio a : b can also be expressed as a/b. So if two items are in the
ratio 2 : 3, we can say that their ratio is 2/3. If two terms are in the ratio
2, it means that they are in the ratio of 2/1, that is, 2 : 1.
TERMS: In
a ratio a : b, a and b are called as the terms of the ratio. The term a is
called as the first term or antecedent. The term b is called as the second term
or consequent.
RULES:
(i) Ratio
of any number of quantities is expressed after removing any common factors that
all the terms of the ratio have. For example, if there are two quantities
having values of 4 and 6, their ratio is 4 : 6, that is 2 : 3 after taking the
common factor 2 between them out. Similarly, if there are three quantities 6, 8
and 18, there is a common factor between all three of them. So, dividing each
of the three terms by 2, we get the ratio as 3 : 4 : 9.
(ii) If
two quantities whose values are A and B respectively are in the ratio a : b,
since we know that some common factor k > 0 would have been removed from A
and B to get the ratio a : b, we can write the original values of the two
quantities as A = ak and B = bk respectively. For example, if the salaries of
two persons are in the ratio 7 : 5, we can write their individual salaries as
7k and 5k respectively.
(iii) A
ratio is said to be a ratio of greater or lesser inequality or of equality
according as antecedent is greater than, less than or equal to the consequent.
In other words,
 The ratio a : b where a > b is
called as ratio of greater inequality.
 The ratio a : b where a < b is
called as ratio of lesser inequality.
 The ratio a : b where a = b is
called as ratio of equality.
(iv) From
the above rule, we can find that a ratio of greater inequality is diminished
and a ratio of lesser inequality is increased by adding the same quantity to
both terms, that is, in the ratio a : b, when we add the same quantity x (
positive ) to both the terms of the ratio, we have the following results
 If a < b then ( a + x ) : ( b
+ x ) > a : b.
 If a > b then ( a + x ) : ( b
+ x ) < a : b.
 If a = b then ( a + x ) : ( b + x
) = a : b.
(v) The
value of a ratio remains unchanged, if each one of its terms is multiplied or
divided by a same nonzero number. For example, 4 : 5 = 8 : 10 = 12 : 15 etc.
COMPOUND
RATIO: Ratios
are compounded by multiplying together the antecedents for a new antecedent and
the consequents for a new consequent. For example, the compounded ratio of the
ratios ( a : b ), ( c : d ) and ( e : f ) is ( ace : bdf ).
 a^{2} : b^{2} is called as the duplicate ratio
of a : b.
 a^{3} : b^{3} is called as the triplicate ratio
of a : b.
 a^{1/2} : b^{1/2} is called as the subduplicate
ratio of a : b.
 a^{1/3} : b^{1/3} is called as the subtriplicate
ratio of a : b.
INVERSE
RATIO: If
a : b is the given ratio, then 1/a : 1/b or b : a is called its inverse or
reciprocal ratio.
PROPORTION: When
two ratios are equal, then the four quantities involved in the two ratios are
said to be proportional, that is, if a / b = c / d, then a, b, c and d are
proportional. In other words, the equality of ratios is called as proportion.
REPRESENTATION: If
the numbers a, b, c and d are said to be in proportion, then it is represented
as ( a : b :: c : d ) and is read as “a is to b (is) as c is to d”. Other ways
of representing the same are, ( a : b = c : d ) or ( a / b = c / d ).
TERMS: If
we have a : b :: c : d, then a, b, c and d are called as terms of the
proportion, where a is the first term, b is the second term, c is the third
term and d is the fourth term. The first and fourth terms, that is a and d are
called as the extremes or end terms of the proportion. The second and third
terms that are b and c are called as the means or middle terms of the
proportion. The fourth term that is d is also called as the fourth
proportional.
RULES:
(a) If
four quantities be in proportion, then the product of the extremes is equal to
the product of the means. In general, if ( a : b :: c : d ), then ( a * d = b * c ).
(b) Three
quantities of the same kind are said to be in continued
proportion when the ratio of
the first to the second is equal to the ratio of the second to the third. The
second quantity is called as the mean
proportional between the
first and the third quantity. The third quantity is called as the third proportional to the first and second terms.
(c) If
a : b = c : d then, b : a = d : c. This relationship is called as INVERTENDO.
(d) If a
: b = c : d then, a : c = b : d. This relationship is called as ALTERNENDO.
(e)
If a : b = c : d then, ( a + b ) : b = ( c + d ) : d. This relationship is
called as COMPONENDO. This is obtained by adding 1 to both sides of the given
relationship.
(f) If
a : b = c : d then, ( a + b ) : b = ( c + d ) : d. This relationship is called
as DIVIDENDO. This is obtained by subtracting 1 to both sides of the given
relationship.
(g) If
a : b = c : d then, ( a + b ) : ( a – b ) = ( c + d ) : ( c – d ). This
relationship is called as COMPONENDO–DIVIDENDO. This is obtained by dividing
the componendo and dividendo relationship.
(h) The
last relationship, that is, ComponendoDividendo is very helpful in simplifying
problems. By this rule, whenever we know a / b = c / d, then we can write ( a +
b ) / ( a – b ) = ( c + d ) / ( c – d ). The converse of this is also true.
(i) If
a/b = c/d = e/f………, then each of these ratios is equal to (a+c+e+…)/(b+d+f+…).
VARIATION: Two
quantities A and B may be such that as one quantity changes in value, the other
quantity also changes in value bearing certain relationship to the change in
the value of the first quantity.
DIRECT
VARIATION:
(a) One
quantity A is said to vary directly as another quantity B if the two quantities
depend upon each other in such a manner that if B is increased in a certain
ratio, A is increased in the same ratio and if B is decreased in a certain
ratio, A is decreased in the same ratio.
(b) This
is denoted as A # B ( A varies directly as B ).
(c) If
A # B then A = k * B, where k is a constant. It is called as constant of
proportionality.
(d) For
example, when the quantity of sugar purchased by a housewife doubles from the
normal quantity, the total amount she spends on sugar also doubles, that is,
the quantity and the total amount increases ( or decreases ) in the same ratio.
(e) From
the above definition of direct variation, we can see that when two quantities A
and B vary directly with each other, then A/B = k or the ratio of the two
quantities is a constant. Conversely, when the ratio of two quantities is a
constant, we can conclude that they vary directly with each other.
(f) If
X varies directly with Y and we have two sets of values of the variables X and
Y, that is, X_{1} corresponding
to Y_{1} and X_{2}corresponding
to Y_{2}, then, since X # Y, we can write down
X_{1}
X_{2}
X_{1} Y_{1}
— =
— or — = —
Y_{1}
Y_{2}
X_{2} Y_{2}
INVERSE
VARIATION:
(a) One
quantity A is said to vary inversely as another quantity B if the two
quantities depend upon each other in such a manner that if B is increased in a
certain ratio, A is decreased in the same ratio and if B is decreased in a certain
ratio, A is increased in the same ratio.
(b) It
is the same as saying that A varies directly with 1/B. It is denoted as, if A #
1/B, that is, A = k/B where k is constant of proportionality.
(c) For
example, as the number of men doing a certain work increases, the time taken to
do the work decreases and conversely, as the number of men decreases, the time
taken to do the work increases.
(d) From
the above definition of inverse variation, we can see that when two quantities
A and B vary inversely with each other, then AB = k or the product of the two
quantities is a constant. Conversely, if the product of two quantities is a
constant, we can conclude that they vary inversely with each other.
(e) If X
varies inversely with Y and we have two sets of values of the variables X and
Y, that is, X_{1} corresponding
to Y_{1} and X_{2}corresponding
to Y_{2}, then, since X # 1/Y, we can write down
X_{1}
Y_{2}
— =
— or X_{1} * Y_{1} = X_{2} * Y_{2}
X_{2}
Y_{1}
JOINT
VARIATION: If
there are three quantities A, B and C such that A varies with B when C is
constant and varies with C when B is constant, then A is said to vary jointly
with B and C when both B and C are varying, that is, A # B when C is constant
and A # C when B is a constant. This implies A # B * C = k * B * C where k is
the constant of proportionality.
 Ratio:
This is a comparison of of the sizes of
two or more quantities of the same kind.
If "p" and "q" are the two quantities of the same kind as well as in the same units, the fraction p/q is called the ratio of "p" to "q".
Thus, the ratio. of "p" to "q" = p/q or p:q. The quantities "p" and "q" are called the terms of the ratio. "p" is called the first term or antecedent "q" is called the second term or consequent.
The ratio of two quantities a and b in the same units, is the fraction and we write it as a : b.
If "p" and "q" are the two quantities of the same kind as well as in the same units, the fraction p/q is called the ratio of "p" to "q".
Thus, the ratio. of "p" to "q" = p/q or p:q. The quantities "p" and "q" are called the terms of the ratio. "p" is called the first term or antecedent "q" is called the second term or consequent.
The ratio of two quantities a and b in the same units, is the fraction and we write it as a : b.
In the ratio a: b, we call a as
the first term or antecedent and b, the second term or consequent.
Eg. The ratio 5 : 9 represents

5

with antecedent = 5, consequent = 9.

9

Rule: The multiplication
or division of each term of a ratio by the same nonzero number does not affect
the ratio.
Eg. 4 : 5 = 8 : 10 = 12 : 15. Also, 4 : 6 = 2 : 3.
 Proportion:
This is another branch of the topic Ratio and Proportion. If two ratios are
equal, then it is called proportion.
For example
Four quantities a,b,c,d are said to be in proportion if a:b=c:d.
And also it can be said as a:b :: c:d or a/b = c/d or ad=bc.
Cross product rule in Proportion
product of extremes = product of means
For example
Four quantities a,b,c,d are said to be in proportion if a:b=c:d.
And also it can be said as a:b :: c:d or a/b = c/d or ad=bc.
Cross product rule in Proportion
product of extremes = product of means
The equality of two ratios is called proportion.
If a : b = c : d,
we write a : b :: c : d and
we say that a, b, c, d are in proportion.
Here a and d are called extremes,
while b and c are called mean terms.
Product of means = Product of extremes.
Thus, a : b :: c : d (b x c) =
(a x d).
 Fourth Proportional:
If a : b = c : d,
then d is called the fourth proportional to a, b, c.
Third Proportional:
a : b = c : d,
then c is called the third proportion to a and b.
Mean Proportional:
Mean proportional between a and b is ab.
 Comparison of Ratios:
We say that (a : b) > (c : d)

a

>

c

.

b

d

 Compounded Ratio:
 The compounded ratio of the
ratios: (a : b), (c : d),
(e : f) is (ace : bdf).
 Duplicate Ratios:
Duplicate ratio of (a : b) is (a^{2} : b^{2}).
Subduplicate ratio of (a : b) is (a : b).
Triplicate ratio of (a : b) is (a^{3} : b^{3}).
Subtriplicate ratio of (a : b) is (a^{1/3} : b^{1/3}).
If

a

=

c

, then

a + b

=

c + d

. [componendo and dividendo]

b

d

a  b

c  d

 Variations:
We say that x is directly proportional to y,
if x = ky for some constant k and
we write, x y.
We say that x is inversely proportional to y,
if xy = k for some constant k and
we write, x

1

.

y


SOME POINTS TO BE REMEMBERED

1.
The ratio a : b represents a fraction a/b. a is called antecedent and b is
called consequent.
2.
The equality of two different ratios is called proportion.
3.
If a : b = c : d then a, b, c, d are in proportion. This is represented by a :
b :: c : d.
4.
In a : b = c : d, then we have a* d = b * c.
5.
If a/b = c/d then (a + b ) / ( a – b ) = ( d + c ) / ( d – c ).
RATIO The ratio of two quantities of the same kind is the fraction that one
quantity is of the other, in other words to say, how many times a given number
is in comparison to another number. A ratio between two nos. A and B is denoted
by A/B
1. The two quantities must be of the same kind.
2. The units of the two quantities must be the
same.
3. The ratio has no measurement.
4. The ratio remains unaltered even if both the
antecedent (A) and the consequent (B) are multiplied or divided by the same no.
5 If two different ratios ( say A /B and C/D)
are expressed in different units, then if we are required to combine these two
ratios we will follow the following rule=
A xC / B xD The required ratio is AC / BD
6 The duplicate ratio of A/B is A^{2}/B^{2} the
triplicate ratio of A/B is A^{3}/B^{3}
7 The sub duplicate ratio of A/B is sq.root of
A/ sq.root of B
8 The sub triplicate ratio of A/B is cube root
of A/ cube root of B
9 To determine which of the given two
ratio A/B and C/D is greater or smaller ,we compare A xD and B xC
provided B>0 and D>0;
if AxC> B
xD then A/B > C/D and vice versa,but if A xC= B xD then A/B =
C/D
Properties of ratios.
1. Inverse ratios of two equal ratios are equal,
if A/B=C/D then B/A = D/C.
2. The ratios of antecedents and consequents of
two equal ratios are equal if A/B=C/D then A/C=B/D
3. If A/B=C/D THEN A+B/B=C+D/D
4. If A/B=C/D THEN AB/B=CD/D
5. If A/B=C/D THEN A+B/AB=C+D/CD
6. If A/B=C/D=E/F.....so on then each of the ratio
(A/B, C/D.....etc) is equal to
sum of the numerators/sum of
the denominators=A+C+E...../B+D+F......=k
PROPORTION
1 Two ratios of two terms is equal to the ratio
of two other terms, then these four terms are said to be in proportion i.e. if
A/B=C/D then A,B,C and D are in proportion.
A,B,C and D are called first, second,
third and fourth proportional’s respectively.
A and D are called Extremes and
B and C are called the Means
and it follows that A xD=B xC
2 Continued proportion: when A/B=B/C then A, B
and C are said to be in continued proportion and B is called the geometric mean
of A and C so it follows,
A xC=B^{2} ,OR
square root of (A xC)=B
3 Direct proportion: if two quantities A and B
are related and an increase in A decreases B and viceversa then A and B are
said to be in direct proportion. Here A is directly proportional to B is
written as AB.when is removed equation comes to be
A = kB,where k is constant.
4. Inverse proportion: if two quantities A and B
are related and an increase in A increases B and viceversa then A and B are
said to be in inverse proportion. Here A is inversely proportional to B is
written as A1/B or, A=k/B,where k is constant.
5 Proportional division:
It simply means a method by
which a quantity may be divided into parts which bear a given ratio to one
another .The parts are called proportional parts.
e.g. divide quantity
"y" in the ratio a:b:c then
first
part= a/(a+b+c)=y
second part=b/(a+b+c)=y third part=c/(a+b+c)=y
6.
If in x liters mixture of Milk and water the ratio of Milk and Water is
a;b, the quantity of water to be added in order to make this ratio c: d is
X(adbc) / c(a+b)
7. A mixture contains milk and water in
the ratio of a;b. If x liters of water is added to the mixture, milk and water
become in the ratio a;c. then the quantity of milk in the mixture is given by
ax / cb and that of water is given by bx /cb , M= x9a+b) /cb
8. If two quantities X and Y are in the
ratio x;y, then X+Y : XY :: x+y : xy
9. If the sum of two numbers is A and
their difference is a, then the ratio of numbers is given by A+a : Aa
EXAMPLES
1. If (x/y) = (2/3) then find the value of (3x+4y)/(4x+3y)
Sol: =(3x+4y)/(4x+3y)
1. If (x/y) = (2/3) then find the value of (3x+4y)/(4x+3y)
Sol: =(3x+4y)/(4x+3y)
Divide
numerator and denominator by “y” ={3(x/y)+4y/y}/{4(x/y)+3y/y}
={3(x/y)+4}/{4(x/y)+3}
Substitute x/y= 2/3
= {3(2/3)+4}/{4(2/3)+3}
= {2+4}/{(8/3)+3}
= 6/{(8+9)/3
= 6/{17/3}
= (6x3)/17
= 18/7
={3(x/y)+4}/{4(x/y)+3}
Substitute x/y= 2/3
= {3(2/3)+4}/{4(2/3)+3}
= {2+4}/{(8/3)+3}
= 6/{(8+9)/3
= 6/{17/3}
= (6x3)/17
= 18/7
2. For what value of ‘m’, will the ratio (7+m)/(12+m) be equal to 5/6?
Sol: Let (7+m)/(12+m)= 5/6
6(7+m)= 5(12+m)
42+6m=60+5m
6m5m=6042, m=18
3.Find the value of "x" if 10:x = 5:4.
Sol: By using cross product rule, we have 5x=10 times 4
5x=40
x=40/5
x=8
4. Find the fourth
proportional to 2/3, 3/7, 4,
Sol: Let the fourth proportional be "x", then 2/3, 3/7, 4, x are in proportion.
Using cross product rule, (2/3)x=(3 times 4)/7
(2/3)x=12/7
x=(12 times 3)/((7 times 2)
x= 36/14
x= 18/7
Sol: Let the fourth proportional be "x", then 2/3, 3/7, 4, x are in proportion.
Using cross product rule, (2/3)x=(3 times 4)/7
(2/3)x=12/7
x=(12 times 3)/((7 times 2)
x= 36/14
x= 18/7
Sol: let 1st no. be 1x,2 nd no. be 2x and 3rd
no. be 3x
their squares x^{2 },
(2x)^{2} and (3x)^{2}
as per the question, x^{2} +
(2x)^{2}+(3x)^{2} = 504
x^{2}+4x^{2}+9x^{2}=504
14x^{2}=504
x^{2}=504/14=36
so, x=6
So the three no. are 1x=6,2x=12 and 3x=18
6. Find the fourth proportional to
the numbers 6,8 and 15?
Sol: let K be the fourht
proportional, then 6/8=15/K
Solving it we get K=(8x15)/6= 20
7. Find the mean mean proportion
between 3 and 75?
Sol. this is related to continued proportion.let x
be the mean proportionalx then we have
x^{2}=3x75 or x=15
8. Divide Rs 1350 into three shares
proportional to the numbers 2, 3 and 4?
Sol: 1st share= Rs 1350x(2/2+3+4)=Rs 300
2nd share =
Rs1350x(3/2+3+4)=Rs 450
3rd share=
Rs1350x(4/2+3+4)=Rs 600
9. A certain sum of money is divided
among A,B and C such that for each rupee A has ,B has 65 paise and C has 40
paisex if C's share is Rs 8, find the sum of money?
Sol: here A:B:C = 100:65:40 =
20:13:8
now 20+13+8=41
As 8/14 of the whole sum=Rs 8
So, the whole sum=Rs 8x41/8=Rs 41
10. In 40 liters mixture of milk and
water the ratio of milk and water is 3:1. how much water should be added in the
mixture so that the ratio of milk to water becomes 2:1.?
Sol: here only amount of water is
changing. the amount of milk remains same in both the mixtures. So, amount of
milk before addition of water =(3/4)X40=30 ltrs. So amount of water is 10 ltrs.
After addition of water the ratio changes to
2:1.here the mixture has two ltrs of milk for every 1 ltr of water. Since
amount of milk is 30 ltrs the amount of water has to be 15 ltr so that the
ratio is 2:1. So the amount of water to be added is 1510=5 liters.
11. A sum of Rs.
427 is to be divided among A, B and C such that 3 times A’s share, 4 tunes B’s
share and 7 times C’s share are all equal. The share of C is
Sol: 3A = 4B = 7C = k,Then A = k/3, B
= k/4 and C= k/7.
A : B : C = k/3 : k/4 : k/7 = 28:21 :12.
Cs share = Rs. [427 x (12/61)] = Rs. 84
A : B : C = k/3 : k/4 : k/7 = 28:21 :12.
Cs share = Rs. [427 x (12/61)] = Rs. 84
12. If a+b : b+c : c+a = 6 : 7
: 8 and a + b + c = 14, then the value of c is
Sol: a/3) = (b/4) = (c/7) then a = 3k,
b = 4k, c = 7k
a+b+c/c = 3k+4k+7k/7k = 14k/7k = 2
a+b+c/c = 3k+4k+7k/7k = 14k/7k = 2
13. The least whole number which when subtracted from both the terms of
the ratio 6 : 7 to give a ratio less than 16 : 21, is.
Sol: Let x is subtracted. Then, ((6 
x)/(7  x)) < 16 / 21
21(6—x) < 16(7—x) ⇒ 5x > 14 = x > 2.8.
Least such number is 3.
21(6—x) < 16(7—x) ⇒ 5x > 14 = x > 2.8.
Least such number is 3.
14. If 15% of x
is the same as 20% of y, then x : y is :
Sol: 15% of x = 2O% of y ⇒ 15x/100 = 20y/100 ⇒ x/y = 4/3
15. The ratio of
income of A to that of B is 5 : 4 and the expenditure of A to that of B is 3:
2. If at the end of the year, each saves Rs, 800, the income of A is: .
Sol: Let the income of A and B be 5x
and 4x and. the expenditures of A and B be 3y and 2y. Then, 5x—3y = 800 and 4x—
2y= 800.
On solving we get: x = 400. As income = 5x = Rs. 2000.
On solving we get: x = 400. As income = 5x = Rs. 2000.
16. An alloy is
to contain copper and zinc in the ratio 9:4. The zinc required (in kg) to be melted
with 24 kg of copper, is 7
Sol: 9:4: 24:x ⇒ 9x = 4 * 24 ⇒ x = (4*24)/9 = 32/3 Kg.
hence `0 and 1/3
17. The ratio of
two numbers is 3 : 4 and their sum is 420. The greater of the two numbers is
Sol: Required number = (420 * (4/7)) =
240.
18. Rs. 730 were
divided among A, B, C in such a way that if A gets Rs. 3, then B gets Rs. 4 and
if B gets Rs. 3.50 then C gets Rs. 3. The share of B exceeds that of C by:
Sol: A:B = 3:4 and B:C = 7/2:3 =
(8/7)*(7/2)*(8/7)*3 = 4:(24/7)
A : B : C = 3 :4: 24/7 = 21 : 28 : 24.
Bs share = Rs. [730 *(28/73)]= Rs. 280.
C’s share = Rs. [730 * (24/73)] = Rs. 240.
Difference of their shares = 40
A : B : C = 3 :4: 24/7 = 21 : 28 : 24.
Bs share = Rs. [730 *(28/73)]= Rs. 280.
C’s share = Rs. [730 * (24/73)] = Rs. 240.
Difference of their shares = 40
19. If 7 : x =
17.5 : 22.5 , then the value of x is:.
Sol: 7*22.5 = x*17.5 ⇒ x = 7 * 22.5/17.5 ⇒ x = 9.
20. What number should be subtracted from both the terms of the
ratio 15 : 19 so as to make it as 3 : 4 ?
Sol: Let x be subtracted. Then,
(15  x) / (19  x) = 3/4 ⇒ 4(15  x) = 3(19  x) x = 3
(15  x) / (19  x) = 3/4 ⇒ 4(15  x) = 3(19  x) x = 3
21. What number should be added to each of the numbers 8, 21, 13 and 31
so that the resulting numbers, in this order form a proportion?
Sol: (8+x)/(21+x) = (13+x)/(31+x)
Then, (8 + x)(31 + x) = (13 + x)(21 + x)
or39x + 248 = 34x + 273 or 5x=25 or x = 5.
Then, (8 + x)(31 + x) = (13 + x)(21 + x)
or39x + 248 = 34x + 273 or 5x=25 or x = 5.
22. If 0.4: 1.4: 1.4:
x, the value of x is
Sol: 0.4 * x = 1.4 * 1.4 ⇒ x = (1.4*1.4)/0.4 = 4.9
23. A dog takes 3
leaps for every 5 leaps of a hare. If one leap of the dog is equal to 3 leaps
of the hare, the ratio of the speed of the dog to that of the hare is:.
Sol: Dog : Hare = (3*3) leaps of hare
: 5 leaps of hare = 9 : 5.
24. The salaries
of A, B, and C are in the ratio of 1 : 2 : 3. The salary of B and C together is
Rs. 6000. By what percent is the salary of C more than that of A?
Sol: Let the salaries of A, B, C hex, 2x
and 3x respectively.
Then,2x + 3x = 6000 = x = 1200. As salary = Rs. 1200, Bs salary = Rs. 2400, and Cs salary Rs. 3600.
Excess of Cs salary over As=[(2400/1200)x100] = 200%.
Then,2x + 3x = 6000 = x = 1200. As salary = Rs. 1200, Bs salary = Rs. 2400, and Cs salary Rs. 3600.
Excess of Cs salary over As=[(2400/1200)x100] = 200%.
25. A certain
amount was divided between Salim and Rahim in the ratio of 4 : 3. If Rahim’s
share was Rs. 2400, the total amount was.
Sol: Let S = 4x and R = 3x. Total
amount = 7x.
Then, 3x = 2400 so x= 800.
Total amount = 7x = Rs. 5600
Then, 3x = 2400 so x= 800.
Total amount = 7x = Rs. 5600
26. A sum of
money is to the divided among F, Q andR in the ratio of 2 : 3 : 5. If the total
share of P andR together is Rs 400 more than that of Q, what is R’s share in it
Sol: Let P = 2x , Q = 3x and R=5x. Now
P+RQ = 400 2x+5x3x = 400 hence x =1OO R = 5x = 500.
27. Pencils, Pens
and Exercise books in a shop are in the ratio of 10: 2 : 3. If there are 120
pencils, the number of exercise books in the shop is:.
Sol: Let Pencils = 10x, Pens = 2x
& Exercise books = 3x. Now, 10x = 120 hence x = 12.
Number of exercise books = 3x = 36.
Number of exercise books = 3x = 36.
28. If p : q = 3 : 4
and q : r= 8 : 9, then p : r is
Sol: p/r = (p/q) * (q/r) = (3/4) *
(8/9) = 2/3 so p : q = 2:3
29. Rs. 120 are
divided among A, B, C such that A’s share is Rs. 20 more than B’s and Rs. 20
less than C’s. What is B’s share..
Sol: Let C = x. Then A = (x—20) and B
= (x—40).
x + x  20 + x  40 = 120 Or x=60.
A:B:C = 40:20:60 = 2:1 :3.
Bs share = Rs. 120*(1/6) = Rs. 20.
x + x  20 + x  40 = 120 Or x=60.
A:B:C = 40:20:60 = 2:1 :3.
Bs share = Rs. 120*(1/6) = Rs. 20.
30. If three numbers in the ratio 3 : 2: 5 be such that the sum of
their squares is 1862, the middle number will be:
Sol: Let the numbers be 3x, 2x and 5x.
Then,
9x + 4x + 25x =1862 ⇒ 38x = 1862 ⇒ x = 49 ⇒ x = 7.
middle number = 2x = 14.
9x + 4x + 25x =1862 ⇒ 38x = 1862 ⇒ x = 49 ⇒ x = 7.
middle number = 2x = 14.
31. In a college,
the ratio of the number of boys to girls is 8 : 5. If there are 160 girls, the
total number of students in the college is:
Sol: Let the number of boys and girls
be 8x and 5x.
Total number of students = 13x = 13 x 32 = 416.
Total number of students = 13x = 13 x 32 = 416.
32. X, Y and Z share
a sum of money in the ratio 7 : 8 : 16. If Z receives Rs. 27 more than X, then
the total money shared was:
Sol: Let X = 7x, Y = 8x & Z = 16x.
Then, total money = 31x.
Now, Z  X = 27 so 16x—7x = 27 that is why x = 3.
Total money 31*x = Rs.93.
Now, Z  X = 27 so 16x—7x = 27 that is why x = 3.
Total money 31*x = Rs.93.
33. An amount of
money is to be distributed among F, Q and R in the ratio 3 : 5 : 7. If Qs share
is Rs. 1500, what is the difference between Ps and Rs shares?.
Sol: Let P = 3x, Q = 5x and R = 7x.
Then, 5x = 1500 ⇒ x = 300. P=900,Q=1500 and R = 21OO.
Hence, (R  p) = (2100  900) = 1200
Then, 5x = 1500 ⇒ x = 300. P=900,Q=1500 and R = 21OO.
Hence, (R  p) = (2100  900) = 1200
34. A profit of
Rs. 30000 is to be distributed among A, B, C in the proportion 3 : 5 : 7. What
will be the difference between B’s and C’s shares?
Sol: Bs share = Rs. 30000 *(5/15) =
Rs.10000.
C’s share = Rs. 30000 * (7/15) = Rs.14000,
Difference in Bs and Cs shares = Rs.4000.
C’s share = Rs. 30000 * (7/15) = Rs.14000,
Difference in Bs and Cs shares = Rs.4000.
35. The
compounded ratio of (2 : 3), (6: 11) and (11 :2) is
Sol; Required ratio = (2/3) * () *
(6/11) * (11/2) = 2/1
36. The ratio of the number of boys and girls in a college is 7: 8.
If the percentage increase in the number of boys and girls be 20% and 10%
respectively, what will be the new ratio?
Sol: Originally, let the number of boys
and girls in the college be 7x and 8x respectively.
Their increased number is (120% of 7x)
and (110% of 8x).


120

x 7x


and


110

x 8x


100

100


42x

and

44x

5

5

The required ratio =


42x

:

44x


= 21: 22.

5

5

37. Salaries of Ravi and Sumit are in the ratio 2 : 3. If the
salary of each is increased by Rs. 4000, the new ratio becomes 40 : 57. What is
Sumit's salary?
Sol: Let the original salaries of Ravi
and Sumit be Rs. 2x and Rs. 3x respectively.
Then,

2x + 4000

=

40

3x + 4000

57

57(2x + 4000) = 40(3x +
4000)
6x = 68,000
3x = 34,000
Sumit's present salary = (3x +
4000) = Rs.(34000 + 4000) = Rs. 38,000
38. The sum of three numbers is 98. If the ratio of the first to
second is 2 :3 and that of the second to the third is 5 : 8, then the second
number is:
Sol: Let the three parts be A, B, C.
Then,
A : B = 2 : 3 and B : C = 5 : 8 =


5 x

3


:


8 x

3


= 3 :

24

5

5

5

A : B : C = 2 : 3 :

24

= 10 : 15 : 24

5

B =


98 x

15


= 30.

49

39. If Rs. 782 be divided into three parts, proportional to : : , then the first part is
Sol: Given ratio = : : = 6 : 8 : 9.
1^{st} part = Rs.


782 x

6


= Rs. 204

23


40. The salaries A, B, C are in the ratio 2 : 3 : 5. If the
increments of 15%, 10% and 20% are allowed respectively in their salaries, then
what will be new ratio of their salaries?
Sol: Let A = 2k, B = 3k and
C = 5k.
A's new salary =

115

of 2k =


115

x 2k


=

23k

100

100

10

B's new salary =

110

of 3k =


110

x 3k


=

33k

100

100

10

C's new salary =

120

of 5k =


120

x 5k


= 6k

100

100

New ratio


23k

:

33k

: 6k


= 23 : 33 : 60

10

10

41.If 40% of a number is equal to twothird of another number,
what is the ratio of first number to the second number?
Sol: Let 40% of A =

2

B

3

Then,

40A

=

2B

100

3


2A

=

2B

5

3


A

=


2

x

5


=

5

B

3

2

3

A : B = 5 : 3.
42. Two number are in the ratio 3 : 5. If 9 is subtracted from
each, the new numbers are in the ratio 12 : 23. The smaller number is:
Sol: Let the numbers be 3x and
5x.
Then,

3x  9

=

12

5x  9

23

23(3x  9) = 12(5x 
9)
9x = 99
x = 11.
The smaller number = (3 x 11) =
33.
43. In a bag, there are coins of 25 p, 10 p and 5 p in the ratio
of 1 : 2 : 3. If there is Rs. 30 in all, how many 5 p coins are there?
Sol: Let the number of 25 p, 10 p and 5
p coins be x, 2x, 3x respectively.
Then, sum of their values = Rs.


25x

+

10 x 2x

+

5 x 3x


= Rs.

60x

100

100

100

100


60x

= 30

x =

30 x 100

= 50.

100

60

Hence, the number of 5 p coins = (3 x
50) = 150.
44.In a mixture 60 liters, the ratio of milk and water 2 : 1. If
the this ratio is to be 1 : 2, then the quantity of water to be further added
is:
Sol: Quantity of milk =


60 x

2

litres = 40 litres.

3

Quantity of water in it = (60 40)
litres = 20 litres.
New ratio = 1 : 2
Let quantity of water to be added
further be x litres.
Then, milk : water =


40


.

20 + x

Now,


40


=

1

20 + x

2

20 + x = 80
x = 60.
Quantity of water to be added =
60 litres.
45.A sum of money is to be distributed among A, B, C, D in the
proportion of 5 : 2 : 4 : 3. If C gets Rs. 1000 more than D, what is B's share?
Sol: Let
the shares of A, B, C and D be Rs. 5x, Rs. 2x, Rs. 4x and Rs. 3x respectively.
Then, 4x  3x = 1000
x = 1000.
B's share = Rs. 2x = Rs. (2 x 1000) = Rs. 2000.
46.Zinc and copper are melted together in the ratio 9 : 11. What
is the weight of melted mixture, if 28.8 kg of zinc has been consumed in it?
Sol: For 9 kg zinc, mixture melted = (9 + 11) kg.





47. Gold is 19 times as heavy as water and copper is 9 times as
heavy as water. In what ratio should these be mixed to get an alloy 15 times as
heavy as water?
Sol: G = 19W and C = 9W.




Let 1 gm of gold be mixed with x gm of
copper to get (1 + x) gm of the alloy.




(1 gm gold) + (x gm copper) = (x +
1) gm of alloy




19W + 9Wx = (x + 1)
x 15W















48 The prices of a scooter and a T.V. are in the ratio 7 : 5. If
the scooter costs Rs. 8000 more than a T.V. set, then the price of a T.V. set
is :
Sol: Let the prices of a scooter and a T.V. set be Rs.
7x and Rs. 5x respectively.




Then, 7x  5x = 8000




2x = 8000




x = 4000








49.A fraction which
bears the same ratio


that


does to


, is equal to:


Sol: Let x :


::


:


. Then, x x


=


x



x =



x


x



=




50.A and B are two alloys of gold and copper prepared by
mixing metals in the ratio 7 : 2 and 7 : 11 respectively. If equal quantities
of the alloys are melted to form a third alloy C, the ratio of gold and copper
in C will be:






52.Two numbers are in the ratio 1 : 2, If 7 is added to both,
their ratio changes to 3 : 5, The greatest number is:
Sol:
53. If 10% of x = 20% of y, then x : y is equal
to:
54. If 0.75 : x :: 5 : 8, then x is equal to:
Sol:
55. The ages of A and B are in the ratio of 3 : 1. Fifteen years
hence, the ratio will be 2: 1. Their present
ages are :
Sol;
57. If a carton containing a dozen mirrors is dropped, which
of the following cannot be the ratio of broken mirrors to unbroken mirrors?
58.The ratio of the incomes of A and B is 5 : 4 and the ratio
of their expenditures is 3 : 2. If at the end of the year, each saves Rs.
1600, then the income of A is :
SoL;
59. If A : B = 2 : 3, B : C = 4 : 5 and C : D = 6 : 7, then A
: B : C : D is:


60. A sum of Rs. 1300
is divided amongst P, Q, R and S such that




Sol:































61. A certain amount was divided between A and B in the ratio 4
: 3. If B’s share was Rs. 4800, the total amount was:
Sol: If B’s share is Rs. 3, total amount = Rs. 7.





62. A sum of Rs. 53 is divided among A, B, C in such a way that A gets
Rs. 7 more than what B gets and B gets Rs. 8 more than what C gets. The ratio
of their shares is:
Sol: Suppose C gets Rs. x.Then, B gets Rs.
(x + 8) and A gets Rs. (x + 15).











63. The ratio of three numbers is 3: 4: 7 and their product is
18144. The numbers are:
Sol: Let the numbers be 3x, 4x and 7x.
Then,




3x x 4x x 7x =
18144




x^{3} = 216,




x^{3} = 6^{3}




x = 6.






64. what least number must be subtracted from each of the
numbers 14, 17, 34 and 42 so that the remainders may be proportional?
Sol: Let the required number be x.




Then, (14 – x) : (17 – x) : :
(34 – x) : (42 – x).

























65. If 76 is divided into four parts proportional to 7, 5, 3, 4.
then the smallest part is
Sol: Given ratio = 7 : 5 : 3 : 4, Sum of ratio
terms = 19.







66.The ratio of the number of boys and girls in a school is 3 :
2. If 20% of the boys and 25% of the girls are scholarship holders, what
percentage of the students does not get the scholarship?
Sol: Let boys = 3x and girls = 2x




Number of those who do not get scholarship = (80% of 3x)
+ (75% of 2x)









67.Two numbers are respectively 20% and 50% more than a third
number. The ratio of the two numbers is:
Sol: Let the third number be x.
















68. If


=


=


, then A : B : C is :


Sol:







69. An amount of Rs. 2430 is divided among A, B and C such that
if their shares be reduced by Rs. 5, Rs.10 and Rs. 15 respectively, the
remainders shall be in the ratio of 3 : 4 : 5. Then, B’s share was
Sol:


Remainder = Rs. [2430 – (5 + 10 + 15)] = Rs. 2400.








70.If


=


=


, then


is equal to :


Sol:






71.The lease whole number which when subtracted from both the
terms of the ratio 6 : 7 gives a ratio less than 16 : 21 is:
Sol:


Let x be subtracted. Then,









72. The speeds of three cars are in the ratio 5 : 4 : 6. The
ratio between the time taken by them to travel the same distance is :
Sol:Ratio of time taken =


:


:


= 12 : 15 : 10




.What is the ratio of
boys to girls in that school?
Sol:







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