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The document discusses the concepts of ratio and proportion, explaining how to express the relationship between two quantities using ratios and how to manipulate them through various mathematical operations. It covers direct and inverse variations, compounding of ratios, and properties of ratios, including methods for comparing and equating them. Additionally, it provides solved examples and applications of these concepts in problem-solving scenarios.
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Save Questions for CAT For Later 4.6 | RATIO, PROPORTION,
oh | BND VARIATION BA tHeory
me kind is the fraction that one quantity is of the other. In
given number is in comparison to another number, or to say
is to state the ratio that holds between them.
ratio
ai of two quantities of the sa
he Myords, £0 Say how many times a
i ira given number isin comparison to another
wha : :
meat of two terms ‘at and ‘bs denoted by a: b (read as ‘a is to b’) and is measured by
rw i+ b. The numerator ‘ais called the ‘antecedent’ and the denominator ‘b’ is called the
‘consequent
we comparing Wo quantities in terms of ratio:
0
fi the units o
the two quantities must be of the same kind.
ff measurement of the two quantities must be the same.
example
ifa = R52 and b = 50 paise, then a: b= 200: 50 = 4:2.
fip the ratio is @ pure number, je., without any Unit of measurement as it denotes how many
times a given quantity is in comparison to another.
red even if both the antecedent and the consequent are multi
ied or
the ratio would stay unalte
Ww)
divided by the same non-zero number.
Example
a, axm _ a/m
b bem b/m*
Compounding of ratios
1, If two different ratios (say
combine these two ratios,
a:b and c; d) are expressed in different units, then, if we require to
we do so through compounding,
Here, the compound of a: band c:dis pyqe Hen ac: bd.
Example
To complete a certain job,
day for 10 days. We want to know how many boys are require!
4. man works for 4 hours a day for 5 days; 1 boy works for 6 hours a
.d to do the same job that 1. man
can do?
To know this, we need to com
The ratio of days worked is 5 : 10, the ratio of hours worked is 4 : 6.
.e two different units, hours and days.
97 @ | cAteputt
ARITHMETIC
The compound ratio (day-hours) is He = 20:60=1:3
So, 1 man can do the job that requires 3 boys.
2. The duplicate ratio of a: b is a? : b?
3. The triplicate ratio of a : b is a? : b?,
4, The subduplicate ratio of a: bis / ;
is Va
5. The subtriplicate ratio of
Order of ratios
To determine which of the two given ratios a : b and c : d is greater or smaller, we compare a
x d and b x c. For b > 0 and d > 0:
als
1. Ifaxd>b xc, then 2. Iaxd b.
atx
btx
a atx a fa
@ We =a then Feb ay
(i) It § > 4, then
Giy FB 4, then B% >
2. Aratio of equality is unaltered, aratio of greater inequality is increased and a ratio of lesser inequality
is diminished if some positive quantity not greater than the smaller term be subtracted from each
of its terms.
Let 2 be the given ratio, x be a positive quantity and x < b.
a-x
= ot 2
@ W224, ten BSE =F Go WB > 4, then B=¥ > 2
Gi) 2 ¢4, then BEE < F-
Properties of ratios
©
4 i, the inverse ratios oftwo equa ratios ae eaual. The property sca
a b
1. We gothens
invertendo.
98 RATIO, PROPORTION, AND vatRTON |
f= Fithen 2 = 9, ic, the ratios of antecodents
, wees 2 = D6, the ratios of antecoclonts and consaquants of to aqua ratios are
qual. This property is called alternendo,
= d
pecthiae die, adding 1 to both sides.
|
lirfqpis property is called componendo,
c
et 3 = q then ., subtracting 2 from both sides.
Lun this property is called dividendo.
+ +
a. atb _ £t4) je, dividing the results of componendo by dividendo.
lr ‘his property is called componendo-dividendo.
6, Equal Ratio
af. 2. __SumofNumerators_ _ a*c*e.
tp d f each ratio = Sym of Denominators = ‘pedtt.
the principle can also be applied after multiplying the Numerator and Denominator of any
fraction by the same number. i
gy B= G = Fe ow then cach rato is eau 10 (sectacrcte ea)’, whee pane
may have any values except that they must not all be zeros.
SOLVED EXAMPLE
sons are in the ratio 3: 5. If each receives an increase of
Q: The monthly salaries of two per
tered to 13 : 21. Find their respective salaries.
Rs.20 in the salary, the ratio is al
A: Let the salaries be 3x and 5x.
ax+20 _ 13,
5x+20 ~ 22
_ 24(3x + 20) = 13(5x + 20)
63x + 420 = 65x + 260 x= 80
“= The required salaries are Rs.240 and Rs.400.
99
us /CATaputt
sees | ARITHMETIC
Concept Builder 2.
eRe
2 and its perimeter js
er IS 94 cm,
s
Wha
te 1
1. The sides of a triangle are in the ratio 3):
is the length of the smallest side?
aitoh 76 ;
If 49° find the value of a: b.
2, a-b
3. Ina ratio, whichis equal to 5 : 6, if the consequent is 42, then the antecedent ig
4. The salaries ofA B and C ae in the ratio 4 : 2:3 Ifthe increments of 205, 10% ang
10% ands,
are alowed, respectively in their salaries, then what will be the new rato of thei at
ie
5. The average age of three girls is 24 years and their ages are in the ratio 3 : 4 .
age of the oldest gil is: i rato S425. ty
‘Answer Key
OFS) E9: pyr ee 1 e ee
£9: phi be oy se € § % ‘Swope 5 AATIO, PROPORTION, ann ASEM Ls
ortion
ee ot oto forms eat tho ratio of two othor tornns, then these f
wat bo 9 propartion, 4a i 0 10th tenia brcrandd era in propetlons drtstauay |
wi gad aonb ed proportion, His alo
ree are calle tho i, gecont third andl fourth proportionals raspactivoly, Th
Bg ae called the extremes, while b and c are called the means m The terme
and ly = ed then ad = Des Ht the product of extromes is equal 10 the product of the 1
ntean Proportion
a, > Si
men 22 Betton a, Band c re said to be in continued proportion and bis called the mean
proportional petween & and c.
iso? = 2% Gr Le B= vac
bis also called the geometric mean of a and ¢. The concept of geometric mean will be covered
in detail in the next chapter of this module.
id b.
qhird Proportional
.¢ the third proportional to a an
is the duplicate ratio of the
se then c is said to be
1° to the 3
a:b=>
tional, then the ratio of the
14.3 quantities are propor
2 to the 2".
le Fora:bubic=a? b
Continued Proportion
weebe ge da. then a, b, © ds On 218 said to be in continued proportion.
Je following situation
‘the concept of continued proportion is useful in th
b,c and d are such that a:b, bi cand: dare known then
con tregardss SORES
1, If four quantities a
alo
d are known, then we can find
a, b,¢ and d are four quantities and 2 |B, BG 6°
a:bie:d.
Example
Wa: 13;bie 5,c:d=7:9 find a:
a az, be = 12:15 (LCM of 3 and 4 is 12)
@: 12:15
a: b:c¢ = 56:84: 105, ¢ 105 : 135 (LCM of 7 and 15 is 105)
aib:c¢:d= 56:84: 105: 135
102
a=, 4 CATapult
bad @ Milne
SOLVED EXAMPLES
in
j i Q: Ihab, be, x and c? are in proportion, find x,
| { \ aX yea,
| MY be Fo
@ : The sum of two numbers is 10 and their geometric mean is 4. Find the numbers,
A: If one number is x, the other number is 10 ~ x,
(GMy? = x(20 - x); a = 10x ~ x?
x7 - 10x +16 = 0
| = & = Kk -2)=0->x=Borx
' :. The numbers are 8 and 2.
| Concept. Builder 2 e oe
4. _R5.262 is divided among A, 8 and C such that for every R.2 that A receives, B receives
Vv Rs.5 and for every Rs.3 that B receives, C receives Rs.7. Find A’s share.
2. The third proportional to (x? - y2) and (x = y) is:
3. WA:B=4:5andB:C=4:5, then A: Cis:
4. Asum of Rs.350is divided asRs.A, Rs.B, Rs.CandRs.D among fourpeoplesuch that
12A = 15B = 20C = 30D. What is the value of (A + C) - (B + D)?
5. The ratio of the incomes of A and B is § : 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:
6. An amount of Rs.735 was divided between A, B and C. If each of them had received Rs.25
less, then their shares would have been in the ratio of 1 : 3: 2. The money received by C
was:
| ‘Answer Key
sve “su "9 ooor'sy *¢ 0g ‘Su 'y
uty T
St
7 TIO, PHOPOWTION, AD YAMIRTHONT 2
iatlon
1 proportion (Varlation)
Mio quantities ‘and ‘b’ aro related, such that any increnne of RnereeaTInC UI arcTent te
‘a’ or vice-versa, then the two quantities ars said to be in
ry
11 WH ionate increase or decrease
roportion.
words, the ratio of ‘a’ and ‘b' is a constant
ment 'y' varies as ‘x’ is symbolically written as y « %-
then y = KX, Where k is a non-zero constant called the constant of variation,
direct P
n other
sre state
tye®
y= hes cated the ection of direct variation.
nverse Proportion (Variation)
if two quantities ‘a’ and’ are related, such that any increase in ‘a’ would lead to a decrease in
iy’ or any decrease in ‘a’ would lead to an increase in ‘b’, then the two quantities ‘a’ and ‘b’ are
said to be in inverse proportion.
he statement ‘y’ varies inversely 35
k, where k, a non-zero constant, is the constant of variation.
+e is symbolically written as y a
k
nya Ey then y= ye be Y=
wy = kis called the equation of indirect variation.
Mixed Variation
4. In this type we have both a direct and inverse relationship between the variables involved.
Ita variable x is dependent on two variables y and z and varies inversely as y (when z is constant)
and directly as z (when y is constant), the relationships can be written as
1
+ and xaz
xa y and xa
We can combine the two as
z kz
xa Z orx= ©, where k is a non-zero constant of proportionality.
vis
Example
It the square of x varies as the cube of y, and x = 2, when y = 3, find the equation between
x and y.
103 | [CATapurt
ARITHMETIC
2. Another type of mixed variation is when the variable x can be expressed as a sum of a
constant ¢ and a variable y (to which it is directly proportional) fe.
x=c+ky
Example
The expenses of a boarding house are partly fixed and partly variable as the number of boarders,
The charge is Rs.70 per head when there are 25 boarders and Rs.60 per head when there are
50 boarders. Find the charge per head when there are 100 boarders.
Expenses = Constant + Variable
:. E= m+ kb, where m and k are constant.
2. 1750 = m + 25k; 3000 = m + 50k
Subtracting the first equation from the second,
1 E= 500 + 50b
= Expense when there are 100 boarders = E = 500 + 50 x 100 = Rs.5500
25k = 1250; k = 50, m = 500
. Charge per head = Rs.55.
Important results of proportion
1. IAaBandBaC, thenAaCc
IfAa Cand Ba, thenA+BaCand VAB aC
2
3. ItAa BC, then Ba & andCa @
4. If Aa B and Ca D, then AC a BD
5. IfAaB, then A’ a B”
If Aa B, then AP a BP, where P is any quantity, variable or constant.
SOLVED EXAMPLE
If © then prove the following
°
i
vo ) 22#50 | 4a=7b
2c+5d ~ dc
») 2atse , 4an7e
2b+5d ~ 4b—7d
104 RATIO, PROPORTION, AuD vAMIAHOH |b
dk. Substituting these values can solve most of the
4a- 7b. Abk- 7). DUAR
ac- 7d ~ Adk- 7d “alae 4 “4
Abk- 74k
4b- 7d
pplcations of Ratios & Proportion in Numbers
‘ven the sum of two unknown variables, the product of the two variables say (a and b) is
righest when they are equal.
L
Example
Given a + b= 10
Sfesible values of (a, b) are (0, 10) (2, 9) (2, 8)
Ox 10 = 0,1x9= 9,2 8 = 16, 5% 5 = 25
"the product of the two numbers is highest when they are equal.
. (8, 2) (10, 0)
(5, 5)
iven that two variables are inversely proportional to each other 1.é the product of the two
variables is constant, the sum of the two variables will be least when they are equal.
Example
Given a x b = 100
Possible values of (a, b) are (2, 100) (2, 50) (4, 25) ..... (20, 10)
1+ 100 = 101, 2 + 50 = 52, 4 + 25 = 29, 10 + 10 = 20
"the sum of the two numbers is least when they are equal.
.. (20, 5) (200, 1)
gx byw eis maximum when 2 = 2 = © provided a + b + c is constant
Example
Find the maximum value a? x b2, where a + b = 10
3b > 2a- 3b=0... @
Now, the product will be maximum when 3 = $ = 2a
Also given a+b = 10 wn. (ii)
Solving () and (ii) we get a = 6 and b= 4
Maximum value of a? x b? = 6? x 4? = 3456
105 @D | cMaputt
| ARITHMETIC
Eee
SOLVED EXAMPLE
Q: The cost of a diamond varies as the square of its weight. A diamond weighing 10 decigrams
costs Rs.1600. Find the loss incurred when it breaks into two pieces whose weights are in the
VV tatio 2:3.
we
A: If Cis the cost and W is the weight, Ca WC
1600
= 10: K = Cy = 1600 _ a6, = 16W?
When C = 1600, W = 20; K = Wr = “agg” = 16; « ¢ = 16W’
The weights of the two pieces (in decigrams) are: $ x 10 and B * 10, ie, 4 and 6.
Cost of the two pieces are 16 x 4?.and 16 x 6%,
1 Total cost = 256 + 576 = Rs.832
-. Loss incurred = 1600 ~ 832 = Rs.768
i Concept Builder 3
2. The electricity bill of a certain establishment is partly fixed and partly varies as the timber ©
. of units of electricity consumed. When in a certain month 480 units are consumed, the —
bill is Rs.1600. In another month when 550 units are consumed, the bill is s.1810, In
another month when 600 units are consumed, the bill for that month would be:
‘Answer Key
096T
ovga'sy
106 a Chapter | QA
AATIO, PROPORTION, AND VARIATION | 1.6
partnership
partnership is an association of two or more persons who invest their money in ordar ta earry on 9
certain business. A partner who manages the business is called the working, partner arid the one
‘who simply invests the money is called the Sleeping Partner or the Dormant Partner, Partnership
ig based on the concept of Ratio and Proportion.
simple & Compound Partnership
if the capitcls of the partners are invested for the same period, the partnership is called simple,
and if for different periods, is called compound.
About Simple Partnership
If the period of investment is the same for each partner,
ratio of their investments.
then the profit or loss is divided in the
4. If A and B are partners in a business, then:
Investment of A. _ ProfitofA op InvestmentofA _ Loss.ot A
Investment of B ~ Profit of B Investment of B ~ Loss of B
2. If A, B and C are partners in a business, then:
Investment of A : Investment of B : Investment of C
= Profit of A : Profit of B ; Profit of C OR Loss of A : Loss of B : Loss of C
‘About Compound Partnership
Monthly Equivalent Investment is the product of the capital invested and the period for which it is
invested.
If the period of investment is different, then the profit or loss is divided in the ratio of their
Monthly Equivalent Investment.
2. If A and B are partners in a business, then,
Monthly Equivalent Investment of A _ Profit of A
Monthly Equivalent Investment of 8 ~ Profit of B
jie, Investment of AxPeriodof Investmentof A _ ProfitofA. op
“Investment of B x Period of Investment of B ~ Profit of B
Monthly Equivalent Investment of A _ Loss of A.
Monthly Equivalent Investment of B ~ Loss of B
ie, Investment of A x Period of Investment of A _ Loss of A
Investment of 8 x Period of Investment of B ~ Loss of B
2. If A, B and C are partners in a business, then,
Monthly Equivalent Investment of A : Monthly Equivalent Investment of 8
Monthly Equivalent Investment of C = Profit of A : Profit of B : Profit of C
OR
Monthly Equivalent Investment of A : Monthly Equivalent Investment of B
: Monthly Equivalent Investment of C = Loss of A : Loss of B : Loss of C
107 CATapult
ARITHMETIC
‘SOLVED EXAMPLES
Q: Three partners A, B and C invest Rs.1600, Rs.1800 and Rs.2300 respectively in a business,
How should they divide a profit of Rs.3997
A: Profit is to be divided in the ratio 16 : 18 : 23
16
16+18+23
18 5
57 * 399 = Rs.126
A's share of profit,
16
x 399 = Gp x 399 = Rs.112
B's share of profit
23
7 * 399 = R161
C's share of profit
@ = A and B enter into a partnership. A puts in Rs.2000 but at the end of 3 months, withdraws
Rs.500 and again at the end of 8 months withdraws Rs.300, Out of a total profit of Rs.900 at
the end of the year, B's share was Rs.400. Find B's capital,
A Ratio of profits = (A's Rs.2000 for 3 months) + (A's Rs.1500 for § months) + (A's Rs.1200 for
4 months) : (B's capital x for 12 months) = (6000 + 7500 + 4800) :
18300 _ 500, | 1525 _ 5
Bex= Dx = 4008 gh x= R520
108 Chapter |QA
RATIO, PROPORTION, AND VARIATION |1.6
1, Sunil and Anil started a business and invested Rs.120000 and Rs.180000 respectively.
After 3 months Anil left and Kapil joined by investing Rs.150000. At the end of the year, %
they realized a profit of Rs.148000. What was Kapil’s share?
p
P, Q and R invest Rs.30000 for a business. P invests Rs.1000 more than Q and Q invests
Rs.4000 more than R. Out of the total profit of Rs.18,000, R receives:
3, A,B and C started a business with their investments in the ratio 2 : 2 : 3. After 6 months,
A invested the same amount as before and B and C withdrew half of their investments. 2
The ratio of their profits at the end of the year is: u
4, A,B and C enter into a partnership. They invest Rs.25000, Rs.50000 and Rs.75000,
respectively. At the end of the first year, B withdraws Rs.25000 while at the end of the 2
second year C withdraws Rs,50000. In what ratio the profits have to be shared at the
end of 3 years? .
5, P,Qand R hire a meadow for Rs.1560. If P puts in 12 cows for 15 days; Q puts 18
cows for 10 days and R puts 20 cows for 8 days, then the rent paid by Q is:
6. P-and Q invest in a business in the ratio 5 : 3. If 10% of the total profit goes to charity
and P's share is Rs.270, the total profit is:
7. Three partners A, B and C started a business. A's capital is equal to twice B’s capital and
B's capital is two-third times C’s capital. Out of a total profit of Rs.14400 at the end of
the year, C’s share is:
8. Two partners X and Y start a business. Twice Y's capital is equal to thrice X’s capital.
‘ut of an annual profit of Rs.40,000 at the end of the year, X's share is:
9. If the profits earned by X, Y and Z are in the ratio of 3: 5 : 9, respectively and the
P period for which each of them invested the amount in business is in the ratio of 3 :
4: 6 for X, Y and Z, respectively, then what is the ratio of the investment of X, Y and
2
10, Raju, Ravi and Ramu invest Rs.15000, Rs.25000 and Rs.30000, respectively in a business.
Alter one year, Raju removed his money but Ravi and Raju continued for one more year.
If the net profit after 2 years is Rs.25200, then Raju’s share in the profit is:
11.. A, B and C started a business, two times of A’s capital is equal to 3 times of B's capital
and equal to 5 times of C's capital. Out of a profit of Rs.1550, A’s share is:
12. Varun and Tarun started a business with capitals in the ratio of 3: 4.
After one year,
Varun's profit is Rs.2100. Then what is the total annual profit?
‘Answer Key
‘006v'sd “ZT. ‘oodst ‘su 8 coe ot
‘ose “su “TT ooa'sy % Geddy G
“vz0e'sd “OT ‘osy'sy ‘oory'syZ
o:siy 6 ‘osu Ss o0009's4 = T
109
