Mastering Ratios and Proportions: Essential Theories and Shortcut Tricks Description:

This comprehensive PDF, "Mastering Ratios and Proportions: Essential Theories and Shortcut Tricks," is an indispensable resource for students, math enthusiasts, and professionals seeking to enhance their understanding of ratios and proportions. It covers essential theories, practical examples, and efficient shortcut techniques for quickly and accurately solving ratio and proportion problems. The guide delves into fundamental concepts such as defining ratios and proportions, understanding their properties, and exploring their applications in real-world scenarios. Theoretical insights include various methods for simplifying ratios, solving proportion equations, and working with direct and inverse proportions. Practical examples illustrate these concepts in contexts like financial analysis, recipe adjustments, and scaling models. Unique shortcut tricks streamline calculations, making problem-solving more efficient. Additionally, the guide explores real-world applications in fields such as economics, physics, and everyday life, highlighting the relevance of ratios and proportions. It also provides problem-solving strategies, common mistakes to avoid, and practice problems with detailed solutions. Perfect for enhancing problem-solving skills and exam preparation, this guide ensures readers can confidently apply their knowledge of ratios and proportions in both academic and practical contexts.

1. PEA215–Lecture#
Ratio & Proportion

2. Intheratioa:b,wecallaasthefirstterm
orantecedentandb,thesecondtermorconsequent.
Eg.Theratio5:9represents5withantecedent=5,
consequent=9
Rule:Themultiplicationordivisionofeachtermofa
ratiobythesamenon-zeronumberdoesnotaffectthe
ratio

3. Types of Ratio

4. 5. Inverse or Reciprocal Ratio:The inverse ratio of a: b is 1/a: 1/b
Example: If 2: 3 is a ratio, then its inverse ratio is (1/2) : (1/3)
6. Compounded Ratio: Compound ratio is the ratio of the
products, of the corresponding terms of two or more simple
ratios.
Example: The compounded ratio of the ratios: (A : B), (C : D), (E :
F) is (ACE : BDF).

5. Proportion
Theequalityoftworatiosiscalledproportion.
Ifa:b=c:d,wewritea:b::c:dandwesaythata,b,c,
dareinproportion.Then,
�/�=�/�
Hereaanddarecalledextremes,whilebandcare
calledmeanterms.
Productofmeans=Productofextremes.
Thus,a:b::c:d(bxc)=(axd)

6. If a/b=c/d, then:
i.Invertendo-b:a = d:c
ii. Alternendo-a:c = b:d
iii. Componendo-(a+b):b = (c+d):d
iv. Dividendo-(a-b):b = (c-d):d
v.Componendo& Dividendo–
(a+b)/(a-b) = (c+d)/(c-d)

7. DirectandInverseProportion/Variation
IfXisdirectlyproportionaltoY,itmeanswhenXincreasesY
increasesandviceversa.IfXisinverselyproportionaltoY,it
meanswhenXincreasesYdecreasesandviceversa.
Example:I=Q/twhereQisthecharge,Iisthecurrentandtis
thetime.
Here,IisdirectlyproportionaltoQwhentimeisconstant,andI
isinverselyproportionaltotwhenchargeQisconstant.

8. Example :
If 0.75 :x:: 5 : 8, thenxis equal to:
A.1.12
B.1.2
C.1.25
D.1.30
Ans: B

9. TypesofProportions
FourthProportional:
Ifa:b=c:d,thendiscalledthefourthproportional
toa,b,c.
ThirdProportional:
a:b=c:d,thenciscalledthethirdproportion
toaandb.
MeanProportional:
Meanproportionalbetweenaandbis√ab.

10. ExamplesonRatioAndProportions
Q1.TheratioofA:B:Cis5:2:3
a)ifthesumofB&Cis75findtheno's
b)ifthesumofA&Bis36morethanC,FindA,B,C
c)IftheproductofB&Cis96findtheno's
d)ifBis18,Findtheno's
Ans: a) 75, 30, 45
b) 45, 18, 27
c) 20, 8, 12
d) 45, 18, 27

11. Q2.TheratioofB&Gis10:3,when36girlmore
joinedtheratiobecomes10:7.Findtheno.ofboys?
A.90
B.100
C.60
D.None
Ans:A,90

12. Q3.TheincomeratioofA&Bis5:8,ifincomeofA
increasesby60000,thenthenewratiois5:4,Find
currentincomeofA.
A.120000
B.240000
C.360000
D.None
Ans:A,1,20,000

13. Q4.Ratioofboysandgirlis4:5when100girlleft
theratiobecomes6:7,findno.ofboys
A.1200
B.600
C.800
D.None
Ans:A,1200

14. Q5.Theratioofthenumberofboysandgirlsina
collegeis7:8.Ifthepercentageincreaseinthe
numberofboysandgirlsbe20%and10%
respectively,whatwillbethenewratio?
A.8:9
B.17:18
C.21:22
D.Cannotbedetermined
Ans:C,21:22

15. Q6. 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?
a)3 : 3 : 10
b)10 : 11 : 20
c)23 : 33 : 60
d)Cannot be determined
Ans: C, 23:33:60

16. Q7.IfRs.782bedividedintothreeparts,
proportionalto1/2:2/3:3/4,thenthefirstpartis:
A.Rs.182
B.Rs.190
C.Rs.196
D.Rs.204
Ans: D, Rs 204

17. Q8.Thefourthproportionalto5,8,15is:
A.18
B.24
C.19
D.20
Ans:B,24

18. Q9.Inabag,therearecoinsof25p,10pand5pin
theratioof1:2:3.IfthereisRs.30inall,how
many5pcoinsarethere?
A.50
B.100
C.150
D.200
Ans:C,150

19. Q10. A bag contains Rs. 600 in the form of one-rupee,
50 paiseand 25 paisecoins in the ratio 3 : 4 : 12. The
number of 25 paisecoins is
A.600
B.900
C.1200
D.1376
Ans: B, 900

20. Q11. If x:y is 2:3, find the value of (3x + 2 y) :(2x+5y)
A.12 : 25
B.11 : 27
C.12 : 19
D.11 : 23
Ans: C, 12 : 19

21. Q12. Ruby stone Rs. 6800 worth was dropped and
broke into three pieces. The weight of the 3 pieces is 5,
7 and 8. The value Ruby is proportional to the square of
its weight. Find the loss.
A.Rs. 4454
B.Rs. 7890
C.Rs. 6785
D.Rs. 4500
Ans: A, Rs4454

23. Q13. If x : y = 3 : 4 and y : z = 8 : 9, z : a is 15 : 16,
find x : y : z : a
A.78 : 82 : 65 : 45
B.30 : 40 : 45 : 48
C.76 : 90 : 56 : 80
D.None of these
Ans: B, 30 : 40 : 45 : 48

24. Q14. An organization reduces the number of interns in
the ratio 7:5 and reduces their stipend by 10:9. State
whether his bill of total stipend increases or decreases in
what ratio?
A.Decreases 12 : 5
B.Increases 9 : 14
C.Decreases 13 : 9
D.Decrease 14 : 9
Ans: D

25. Q15. What should be subtracted from each of the
numbers 54, 71, 75, 99 so that the remainders are in
continuous proportion?
A.3
B.7.5
C.9
D.10
Ans: A, 3

26. Q16. Jyothi got thrice as many marks in English as in
Science. The proportion of her marks in English and
History is 4:3. If his total marks in English, Science and
History are 250, what are his marks in Science?
A.60
B.40
C.100
D.120
Ans: B. 40

27. Q17. Rs. 1400 is divided among Arun, Bheem, Chirag
so that Arunreceives half as much as Bheemand
Bheemhalf as much as Chirah. Then Chirag's share is
A.Rs. 500
B.Rs. 600
C.Rs. 700
D.Rs. 800
Ans: D, Rs. 800

28. Q18. There are 12 animals in a zoo, what is their head to
leg ratio if there are 4 goats, 2 ducks and 6 gorillas?
A.8:3
B.3:8
C.4:5
D.7:9
Ans: B, 3:8

29. Q 20. In a bowl there is 30 litre mixtures of milk and
water. The ratio of milk and water is 7:3. How much
water must be added to it so that the ratio of milk to
the water be 3:7?
A.20 Ltr
B.40 Ltr
C.60 Ltr
D.50 Ltr
Ans: B, 40 Ltr

30. Q21. A sum of money is shared between Kamal, Vimal,
Keyar and Nishant in the ratio 5:2:4:3. If Keyar gets Rs.
1000 more than Nishant, What is Vimal's share?
A.Rs. 2000
B.Rs. 1000
C.Rs. 700
D.Rs. 600
Ans: A, Rs. 2000

31. Q22. The amount Rajesh and Ajjuearns are respectively
20% and 50% more than what Kiranearns. What is the
ratio at which Rajesh and Ajjuearn?
A.2:5
B.3:4
C.4:5
D.8:9
Ans: C, 4:5