This document contains information about ratios, proportions, and variations. It begins with introducing ratios and ratio terminology. It then discusses various properties of ratios, including how ratios change when multiplied or divided by the same number. It also covers combining multiple ratios. The document next discusses properties of proportions, including the relationship between extremes and means. Finally, it addresses different types of variations like direct, inverse, and joint variations. Examples are provided throughout to illustrate the concepts.

1. www.georgeprep.com
1
Ratio, Proportion and Variation

2. www.georgeprep.com
2
Chapter Pathway
02
03
04
01
05
Ratio – Introduction and Terminology
Ratio – Properties
Combining Ratios
Proportion - Properties
Variation - Properties

3. www.georgeprep.com
3
Fraction Comparison
Equating the Denominator
Equating the Numerator
Difference Method
Cross Multiplication Method
Supplement Videos
Decimal Approximation Method

4. www.georgeprep.com
4
Ratio - Introduction and
Terminology

5. www.georgeprep.com
5
Ratio - Introduction
Ratio: Comparison of magnitude of two
or more quantities.
Example Problem:
The ratio of the expenses of A and B is 5:9. If the sum of
their expenses is Rs. 28000.
-What is A’s expense?
-By how much is B’s expense greater than A’s expense?

6. www.georgeprep.com
6
Ratio
What is the value of A:B:C ? If,
a) A:B:C =
1
3
∶
1
2
∶
1
5
b)
A
3
=
B
2
=
C
5

7. www.georgeprep.com
7
Ratio - Terminology
• Antecedent and consequent
Ratio of a and b 𝒂 ∶ 𝒃
Duplicate ratio of a and b 𝑎2
∶ 𝑏2
Triplicate ratio of a and b 𝑎3
∶ 𝑏3
Sub-duplicate ratio of a and b 2
𝑎:
2
𝑏
Sub-tuplicate ratio of a and b 3
𝑎:
3
𝑏
Antecedent Consequent
𝑎 ∶ 𝑏

8. www.georgeprep.com
8
Problems
If 5𝑎 + 2 ∶ 12𝑎 + 4 is the triplicate ratio of 3: 4, what is the value of 𝑎?
Answer: 5

9. www.georgeprep.com
9
Ratio - Properties

10. www.georgeprep.com
10
Ratio - Introduction
Value of ratios when multiplied or divided by the
same number remains the same
For Example:
3 ∶ 4 ~ 6 ∶ 8

11. www.georgeprep.com
11
Ratio - Properties
Change of values of proper and improper fractions when
the same value is added or subtracted to the Numerator
and the Denominator.
Proper Fraction
(fractions of the form 𝑝/𝑞, where 𝑝 < 𝑞)
Improper Fraction
(fractions of the form 𝑝/𝑞, where 𝑝 ≥ 𝑞)
𝑝
𝑞
<
𝑝 + 𝑥
𝑞 + 𝑥
, 𝑥 > 0
𝑝
𝑞
>
𝑝 + 𝑥
𝑞 + 𝑥
, 𝑥 > 0
𝑝
𝑞
>
𝑝 − 𝑥
𝑞 − 𝑥
, 𝑥 > 0
𝑝
𝑞
<
𝑝 − 𝑥
𝑞 − 𝑥
, 𝑥 > 0

12. www.georgeprep.com
12
Ratio - Properties
If
𝑎
𝑏
=
𝑐
𝑑
=
𝑒
𝑓
……. then
𝑎 + 𝑐 + 𝑒 +⋯
𝑏+𝑑+𝑓+⋯
= each of
𝑎
𝑏
,
𝑐
𝑑
… as long as
(𝑏 + 𝑑 + 𝑓 + ⋯ ) ≠ 0

13. www.georgeprep.com
13
Ratio – Properties (Summary)
• Value of ratios when multiplied or divided by the same number
remains the same
• Change of values of proper and improper fractions when the same
value is added or subtracted to the Nr and the Dr.
• If
𝑎
𝑏
=
𝑐
𝑑
=
𝑒
𝑓
……. then
𝑎 + 𝑐 + 𝑒 +⋯
𝑏+𝑑+𝑓+⋯
= each of
𝑎
𝑏
,
𝑐
𝑑
… As long as (𝑏 +
𝑑 + 𝑓 + ⋯ ) ≠ 0
• Given
𝑎
𝑏
,
𝑐
𝑑
,
𝑒
𝑓
, the value of
𝑎+𝑐+𝑒
𝑏+𝑑+𝑓
will be between the lowest and
highest fraction.

14. www.georgeprep.com
14
Problems
If
𝑎+𝑏
𝑐
=
𝑏+𝑐
𝑎
=
𝑎+𝑐
𝑏
= p, what is the value of p?
1) 2
2) -1
3) 3
4) Either option 1 or option 2
Answer: option 4

15. www.georgeprep.com
15
Ratio – Combining Ratios

16. www.georgeprep.com
16
Combining Ratios - Technique
𝐴 ∶ 𝐵 = 1 ∶ 2
𝐵 ∶ 𝐶 = 3 ∶ 4
𝐴 ∶ 𝐶 = ?
How about combining more than two ratios?

17. www.georgeprep.com
17
Problems
If A:B = 3:1, C:B = 2:1, C:D = 1:3 what is the value of A:D?
1) 1 : 2
2) 2 : 1
3) 1 : 3
4) None of these
Answer: option 1

18. www.georgeprep.com
18
Problems
A has thrice the amount of money that B has, B has twice the amount
of money that C has and thrice the amount of money with C is equal
to four times the amount of money with D. The amount of money with
D is what percentage of the amount of money with A?
1) 200
2) 25
3) 12.5
4) None of these
Answer: option 3

19. www.georgeprep.com
19
Proportion - Properties

20. www.georgeprep.com
20
Proportion - Properties
𝑎, 𝑏, 𝑐 and 𝑑 are in proportion if
Fourth proportional
𝑎: 𝑏 = 𝑐: 𝑑
𝑎: 𝑏 = 𝑐: 𝑑
Extremes
Means
Product of extremes = Product of Means

21. www.georgeprep.com
21
Proportion – Continued Proportion
𝑎, 𝑏 and c are in continued proportion if
Third proportional
𝑎: 𝑏 = 𝑏 ∶ 𝑐
𝑏 = 2
𝑎𝑐
Second Proportional (or)
Mean Proportional

22. www.georgeprep.com
22
Proportion - Properties
If
𝑎
𝑏
=
𝑐
𝑑
, then
Invertendo
𝑏
𝑎
=
𝑑
𝑐
Alternendo
𝑎
𝑐
=
𝑏
𝑑
Componendo
𝑎 + 𝑏
𝑏
=
𝑐 + 𝑑
𝑑
Dividendo
𝑎 − 𝑏
𝑏
=
𝑐 − 𝑑
𝑑
Componendo - Dividendo
𝑎 + 𝑏
𝑎 − 𝑏
=
𝑐 + 𝑑
𝑐 − 𝑑

23. www.georgeprep.com
23
Problems
Ratio of 𝑝: 𝑞 is the same as that of 𝑞: 𝑟, where 𝑝, 𝑞 and 𝑟 are positive
numbers. If 𝑝 = 16 and 𝑟 = 100, find 𝑞.
1) 40
2) 64
3) 24
4) None of these
Answer: option 40

24. www.georgeprep.com
24
Problems
Find the fourth proportional to 1, 3 and 4
1) 12
2) 24
3) 18
4) None of these
Answer: option 12

25. www.georgeprep.com
25
Variation - Properties

26. www.georgeprep.com
26
Variation - Properties
Direct Relation Vs Direct Variation

27. www.georgeprep.com
27
Inverse Variation
Inverse Variation

28. www.georgeprep.com
28
Joint Variation
Joint Variation

29. www.georgeprep.com
29
Problems
4 men working 5 hours a day can complete 4 units of work in 48 days. In how
many days will 12 men working 6 hours a day complete a work of 6 units?
1) 20
2) 15
3) 24
4) 16
Answer: option 1

30. www.georgeprep.com
30
Problems
A garrison of 2200 men has ration for 16 weeks at the rate of 60gms per day
per man. How many must leave the garrison if the ration should last for 32
weeks at the rate of 44gms per day per man?
1) 200
2) 750
3) 700
4) 6
Answer: option 3

31. www.georgeprep.com
31
Problems
In a hostel of 400 students, there was ration for 15 days. After 5 days, 100
more students join the hostel. If each student consumes at the same rate, for
how man days will there be a shortage of ration?
1) 2
2) 4
3) 1
4) 3
Answer: option 1

32. www.georgeprep.com
32
Problems
70 cows can graze on a field for 24 days and 30 cows can graze on the same
field for 60 days. Some grass is already there and grass grows at a fixed rate.
How many cows can graze on the field for 96 days?
1) 20
2) 25
3) 18
4) None of these
Answer: option 1

33. www.georgeprep.com
34
Problems
A boy has a few coins of denominations 50 paise, 25 paise and 10 paise in the
ratio 1: 2: 3. If the total amount of the coins is Rs. 6.50, the number of 50
paise coins is
1) 5
2) 10
3) 15
4) 20
5) 25
Answer: option 1

34. www.georgeprep.com
35
Problems
In a mixture 60 litres, 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
1) 20 liters
2) 30 liters
3) 50 liters
4) 60 liters
5) 80 liters
Answer: option 4