ratio and proportion

1. Aptitude presentation on
Ratio and proportion
Submitted to-
Mrs. Shubra Mishra
Submitted by-
Sourabh (GF/2020/3377)
Dibiyanshu Gautam (GF/2020/7432)
Shivani (GF/2020/6871)
Arti Sharma (GF/2020/2840)
Anuradha Sharma (GF/2020/4751)
-B.Tech. Biotechnology / (SRP)
- 3RD Year (6th Semester)

2. Ratio -
A ratio, which is a comparison of two numbers by division of the same unit, is the quotient
obtained when the first number is divided by the second, nonzero number in a definite order.
Hence the ratio of a to b can be expressed as
𝑎
𝑏
or a : b. In the ratio a : b, we call a the first
term or antecedent and b is called the second term or consequent.
For example, Let us say salary of A is Rs5000 and salary of B is Rs3000. if we are to compare
salary of A and B we can say that,
The Ratio of salary of A to that of B =
𝑆𝐴𝐿𝐴𝑅𝑌 𝑂𝐹 𝐴
𝑆𝐴𝐿𝐴𝑅𝑌 𝑂𝐹 𝐵
=
5000
3000
=
5
3
The ratio of A to B is usually written as A : B. The quantities A and B are called the terms of
ratios. Here we have, (salary of A) : (salary of B)= 5 : 3.
Note :
1. If ratio of A to B is 5 : 3 then we cannot say that the value of A is 5 and the value of B is 3.
Referring to the example above we say the salary of a is not Rs5 but it is Rs5000.

3. Basic properties-
1. If we multiply the numerator and the denominator of a ratio by the same number, the ratio
remains unchanged.
2. If divide the numerator and denominator of a ratio by the same number, the ratio remains
unchanged. So,
𝑎
𝑏
=
𝑎/𝑑
𝑏/𝑑
2. If the ratio of A to B is 5 : 3 then we can see that the value of A is 5X and the value of B is
3x. (Assuming that X is the common factor with which the terms of the ratios is divided.)
Referring to the above example we say that the value of X is 1000.
𝑆𝑎𝑙𝑎𝑟𝑦 𝑜𝑓 𝐴 = 5𝑥 = 5 × 1000 = 5000 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑠𝑎𝑙𝑎𝑟𝑦 𝑜𝑓 𝐵 = 3𝑥 = 3 × 1000 = 3000.
3. If the total salary of A and B is 8000 and the ratio of the salaries = 5 : 3, then we can find
that the salary of A and B, i.e. the total salary rs 8000 is divided into 8 parts. out of 8 parts 5
part is A and 3 parts is B.
Hence,
5𝑡ℎ
8
of 8000, i.e.
5
8
× 8000 = 5000 is the salary of A and
3𝑡ℎ
8
of 8000,
i.e.
3
8
× 8000 = 3000 is the salary of B.

4. 3. a. 𝑎2
: 𝑏2
is called the duplicate ratio of a : b. It is a compound ratio of two equal ratios.
b. 𝑎3 : 𝑏3 is called the triplicate ratio of a : b. It is the compound ratio of two equal ratio.
c. 𝑎 : 𝑏 is called sub-duplicate ratio of a : b
d. 𝑎1/3 : 𝑏1/3 is called the sub-triplicate ratio of a : B. The ratio between the cube roots of
two numbers is called the sub-triplicate ratio of two numbers.
e. If a : b , and c : d are two ratios, then ac : bd is called the ratio compound of the given
ratio.
4. Two or more ratios can be compared by reducing the equivalent fractions
through a common denominator.
If
a
b
and
x
y
are the two ratios to be compared, then we have
a
b
=
ay
by
and
x
y
=
xy
by
. now after reducing to the common denominator we can say that
if ay > bx, then
a
b
>
x
y
and if ay < bx, then
a
b
<
x
y
.
5. The ratio of two fractions can be expressed as a ratio of two integers.
a
b
:
c
d
then this ratio is equal to ad ∶ bc

5. Proportion-
When two ratios are equal, the four quantities composing them are said to be
proportional. If we say that a, b, c, d are in proportion and, we write, a : b :: c : d. Here
a and d are known as extremes and b, and c are known as means. If any proportion,
the product of the extremes is equal to the product of the means.
If the proportion Is
𝑎
𝑏
=
𝑐
𝑑
, then 𝑎𝑑 = 𝑏𝑐.
Hence the 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑚𝑒𝑎𝑛𝑠 = 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑠.
Basic properties-
1. In a :b :: c : d , we say that d is the fourth proportional to a, b, and c.
2. If x is the third proportional to a, b then a : b :: c : x
3. If a, b, c, are such that
𝑏
𝑎
=
𝑐
𝑏
, then b is called the geometric mean between a and c and
𝑏2 = 𝑎𝑐, that is 𝑏 = 𝑎𝑐 = ±(𝑎𝑐)1/2 .
4. If
𝑎
𝑏
=
𝑐
𝑑
then
𝑎+𝑏
𝑎−𝑏
=
𝑐+𝑑
𝑐−𝑑
and
𝑎−𝑏
𝑎+𝑏
=
𝑐−𝑑
𝑐+𝑑

6. 5. INVERTENDO:
If
𝑎
𝑏
=
𝑐
𝑑
, 𝑡ℎ𝑒𝑛
𝑏
𝑎
=
𝑑
𝑐
, 𝑖. 𝑒. the inverse ratio of two equal ratios are equal.
This property is called 𝑖𝑛𝑣𝑒𝑟𝑡𝑒𝑛𝑑𝑜.
6. ALTERNENDO:
If
𝑎
𝑏
=
𝑐
𝑑
, 𝑡ℎ𝑒𝑛
𝑎
𝑐
=
𝑏
𝑑
, 𝑖. 𝑒. the ratio of antecedents and consequents of two equal ratios
are equal. This property is called 𝑎𝑙𝑡𝑒𝑟𝑟𝑛𝑒𝑛𝑑𝑜.
7. COMPONENDO:
If
𝑎
𝑏
=
𝑐
𝑑
, then
𝑎−𝑏
𝑏
=
𝑐+𝑑
𝑑
, i.e. adding 1 to both sides. This property is called 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑑𝑜.
8. DIVIDENDO:
If,
𝑎
𝑏
=
𝑐
𝑑
,then
𝑎−𝑏
𝑏
=
𝑐−𝑑
𝑑
,i.e.subtracting 1 from both sides.This property is called 𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑𝑜.
9. COMPONENDO – DIVIDENDO:
If
𝑎
𝑏
=
𝑐
𝑑
, then
𝑎+𝑏
𝑎−𝑏
=
𝑐+𝑑
𝑐−𝑑
, i.e. dividing the results of componendo by dividendo, we get
𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑑𝑜 − 𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑𝑜.

7. • To find a : b : c : d, if a : b and b : c and c : d, is given
if a : b and b : c and c : d is given then a : b : c : d can be solved like the following
multiplication
a : b a b c d
b : c a : b : c : d = ↓ ↙ ↓ ↓
c : d ↓ ↓ ↙ ↓
For example, a : b = 1 : 2 and b : c = 3 : 4 and c : d = 5 : 6 , then a : b : c : d will be
given by
1 : 2 1 2 2 2
↓ ↙ ↓ ↓
3 : 4 a=3 b=3 c=4 d=4
↓ ↓ ↙ ↓
5 : 6 5 5 5 6
So, a : b : c : d = (1 × 3 × 5) : (2 × 3 × 5) : (2 × 4 × 5) : (2 × 4 × 6) = 15 : 30 : 40 : 48

8. • To find a : b : c, if a : b and b : c is given
If a : b and b : c is given then a : b : c can be found by the following
multiplication:- a b
↓↗↓
b c
For example, if a : b = 1 : 2 and b : c = 3 : 4, then a:b:c is given by
a:b:c = 1 2 = (13) : (3×2) : (2×4) = 3 : 6 : 8
↓↗↓
3 4

9. Examples-
Question 1. The sides of a triangle are in the
1
2
∶
1
3
∶
1
4
and its perimeter is 104 centimeters.
Find length of the longest side?
Answer – multiplying each term with 12, we get ratio of sides
=
1
2
× 12 ∶
1
3
× 12 ∶
1
4
× 12 = 6 ∶ 4 ∶ 3
let the sides be 6𝑥, 4𝑥, 3𝑥
perimeter (104) = sum of all sides
104 = 6𝑥 + 4𝑥 + 3𝑥
104 = 13𝑥
𝑥 = 104/13
𝑥 = 8cm
longest side 6𝑥 = 6 × 8 = 48𝑐𝑚.
other sides, 4𝑥 = 4 × 8 = 32𝑐𝑚.
3x = 3 × 8 = 24𝑐𝑚.

10. Question 2. 20 boys and 32 girls form a group for social work. During their membership drive
same number of boys and girls have joined the group. How many members does
this group have now if the ratio of boys to girls is 3 : 4 respectively?
Answer- Let 𝑥 boys and 𝑥 girls have joined
according to the given information, we have
20+𝑥
32+𝑥
=
3
4
𝑥 = 16
Now, present strength of the social group = 20 + 32 + 16 + 16 = 84 .
Question 3.Arrange the following ratios in descending order of magnitude (6:7), (7: 8), (9: 10),
(11: 12)?
Answer- 6/7, 7/8, 9/10, 11/12 , applying the property of ratio of lesser inequality, we have
In the ratio a/b if a < b then
𝑎+𝑥
𝑏+𝑥
>
𝑎
𝑏
(x > 0)
The given ratios could be arranged in the following way:
6
7
,
6+1
7+1
,
6+3
7+3
,
6+5
7+5
then
6
7
<
7
8
<
9
10
<
11
12

11. Question 4. A jar is filled with water and alcohol in the ratio 2:1. when six liters of the mixture
is removed and replaced with six liters of water the ratio of the water to alcohol
becomes 4 : 1. what is the capacity of the jar?
Answer- Since the ratio of the water to alcohol is 2 : 1 let the amount of water and alcohol in
the jar be 2𝑥 and 𝑥 liters.
From the mixture, 6 litres of the mixture is replaced with water.
In this 6 litres, the amount of alcohol =
1
3
× 6 = 2𝑙𝑖𝑡𝑟𝑒𝑠.
The amount of water =
2
3
× 6 = 4 𝑙𝑖𝑡𝑟𝑒𝑠.
It is given that,
2𝑥−4+6
𝑥−2
=
4
1
2𝑥+2
𝑥−2
=
4
1
2𝑥 + 2 = 4𝑥 − 8
2𝑥 = 10 → 𝑥 = 5
The capacity of the jar = 2𝑥 + 𝑥 = 3𝑥 = 3 × 5 = 15 litres.

12. Question. 5 The cost of precious stone is directly proportional to the cube of its weight. A10
gram stone cost 25,000. The stone falls and breaks into three pieces whose
weights are in the ratio 1 : 1 : 3. what is the difference in the cost between the
largest of the three pieces and the smallest?
Answer- It is given that the cost of the stone is directly proportional to cube of its weight
i.e. Cost ∝ 𝑤3
Cost = 𝑘 × 𝑤3, where 𝑘 is the constant of proportionality.
25000= 𝑘 × 103
𝑘 =
25000
1000
= 25
Since the weights of the stones are in the ratio, 1 : 1 : 3, weight of the
largest stone =
3
5
× 10 = 6𝑔𝑚 and weight of the smallest stone=
1
5
× 10 = 2𝑔𝑚.
The difference in cost of the stones= 𝑘 × 63 − 𝑘 × 23 = 216𝑘 − 208𝑘
= 208𝑘 → 208 × 25
= 𝑅𝑠. 5200

13. Question 6. If
𝑎
𝑏
=
7
8
then find the ratio 4𝑎 −
𝑏
2
: 4𝑎 +
𝑏
2
?
Answer-
𝑎
𝑏
=
7
8
𝑎 =
7𝑏
8
4𝑎−
𝑏
2
4𝑎+
𝑏
2
=
4
7𝑏
8
−
𝑏
2
4(
7𝑏
8
)+
𝑏
2
=
7𝑏
2
−
𝑏
2
7𝑏
2
+
𝑏
2
=
6𝑏
8𝑏
=
3
4

14. Question 7. Rupees 1840 is divided into four parts such that a third of the first part, a fifth of
the second part, an eight of the third part, and a seventh of the fourth part are all
equal. Find the difference between the second and the fourth part?
Answer- Let a, b, c, d be the four parts
a+ b+ c+ d = 1840 (1)
Also
𝑎
3
=
𝑏
5
=
𝑐
8
=
𝑑
7
= 𝑘
i.e. a𝑎 = 3𝑘, 𝑏 = 5𝑘, 𝑐 = 8𝑘, 𝑑 = 7𝑘
substituting the above in equation (1) we get,
3𝑘 + 5𝑘 + 8𝑘 + 7𝑘 = 1840
23𝑘 = 1840
𝑘 = 80
the difference between the second and the fourth part is = 7𝑘 − 5𝑘 = 2𝑘
= 2 × 80
= 𝑅𝑠. 160

15. THANK YOU