2. If ‘a’ and ‘b’ are magnitudes of same kind , expressed in
same units , then the unit a/b is called the ratio ‘a’ to ‘b’
and is denoted by a:b
3. A ratio says how much of one thing there is compared to
another thing.
Use the ":" to separate the
values:
3 : 1
Or we can use the word "to": 3 to 1
Or write it like a fraction: 31
4.
5. 1 . Ratio is pure Number i.e it has no units .
2 . In the ratio a:b a is called as antecedant and b is
called as consequent .
3 . If we multiply the numerators and denominator ina
ny ratio by the same (non zero )number , the ratio
remains the same .
6. 1. Two numbers are in the ratio 7:8 and their sum is 195
. Find the numbers .
Solution :-
7x+ 8x= 195
15x=195
Ans :- x = 13
2 . Two numbers are in the ratio 8:4 and their sum is 192
Find the numbers .
7. 3 . If a:b =4:7 and b:c = 9:5, find a:c
Solution :- a : b :c
4 : 7
9 : 5
36 :63: 35
Find a
Solution :- 4 X 9 = 36
Find b
Solution :- 7X9 = 63
Find c
Solution :- 7X5 = 35
8. 4 . The monthly salaries of two person are in the ratio
3:5 . If each receives an increase of 400 rs in monthly
salary , the new ratio is 13:21 find their original
salaries .
Solution:- Let the original salaries be 3x and 5x .
due to increase in salaires , the revised salaries
are rs(3x + 400 )and (5x + 400 ) it is given that
3x + 400 = 13
5x + 400 = 21
Do cross Multiply
21(3x + 400) = 13(5x + 400)
63x + 8400 = 65x + 5200
Solve :- 8400-5200 = 65x – 63x
3200 = 2x x= 3200
2
Original Salaries were Rs 4800 and Rs 8000
X= 1600
9. 5 . The ratio of prices of two houses was 4 : 5 Two years later
when the price of first had risen by 10 % and that of the
second by 6000 the ratio became 11 : 15 . Find the new price
of the house.
Solve it :-
10.
11. Proportion is an equation which defines that the two
given ratios are equivalent to each other.
If two ratios are equal then the four quantities given by
them are said to be in proportion i.e If the ratio a :
B and c :d are equal then a, b ,c ,d are said to be in
proportion and we write it as a : b : c : d .
Here b and c are called means while and d are called as
externs .
Note if a, b, c, d are in proportion a = c
b d
12. that means
Product of externes = Product of means
Proportion says that two ratios (or fractions) are equal
Types of Proportion
1 . Continued Proportion :- If a , b , c are the 3 quantities of the
same kind if a/b and b/c then a , b, c are in continued proportion.
In this case b is called as mean proportional to a and c .
13. 2 Direct Proprtion :- When two variables are so related that an
increase or (reduction )in one cause an increase in the other in
same ration then the proprtion is known as direct proportion .
14. 3. Inverse Proportion :-If two variables are so related that an
increase (or reduction) in one case causes a reduction (or increase
) in the same ratio in the other , then they are said to be in inverse
proportion .
15.
16.
17. Direct variations :-
If two variables x and y are in direct proportion , we write it as x α y
, then , x=ky , where k is called constant proportionality .
If a value of x and corresponding value of y are known as , then this
constant can be obtained at once .
Inverse Varitation :- In inverse or indirect variation the variables
change disproportionately or when one of the variables increases,
the other one decreases. So behavior of the variables is just the
opposite of direct variations. That is why it is called as Inverse or
indirect variation. If X is in indirect variation with Y, it can be
symbolically written as X α 1Y1Y.
18. Joint Variation: If more than two variables are related
directly or one variable changes with the change product
of two or more variables it is called as joint variation.
If X is in joint variation with Y and Z, it can be
symbolically written as X α YZ.
Combined Variation:
Combined variation is a combination of direct or
joint variation, and indirect variation. So in this case
three or more variables exist. If X is in combined
variation with Y and Z, it can be symbolically written as
X α YZYZ or X α ZYZY.
19. Partial Variation: When two variables are related by a formula or a
variable is related by the sum of two or more variables then it is
called as partial variation. X = KY + C (where K and C are constants)
is a straight line equation which is a example of partial variation.
Examples :-
1. Find x , if
1. 6:15 :: 2:x.
2. 15:27 :: x:45.
20. 2 . Ages of Raghu , madhu and shamu are in continued
proportion . If raghu is 4 yrs old and shamu is 9 yrs old , what
is the age of madhu .
Solution :-
Let Madhu be n years old .
4 : n :: n :9 (since they are in continued proportion )
4 = n
n 9
n X n = 36
n =6
Hence the age of Madhu is 6 yrs old
21. Example :- If A α B and A = 4 when B =6 , find the value of
A when B = 27
Solution :-
Direct variation
x = ky
A = kB
When 1) A= 4 & B = 6
4 = k6
4 / 6= k
k = 2/3 ---------------from (1)
when B = 27
Follow A =kB
A = 2/3 X 27 * A = 18
Hence the value of A is 18
22. Example :-If x varies Directly as y and inversely as z
and x =12 when y = 9 and z = 16 find y when x = 9
and z = 24 .
Solve BY yourself
23. S.No Ratio Proportion
1
The ratio is used to compare
the size of two things with
the same unit
The proportion is used to
express the relation of two
ratios
2
It is expressed using a colon
(:), slash (/)
It is expressed using the
double colon (::) or equal to
the symbol (=)
3 It is an expression It is an equation
4
Keyword to identify ratio in a
problem is “to every”
Keyword to identify
proportion in a problem is
“out of”
Difference Between Ratio and Proportion
To understand the concept of ratio and proportion,
go through the difference between ratio and proportion given here.
24. Example :- Two numbers are in the ratio 2 : 3. If the sum of
numbers is 60, find the numbers.
25. Solution:
Given, 2/3 is the ratio of any two numbers.
Let the two numbers be 2x and 3x.
As per the given question, the sum of these two
numbers = 60
So, 2x + 3x = 60
5x = 60
x = 12
Hence, the two numbers are;
2x = 2 x 12 = 24
and
3x = 3 x 12 = 36
24 and 36 are the required numbers
28. 1. What is 30 % of 80 ,
35% of 79,
40% of 100?
Solution:
30 % of 80
= 30/100 × 80
= (30 × 80)/100
= 2400/100
= 24
3. Ron scored 344 marks out of 400 marks and his elder brother Ben scored 582 marks out
of 600 marks. Who scored percentage is better?
4
29. 2 . 27.65
3 . 40
3 . Solution:
Percentage of marks scored by Ron = (344/400 × 100) %
= (34400/400) %
= (344/4) %
= 86 %
Percentage of marks scored by Ben = (582/600 × 100) %
= (58200/600) %
= (582/6) %
= 97 %
Hence, the percentage marks scored by Ben is better.
30. 4. In an election, candidate A got 75% of the total valid votes. If 15% of the total votes
were declared invalid and the total numbers of votes is 560000, find the number of valid
vote polled in favour of candidate.
Solution:
Total number of invalid votes = 15 % of 560000
= 15/100 × 560000
= 8400000/100
= 84000
Total number of valid votes 560000 – 84000 = 476000
Percentage of votes polled in favour of candidate A = 75 %
Therefore, the number of valid votes polled in favour of candidate A = 75 % of 476000
= 75/100 × 476000
= 35700000/100
= 357000
31. 5. A shopkeeper bought 600 oranges and 400 bananas. He found 15% of oranges and 8% of
bananas were rotten. Find the percentage of fruits in good condition.
Solution:
Total number of fruits shopkeeper bought = 600 + 400 = 1000
Number of rotten oranges = 15% of 600
= 15/100 × 600
= 9000/100
= 90
Number of rotten bananas = 8% of 400
= 8/100 × 400
= 3200/100
= 32
Therefore, total number of rotten fruits = 90 + 32 = 122
Therefore Number of fruits in good condition = 1000 - 122 = 878
Therefore Percentage of fruits in good condition = (878/1000 × 100)%
= (87800/1000)%
= 87.8%
32. 6. Aaron had $ 2100 left after spending 30 % of the money he took
for shopping. How much money did he take along with him?
Solution:
Let the money he took for shopping be m.
Money he spent = 30 % of m
= 30/100 × m
= 3/10 m
Money left with him = m – 3/10 m = (10m – 3m)/10 = 7m/10
But money left with him = $ 2100
Therefore 7m/10 = $ 2100
m = $ 2100× 10/7
m = $ 21000/7
m = $ 3000
Therefore, the money he took for shopping is $ 3000.
