This document discusses ratios and proportions. It provides examples of how to calculate ratios when comparing two quantities. It also discusses equivalent ratios, and how for ratios to be equivalent their least values must be equal. It explains that for quantities to be in proportion, the ratios of the extreme terms must be equal to the ratios of the middle terms. Examples are provided to demonstrate calculating ratios, equivalent ratios, and determining if quantities are in proportion.

2.  Introduction
 Ratio
 Equivalent Ratio
 Related Examples
 Common mistakes related to Ratio
 Some Examples with solution

3. In our daily life many times we compare the two quantities
of same type.
Comparison by taking difference :
For comparing quantities, we commonly use the method of
taking difference between the quantities.
For example : Leena and Reema playing marbles, Leena
has 30 marbles and Reema has 18 marbles.
Difference between the marbles = 30 – 18 = 12
Then we can say that Leena has 12 marbles more than
Reema. OR Reema has 12 marbles less than the
marbles Leena has.
This is the way of comparison by taking difference.

5.  To understand the ratio, first we take one example :
 Age of Rahul is 10 years and age of Meena is 5 years.
 Rahul : 10 years Meena : 5 years
 How can we compare their ages ?
 By above taking difference method,
 we can find the difference of ages = 10 – 5 = 5 years.
 So age of Rahul is 5 years more than age of Meena.

6.  There are one other method for comparison, that is Division
Method .
Therefore, Age of Rahul = 2 x Age of Meena
Here is Ratio of the age of Rahul to age of Meena.
Ratio is denoted by symbol “ : ”
Note that : A Ratio is treated as a fraction.
So we can write,
Age of Rahul : Age of Meena = 2 : 1
Similarly Age of Meena : Age of Rahul = 1 : 2
Here 2 : 1 ≠ 1 : 2

7.  If we take age of Rahul in months and age of Meena in
years
 Age of Rahul = 10 years = 10 x 12 = 120 months
and Age of Meena = 5 years
So, Ratio is,
Age of Rahul : Age of Meena = 120 : 5 = 24 : 1
It means age of Rahul is 24 times of age of Meena, which
is wrong.
Therefore, To find the Ratio between two quantities
their unit must be same.

8. If we find age of Rahul and Meena in months then,
Age of Rahul 10 years = 10 x 12 = 120 months
And Age of Meena = 5 years = 5 x 12 = 60 months
Therefore, Age of Rahul : Age of Meena = 120 : 60
= (120 ÷60) : (60 ÷ 60)
= 2 : 1
(HCF of 120 & 60 is 60)
So, 120 : 60 = 2 : 1
Therefore 10 : 5 = 120 : 60 = 2 : 1
It proves that, to find the Ratio always both quantities are in the same
units.

9.  We know that in a fraction we can multiply or divide by the same
number in its numerator and denominator and it gives same value.
 Example ;

 So, 10 : 5 = 120 : 60 = 2 : 1
 Now,
Here 2 :1 = 10 : 5 = 120 : 60 = 20 : 10 = 30 : 15
The least value of all above ratios are 2 : 1.
Value of all above ratios are equal.
Therefore, these ratios are called Equivalent Ratios.
that is, Ratios having equal least value are called equivalent ratio.

10.  In above Example :
Here years is common in numerator and denominator of
fraction, so it can be canceled.
 Therefore, we can say that, Ratio has no unit.

11.  We see that many times some students make common mistakes.
 We discuss this by taking some examples.
Monthly income of Dinesh is Rs. 12000. He saves
 Rs. 5000. Find the ratio of his savings to his expenditure?
 Mistake: Some students find simply ratio of given number.
12000 : 5000 = 12 : 5
Correct Solution : Actually we have to find first expenditure.
Expenditure = 12000 – 5000 = Rs. 7000
Now Ratio of saving to expenditure = 5000 : 7000 = 5 : 7
Mistake : Some student find the ratio 7000 : 5000 = 7 : 5
This is the ratio of expenditure to saving.
So, correct ratio of saving to expenditure is 5000 : 7000 = 5 : 7

12. Anu has a ribbon of length 6 m and width
30 cm. Find the ratio of length of ribbon to its width?
Mistake : Length of ribbon = 6 m
width of ribbon = 30 cm
length of ribbon : width of ribbon = 6 : 30 = 1 : 5
Here length is less than width which is wrong.
Correct solution ; Here first we convert both quantities in
same unit.
length of ribbon = 6 m = 600 cm ( 1 m = 100 cm )
width of ribbon = 30 cm
Therefore length of ribbon : width of ribbon = 600 : 30
= 20 : 1

13. Some student think that ratio is the actual quantities.
For example:
Ratio of number of candies if Meena to Mahesh = 4 ; 5
Some students think that :
Number of candies with Meena = 4
Number of candies with Mahesh = 5
which is not correct because ratios 8 : 10 , 12 : 15 , 16 : 20 give
same least value 4 : 5.
Therefore, values in numerator and denominator of a ratio are not
actual quantities it is the least form of actual quantities.
Actual quantities are the multiple of the least form of ratio.

14. Find ratio of 90 cm to 1.5 m.
 Solution : 1 m = 100 cm
1.5 m = 150 cm
Therefore, Ratio of 90 cm to 1.5 m
= Ratio of 90 cm to 150 cm
= 90 : 150 = 3 : 5
Give two equivalent ratios of 10 : 6
Solution : Ratio
and Ratio
20 : 12 and 5 : 3 are two equivalent ratios of 10 : 6

15. Find missing values :
 Solution :
and
Therefore

17.  What is Proportion?
 Explanation with Example.
 Numbers are in Proportion (General Form)
 Examples of Proportion.
 Unitary Method.
 Examples of Unitary method.
 What we have learnt.
 Assignments : Multiple choice question
Practice Questions
PISA Based Question

18.  Proportion means equal or even ratio.
 Now we will understand it with the help of example-
A nature drawing drawn in a rectangular
sheet of length 5 inch and breadth 7 inches.
If we want to expand the picture we can increase its length or
breadth. But if we increase only its length 2 times or only its breadth
2 times then the picture is not like the original one.

19. How can we get expanded picture?
If we expand length and breadth both 2 times then we get expanded
picture of original picture.
To get expanded picture looking like original, we have to increase its
length and breadth both in equal ratio.
OR Ratio of length of the original picture to its breadth is equal to
ratio of length of expanded picture to its breadth.

20. Ratio of length and breadth of given picture = 5 : 7
If we multiply length and breadth with same number then we get the
expanded picture looking like original one.
So 5 : 7 = 10 : 14
If two ratios are equal then we say that they are in proportion.
The symbol = OR : : is use to show the proportion or equal ratio.
So we can say that 5, 7, 10, 14 are in same proportion.
5 : 7 : : 10 : 14
We cannot write, 5 : 7 : : 14 : 10
because 14 / 10 ≠ 5 / 7
In proportion sequence of numbers are very important.

21.  If a, b, c, d four numbers are in proportion.
 Then we can write,
a : b = c : d OR a : b : : c : d OR a / b = c / d
 Here a, b, c and d are called Terms.
 In which ‘a’ and ‘d’ called extreme terms or end terms
and ‘b’ and ‘c’ are called middle terms.
 If four numbers a, b, c and d are in proportion,
So, Product of extreme terms = Product of middle terms
a x b = b x c

22. To understand proportion very clear, let us solve another example.
Example : Leena and Sheena made 7 garlands. Out of 7 garlands
Sheena made 4 garlands and Leena made 3 garlands. By selling
these garlands they got Rs. 35. How much money got Sheena and
Leena?
Solution :
Sheena’s garland Leena’s garland
Out of 7 garlands, Sheena works 4/7 part and Leena works 3/7 part.
Therefore, ratio of work of Sheena to Leena = 4 : 3
In this proportion,
Amount got by Sheena =
Amount got by Leena =
Ratio of amount for Sheena to Leena = 20 : 15 = 4 : 3
Therefore 4 : 3 = 20 : 15 OR 4 : 3 : : 20 : 15
So numbers 4, 3, 20 and 15 are in proportion.

23. 1. Are the ratios 1500 gm : 3 kg and 300 cm : 6 m in
proportion?
Ratio 1500 gm : 3 kg Ratio 300 cm : 6 m
= 1500 gm : 3000 gm = 300 cm : 600 cm
= 1500 : 3000 = 300 : 600
= 1 : 2 = 1 : 2
Both ratios are equal.
Therefore 1500 gm : 3 kg = 300 cm : 6 m
1500 gm, 3 kg, 300 cm and 6 m are in proportion.

24. 2. Are the ratio 15 cm to 2 m and 10 sec to 3 min are in
proportion?
Ratio 15 cm : 2 m Ratio 10 sec : 3 min
= 15 cm : 200 cm =10 sec : 180 sec
(1m = 100cm) (1min = 60 sec)
= 15 : 200 = 10 : 180
= 3 : 40 = 1 : 18
Here 3 : 40 ≠ 1 : 18
Ratios are not equal
Therefore 15 cm, 2 m, 10 sec and 3 min are not in
proportion.

25.  Unit means finding for one.
 In daily life we see many examples that, If we know one value of an
object then we can find values for required object.
:
If price of 1 dozen banana = Rs. 40
1 dozen ½ dozen 2 dozen
Rs. 40 Rs. 20 Rs. 80
Then we can find price of half dozen or two dozen banana’s with the
help of given price.
So, “The method in which first we find the value of one unit and
then the value of required number of units is known as Unitary
method.”

26. Rohan fence his house with wire. He made 3 round wire fence of
expenditure Rs. 9072. Then find the expenditure of 5 round wire.
House with wire fence
Solution- Given,
Expenditure for making fence for 3 rounds = Rs. 9072
Therefore, expenditure for fence of 1 round of wire = 3072 ÷ 3
= Rs. 3024
Therefore, expenditure for fence of 5 round of wire = 5 x 3024
= Rs. 15120.

27.  * When we go to a shop to purchase pen or pencil, shopkeeper told
that the price of one pen or pencil.
With the help of this we can purchase what we want.
* Like wise a vegetable seller told us price of 1 kg of any vegetable.

28. * To find the required distance covered in given quantity of petrol, a
scooter driver see that his scooter goes how many distance in 1 litre
petrol.
In this way,
“ To find the value of required units, first we find the
value of one unit, this method is called Unitary Method.”

29. A car goes 50 km in 2 hours, how many distance covered
by car in 6 min.
To solve the question first we see that all value are in
same ratio.
Given, Distance covered by car in 2 hours = 50 km
1 hour = 60 min, So 2 hours = 120 min
so, distance covered by car in 120 min = 50 km
Therefore, Distance covered by car in 1 min = 150 ÷ 120
=
Therefore distance covered by car in 6 min =

30.  Comparison of two quantities by division is called ratio.
 If least value of two ratios are equal then they are called equivalent
ratio.
 The ratio may occur in different situations.
 The ratio 3 : 2 is different from 2 : 3. Thus the order in which
quantities are taken to express their ratio is important.
 A ratio may be treated as fraction. Ex. 10 : 3 may treated as 10/3
 To find the ratio of two quantities their units should be same.
 Four quantities are in proportion. If the ratio of first and second
quantities are equal to the ratio of third and fourth quantities.
 The order of terms in proportion is Important.
 The method in which we first find the value of one unit and then find
the value of required number of units is known as unitary method.

32. 1. To make a cup of tea ratio of water to milk is 3 : 1. So, to
make 4 cups of tea the ratio of water to milk is :
(a) 4 : 1 (b) 4 : 2 (c) 12 : 4 (d) 7 : 5
2. A car travels 81 km in 3 hours. Distance travelled by car in 5
hours is :
(a) 27 km (b) 135 km (c) 45 km (d) none of these
3. In the word “ MATHEMATICS” the ratio of numbers of
consonants to the numbers of vowel is :
(a) 1 : 7 (b) 1 : 4 (c) 7 : 4 (d) 5 : 3
4. The ratio of complete angle to right angle is :
(a) 4 : 1 (b) 1 : 4 (c) 1 : 2 (d) 2 : 1
5. In the simplest form of the ratio of 72 to 180 is :
(a) 4 : 10 (b) 18 : 45 (c) 2 : 5 (d) 4 : 5

33. 1. Find the ratio of 30 min to 2 hours.
2. Check 7, 56, 13, 104 are in proportion.
3. For 25 : 10, 10 : 4 find the mean proportion.
4.In the given proportion 9 : 3, 36 : 12 extremes are?
5.12 Sarees costs Rs. 3600. Find cost of 1 saree.
6.Cost of 12 apples is Rs. 96. Then what is the cost of 15 apples.
7.Cost of 5 kg wheat is Rs. 80. Then what will be cost of 8 kg of
wheat?
8.Mohit earns Rs. 7650 and saves Rs. 918 per month. Find the ratio
of (1) his income and saving.
(2) his expenditure and saving.
9. Out of 30 students in a class 12students like football , 10 students
like cricket and the remaining students like tennis. Find the ratio of
(a) Number of student liking football to number of students liking
tennis.
(b) Number of students liking cricket to total number of students.
10. Divide 20 pens between Pooja and Riya in the ratio of 2 : 3.

34.  On a particular day the sales (in Rs.) of
different items of a Baker’s Shop are given
below:

35. ITEMS SALES
Ordinary Bread 160
Cakes and Pastries 40
Biscuits 80
Fruit bread 60
Others 20
TOTAL 360

36. From above information, give answers of the followings :
1. Find the ratio of sales of biscuits to the sales of the fruit
bread.
2. Find the ratio of the sales of ordinary bread to the total
sales .
3. Find the ratio of sales of cakes and pastries to biscuits
and the ratio of biscuits to the sales of ordinary bread.
Are they in proportion?
4. The ratios of cakes and pastries to ordinary bread and
ratios of sales of biscuits to the sales of others. Are they
in proportion?
5. If there are 5 fruit bread packet sold on that day then
find the price of 8 fruit bread packets?