CLASS
CBSE-i
VII
UNIT-6
Student's Section
RATIO
AND PROPORTION
Shiksha Kendra, 2, Community Centre,
Preet Vihar, Delhi-110 092 India
CBSE-i
RATIO
AND PROPORTION
VII
UNIT-6
Shiksha Kendra, 2, Community Centre, Preet Vihar, Delhi-110 092 India
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Preface
The Curriculum initiated by Central Board of Secondary Education -International (CBSE-i) is a progressive step
in making the educational content and methodology more sensitive and responsive to the global needs. It
signifies the emergence of a fresh thought process in imparting a curriculum which would restore the
independence of the learner to pursue the learning process in harmony with the existing personal, social and
cultural ethos.
The Central Board of Secondary Education has been providing support to the academic needs of the learners
worldwide. It has about 11500 schools affiliated to it and over 158 schools situated in more than 23 countries. The Board
has always been conscious of the varying needs of the learners in countries abroad and has been working towards
contextualizing certain elements of the learning process to the physical, geographical, social and cultural environment
in which they are engaged. The International Curriculum being designed by CBSE-i, has been visualized and
developed with these requirements in view.
The nucleus of the entire process of constructing the curricular structure is the learner. The objective of the curriculum is
to nurture the independence of the learner, given the fact that every learner is unique. The learner has to understand,
appreciate, protect and build on values, beliefs and traditional wisdom, make the necessary modifications,
improvisations and additions wherever and whenever necessary.
The recent scientific and technological advances have thrown open the gateways of knowledge at an astonishing pace.
The speed and methods of assimilating knowledge have put forth many challenges to the educators, forcing them to
rethink their approaches for knowledge processing by their learners. In this context, it has become imperative for them
to incorporate those skills which will enable the young learners to become 'life long learners'. The ability to stay current,
to upgrade skills with emerging technologies, to understand the nuances involved in change management and the
relevant life skills have to be a part of the learning domains of the global learners. The CBSE-i curriculum has taken
cognizance of these requirements.
The CBSE-i aims to carry forward the basic strength of the Indian system of education while promoting critical and
creative thinking skills, effective communication skills, interpersonal and collaborative skills along with information
and media skills. There is an inbuilt flexibility in the curriculum, as it provides a foundation and an extension
curriculum, in all subject areas to cater to the different pace of learners.
The CBSE has introduced the CBSE-i curriculum in schools affiliated to CBSE at the international level in 2010 and is
now introducing it to other affiliated schools who meet the requirements for introducing this curriculum. The focus of
CBSE-i is to ensure that the learner is stress-free and committed to active learning. The learner would be evaluated on a
continuous and comprehensive basis consequent to the mutual interactions between the teacher and the learner. There
are some non-evaluative components in the curriculum which would be commented upon by the teachers and the
school. The objective of this part or the core of the curriculum is to scaffold the learning experiences and to relate tacit
knowledge with formal knowledge. This would involve trans-disciplinary linkages that would form the core of the
learning process. Perspectives, SEWA (Social Empowerment through Work and Action), Life Skills and Research
would be the constituents of this 'Core'. The Core skills are the most significant aspects of a learner's holistic growth and
learning curve.
The International Curriculum has been designed keeping in view the foundations of the National Curricular
Framework (NCF 2005) NCERT and the experience gathered by the Board over the last seven decades in imparting
effective learning to millions of learners, many of whom are now global citizens.
The Board does not interpret this development as an alternative to other curricula existing at the international level, but
as an exercise in providing the much needed Indian leadership for global education at the school level. The
International Curriculum would evolve on its own, building on learning experiences inside the classroom over a period
of time. The Board while addressing the issues of empowerment with the help of the schools' administering this system
strongly recommends that practicing teachers become skillful learners on their own and also transfer their learning
experiences to their peers through the interactive platforms provided by the Board.
I profusely thank Shri G. Balasubramanian, former Director (Academics), CBSE, Ms. Abha Adams and her team and
Dr. Sadhana Parashar, Head (Innovations and Research) CBSE along with other Education Officers involved in the
development and implementation of this material.
The CBSE-i website has already started enabling all stakeholders to participate in this initiative through the discussion
forums provided on the portal. Any further suggestions are welcome.
Vineet Joshi
Chairman
Acknowledgements
Advisory
Conceptual Framework
Shri Vineet Joshi, Chairman, CBSE
Dr. Sadhana Parashar, Director (Training),
Shri G. Balasubramanian, Former Director (Acad), CBSE
Ms. Abha Adams, Consultant, Step
Dr. Sadhana Parashar, Director (Training),
Ideators VI-VIII
Ms Aditi Mishra
Ms Guneet Ohri
Ms. Sudha Ravi
Ms. Himani Asija
Ms. Neerada Suresh
English :
Ms Neha Sharma
Ms Dipinder Kaur
Ms Sarita Ahuja
Ms Gayatri Khanna
Ms Preeti Hans
Ms Rachna Pandit
Ms Renu Anand
Ms Sheena Chhabra
Ms Veena Bhasin
Ms Trishya Mukherjee
Ms Neerada Suresh
Ms Sudha Ravi
Ms Ratna Lal
Ms Ritu Badia Vashisth
Ms Vijay Laxmi Raman
Chemistry
Ms. Poonam Kumar
Mendiratta
Ms. Rashmi Sharma
Ms. Kavita Kapoor
Ms. Divya Arora
Ms. Sugandh Sharma, E O
Mr. Navin Maini, R O
(Tech)
Shri Al Hilal Ahmed, AEO
Ms. Anjali, AEO
Shri R. P. Sharma,
Consultant (Science)
Mr. Sanjay Sachdeva, S O
Ms Preeti Hans
Ms Neelima Sharma
Ms. Gayatri Khanna
Ms. Urmila Guliani
Ms. Anuradha Joshi
Ms. Charu Maini
Dr. Usha Sharma
Prof. Chand Kiran Saluja
Dr. Meena Dhani
Ms. Vijay Laxmi Raman
Material Production Groups: Classes VI-VIII
Physics :
Mathematics :
Ms. Vidhu Narayanan
Ms. Deepa Gupta
Ms. Meenambika Menon
Ms. Gayatri Chowhan
Ms. Patarlekha Sarkar
Ms. N Vidya
Ms. Neelam Malik
Ms. Mamta Goyal
Biology:
Mr. Saroj Kumar
Ms. Rashmi Ramsinghaney
Ms. Prerna Kapoor
Ms. Seema Kapoor
Mr. Manish Panwar
Ms. Vikram Yadav
Ms. Monika Chopra
Ms. Jaspreet Kaur
Ms. Preeti Mittal
Ms. Shipra Sarcar
Ms. Leela Raghavan
Ms. Chhavi Raheja
Hindi:
Mr. Akshay Kumar Dixit
Ms. Veena Sharma
Ms. Nishi Dhanjal
Ms. Kiran Soni
CORE-SEWA
Ms. Vandna
Ms.Nishtha Bharati
Ms.Seema Bhandari,
Ms. Seema Chopra
Ms. Madhuchhanda
MsReema Arora
Ms Neha Sharma
Coordinators:
Dr Rashmi Sethi, E O
Dr. Srijata Das, E O
(Co-ordinator, CBSE-i)
Ms. Madhu Chanda, R O (Inn)
Mr. R P Singh, AEO
Ms. Neelima Sharma, Consultant
(English)
Ms. Malini Sridhar
Ms. Leela Raghavan
Dr. Rashmi Sethi
Ms. Seema Rawat
Ms. Suman Nath Bhalla
Geography:
Ms Suparna Sharma
Ms Aditi Babbar
History :
Ms Leeza Dutta
Ms Kalpana Pant
Ms Ruchi Mahajan
Political Science:
Ms Kanu Chopra
Ms Shilpi Anand
Economics :
Ms. Leela Garewal
Ms Anita Yadav
CORE-Perspectives
Ms. Madhuchhanda,
RO(Innovation)
Ms. Varsha Seth, Consultant
Ms Neha Sharma
Ms.S. Radha Mahalakshmi,
EO
Contents
1.
Study Material
2.
Student's support material
C
SW 1: Warm Up Activity W1
1
27
28
Recall Ratios
C
SW 2: Warm Up Activity W2
31
Comparison of Ratios
C
SW 3: Pre Content Worksheet PC1
36
Proportions
C
Sw4: Pre Content Worksheet PC2
37
More Proportions
C
SW5: Content Worksheet CW1
38
Mean Proportion
C
SW 6: Content Worksheet CW2
40
Unitary Method
C
SW 7: Content Worksheet CW3
42
Percentage Fractions and Decimals
C
SW 8: Content Worksheet CW4
44
Fractions and Decimals as Percentage
C
SW 9: Content Worksheet CW5
45
Percentage of a given Quantity
C
SW 10: Content Worksheet CW6
48
C
Application of Percentage
C
SW 11: Content Worksheet CW7
50
Percentage Drill
C
SW 12: Content Worksheet CW8
53
Increase and Decrease Percent
C
SW 13: Content Worksheet CW9
Profit and Loss
56
C
SW 14: Content Worksheet CW10
58
Profit and loss as Percentage
C
SW 15: Content Worksheet CW11
61
Simple Interest
C
SW 16: Content Worksheet C12
63
Finding Rate
C
SW 17: Content Worksheet CW13
65
Finding Time and Principal
C
SW 18: Post Content Worksheet PCW1
67
Assess Ratios and proportion
C
SW 19: Post Content Worksheet PCW2
74
Test your progress
C
Suggested videos/ links/ PPT's
77
STUDY
MATERIAL
1
Introduction
In Class VI, you were introduced to the concept of a ratio as method of „comparison by
division‟ and also as an extension of a fraction and the concept of a proportion as an
equality of two ratios.
In this unit, we shall first recall these concepts and extend the study to include some
more related concepts such as equivalent ratios, comparison of ratios, continued
proportion and mean proportions. We shall also recall the concept of unitary method
introduced in Class VI and learn some more applications of the method in solving
problems of daily life. In addition to these, the concept of percentage will be introduced
as a fraction with denominator 100. In this unit, the process of converting fractions and
decimals into percentages and vice-versa will also be discussed. Finally, use of
percentages in solving problems related to daily life, will be explained.
1. Ratio:
Recall that we can compare two quantities by
(i)
taking their difference
(ii)
division
In the comparison by division, we compare two quantities in terms of „how many
times‟.
This comparison is known as a ratio.
We use the symbol „:‟ to denote a ratio.
For example, if cost of pen is ` 20 and that of a pencil is ` 6, then we say that the ratio of
the cost of a pen to the cost of a pencil is = or 10 : 3
Similarly, the ratio of 5kg to 500g is
=
: or 10 : 1 [1kg=1000g]
Note that to compare two quantities, the units must be same. Therefore, ratio can
only be found when the two quantities are in the same units.
In the above case, the ratio of 500 g to 5kg is 1:10.
Note that the ratio 10:1 is not the same as 1:10.
In the ratio 10:3, 10 is the first term and 3 is the second term.
Similarly, in the ratio 10:1, 10 is the first term and 1 is the second term.
In the ratio 1:10, 1 is the first term and 10 is the second term.
2
In general, in the ratio a:b, a is the first term and b is the second term.
Simplest form of a Ratio.
In the above example, the ratio of cost of a pen and cost of a pencil is
You have also seen that
=
or 20:6
or 10:3.
In the ratio 10:3, there is no common factor (other than 1) in the terms 10 and 3.
We say that the ratio 10 : 3 is in simplest form or lowest form or lowest terms:
You may recall that a fraction is said to be in the simplest form or lowest form if
there is no common factor (other than one) between its numerator and denominator.
In the same way, we say that a ratio a:b is in its simplest form if their is no common
factors (other than 1) between its terms a and b. (i.e., if a and b are co primes)
Equivalent Ratios
Recall that two or more fractions are said to be equivalent if they can be expressed
in the same simplest form or lowest form.
For example, ,
,
are equivalent fractions because
=
=
=
i.e., each of these fractions is expressed in the same simplest form .
Since, a ratio is a type of fraction, two or more ratios are called equivalent ratios if
they can be expressed in the same simplest form.
For example, the ratios 4:6, 12:18 and
expressed in the same simplest form 2 : 3.
are equivalent ratios as they can be
Similarly, the ratios 40:30, 16:12, 24:18, 4:3 and 44:33 are equivalent ratios as each of
these can be expressed in the same simplest form 4:3.
Note that ratios equivalent to a given ratio can be obtained by multiplying or
dividing the terms of the given ratio by a nonzero number
Example1: Find which of the following ratios are equivalent:
3
(i)
25 : 30, 10 : 12
(ii)
36 : 28, 56 : 72
(iii)
18 : 22, 36 : 44, 63 : 77
(iv)
8 : 12, 4 : 6, 18 : 24
Solution
(i)
25 : 30 =
= = 5: 6
10 : 12 =
= =5:6
same simplest form
So, 25 : 30 and 10 : 12 are equivalent.
(ii)
36 : 28 =
= =9:7
not same simplest form
56 : 72 =
= =7:9
So, the ratios are not equivalent
(iii)
18 : 22 =
=
= 9 : 11
36: 44 =
=
= 9 : 11
63 : 77 =
=
= 9 : 11
same simplest form
So, the ratios are equivalent.
(iv)
8 : 12 = 2 : 3
4:6=2:3
not same simplest form
18 : 24 = 3 : 4
So, the ratios are not equivalent.
Example 2: Find two equivalent ratios of
(i)
75 : 125
(ii)
17 : 13
Solution :
(i)
75 : 125 =
=
=
= 15 : 25
4
75 : 125 =
=
=
= 225 : 375
Thus, two equivalent ratios of 75 : 125 of are 15 : 25 and 225 : 375
(ii)
17 : 13 =
=
=
Also 17 : 13 =
=
= 34:26
= 51 : 39
Thus, two equivalent ratios of 17 : 13 are 34 : 26 and 51 : 39.
Note: You may obtain as many equivalent ratios to a given ratio as you wish by
multiplying or dividing the terms of the given ratio by a non-zero number.
Comparison of Ratios
We can compare two ratios in the same way as we compare two fractions. We explain it
through examples.
Example 3: Compare the following ratios:
(i)
1 : 2 and 2 : 3
(ii)
45 : 30 and 48 : 32
(iii)
12 : 15 and 16 : 21
(iv)
7 : 6 and 10 : 9
Solution:
(i)
1 : 2 = and 2 : 3 =
=
=
As
< , so,
<
Hence , 1 : 2 < 2 : 3 or 2 : 3 > 1 : 2.
(ii)
45 : 30 =
=
48 : 32 =
=
So, 45 : 30 = 48 : 32
5
(iii)
12 : 15 =
=
16 : 21 =
=
=
=
=
As
(iv)
>
, so, 12 : 15 > 16 : 21
7:6=
10 : 9 =
Now =
=
As
=
=
>
, so, 7 : 6 > 10 : 9.
2. Proportion
Recall that if two ratios are equal, we say that they form a proportion.
For example, the ratios 2 : 10 and 3 : 15 form a proportion as
2:10 = 3:15
Clearly, the ratios 10:2 and 3:15 do not form a proportion, as
10:2 = 5:1
and 3:15 = 1:5
and 5:1
1 : 5.
We use the symbol „: :‟ or “=” to denote a proportion.
Thus, 2:10 : : 3:15 or 2:10 = 3:15.
Similarly, 10 : 15 : : 4:6 or 10:15 = 4:6
[10 : 15 = 2 : 3, 4 : 6 = 2 : 3]
In general, if two ratios a:b and c:d are equal, we say that they form a proportion,
i.e., a : b : : c : d or a : b = c : d
6
In the proportion
a=b::c:d
a and d are called extreme terms (or extremes) and b and c are called middle terms
(or means). In such a case, we say that the numbers a, b, c and d are in proportion.
Extreme terms (or extremes)
a : b : : c : d
middle terms (or means)
Conversely, four numbers a, b, c and d are in proportion if ad = bc,
i .e., if product of first term and fourth term
= product of second term and third term
Observe the following proportions:
(i)
2 : 10
::
3 : 15
(ii)
10 : 15
::
4:6
(iii)
2:9
::
18 : 81
In (i), product of extremes
Product of means
= 2 x 15 = 30
= 10 x 3 = 30
Thus, product of extremes
= product of means
In (ii), product of extremes
= 10 x 6 = 60
Product of means
= 15 x 4 = 60
Thus, product of extremes
= product of means.
In (iii), product of extremes
= 2 x 81 = 162
Product of means
Thus, product of extremes
= 9 x 18 = 162
= product of means.
7
In general, in a proportion
a:b::c:d
product of extremes = product of means
i.e.,
bxc=axd
Example 4: Do following ratios form a proportion?
(i)
16 : 24 and 20 : 30
(ii)
2 : 3 and 28 : 12
(iii)
250g : 300g and ` 40 : ` 48
Solution:
(i)
16 : 24 =
= =2:3
20 : 30 =
= =2:3
So, 16 : 24 and 20 : 30 form a proportion.
(ii)
2:3= =2:3
(iii)
28 : 12 =
=7:3
As, 2 : 3
7 : 3, so, 2 : 3 and 28 : 12 do not form a proportion.
250g : 300g = 5 : 6
`40 : `48 = 5 : 6
So, the ratios 250g : 300 g and `40: `48 form a proportion.
Example 5 : Check whether the following numbers are in proportion:
(i)
2, 3, 4, 6
(ii)
3, 4, 15, 20
(iii)
12, 18, 28, 35
Soluton:
(i)
Product of first and fourth number = 2 x 6 = 12
8
Product of second and third number = 3 x 4 = 12
So, ad = bc.
Thus, 2, 3, 4 and 6 are in proportion
(ii)
We have
3x20 = 60
4x15 = 60
i.e., ad = bc.
So, 3, 4, 15 and 20 are in proportion
(iii)
Here, ad = 12 x 35 = 420
bc = 18 x 28 = 504
As ad
bc,
So, 12, 18, 28 and 35 are not in proportion.
3. Continued Proportion
Consider the ratio 4 : 6 and the ratio 6 : 9
You may observe that 4 : 6 : : 6 : 9
[4 : 6 = 2 : 3 and 6 : 9 = 2 : 3]
In the above proportion, second and third terms are the same. Such proportion is
known as a continued proportion.
We also say that 4, 6 and 9 are in continued proportion.
The middle number 6 is called the mean proportional of 4 and 9.
Observe that 62 = 4 x 9.
Similarly, 2 . 5 : 5 : : 5 : 10 is a continued proportion and the numbers 2.5, 5, 10 are in
continued proportion as
52 = 2.5 x 10 = 25
The middle term 5 is the mean proportional.
In general, if a : b : : b : d, then we say that a, b and d are in continued
proportion. b is called the mean proportional of a and d. Also b2 = a x d.
9
Example 6: check whether the following numbers are in continued proportion. Find the
mean proportional, if they are in continued proportions.
(i)
25, 100, 400
(ii)
81, 9, 1
(iii)
36, 18, 4
Solution:
(i)
25 : 100 = 1 : 4
100 : 400 = 1 : 4
So, 25 : 100 : : 100 : 400
Hence, 25, 100 and 400 are in continued proportion.
Clearly, mean proportional is 100.
(ii)
81 : 9 = 9 : 1
9:1=9:1
So, 81 : 9 : : 9 : 1 and the numbers 81, 9 and 1 are in continued proportion,
Clearly, mean proportional is 9.
(iii)
36 : 18 = 2 : 1
18 : 4 = 9 : 2
Since 36 : 18
18 : 4,
So, 36 , 18 and 4 are not in continued proportions.
4. Unitary Method
You are familiar with unitary method from Class VI.
Recall that in this method, we first find the value of one unit and then find the value
of required number of units.
For example, if the cost of 3 pens is ` 18, then we find the cost of 10 pens as follows:
Cost of 3 pens = ` 18
Cost of 1 pen = `
=`6
[we find cost of one unit]
So, cost of 10 pens = ` 6 x 10 = ` 60. [we find the cost of required number of units]
We now take some examples.
10
Example 7: A scooter requires 3 litres of petrol to cover a distance of 146 km. How
much distance will it cover in 5.4 litres?
Solution:
In 3 litres of petrol, distance covered
= 146 km
In 1 litre of petrol, distance covered
=
km
In 5.4 litres of petrol, distance covered =
x 5.4 km
= 146 x 1.8 km
= 262.8 km
Example 8: One dozen of bananas cost `46. Find the cost of 21 bananas. How many
bananas can be purchased for ` 161?
Solution: One dozen = 12
Now, cost of 12 bananas = `46
cost of 1 banana = `
cost of 21 bananas = `
x 21 = `
= ` 80.50
Further,
In `46, the number of bananas purchased = 12
In `1, the number of bananas purchased =
In `138, the number of bananas purchased =
x 161= 42
Example 9: 5 kg of wheat flour cost `124 , and 8 kg of rice cost `336. George buys 8kg
of wheat and 5kg of rice. How much money will he pay?
Solution:
Cost of 5kg of wheat flour = ` 336
Cost of 1kg of wheat flour = `
=`
So, cost of 8kg of wheat flour
= ` 198.40
11
x8
Cost of 8kg of rice = ` 336.
Cost of 1kg of rice = `
So, cost of 5kg of rice = `
= ` 210
Total amount to be paid by George = ` 198.40 + ` 210
= `408.40
5. Percentage
You are already familiar with fractions.
Observe the following fractions:
, , , ,
,
, ,
,
,
,
,
In the above fractions, some fractions have 100 in their denominators such as
,
,
,
Such fraction with denominator 100 is called a percent.
The word percent is symbolically denoted by „%‟
Thus,
is written as 9%.
Similarly,
and
is written as 101%.
is written as 1234%.
The word percent is derived from Latin word „per centum‟ meaning „per hundred‟
or „on hundred‟ or „hundredths‟
For example, by the phrase “price of sugar has increased by 5%”,
we mean that if the price of sugar is `100, then increase in price is `5. We also say
that percentage increases in price of sugar is 5
Thus, 9% means the fraction
1% means the fraction
20% means the fraction
.
.
.
12
210% means the fraction
.
and so on.
6. Conversion of Fractions and Decimals into Percentages and vice-versa
Converting fractions to percents
You have seen that a fraction with denominator 100 is called a percent. However,
if a fraction has denominator other than 100, it can also be converted into an
equivalent fraction with denominator 100. This way, we can convert a fraction to
percent. We explain it through examples.
Example 10: Convert
, , , to percents.
Solution:
= x
=
= 50%
percent
We can also write:
= x
%=
Also,
%
= x 100%
Example 11: Write
and
= 50%
=
% = 50% [Note that 100% =
as percents.
Solution: We have
=
=
=
x 100% =
=
%=
% = 162 %
%
%=
% = 187 %
Converting percents to fractions
13
We explain conversion of percents to fractions through examples.
Example 12: Convert (i) 25%, (ii) 140%, (iii) 8 % to fractions.
Solution:
(i)
25% = 25 x
(ii)
140% =
(iii)
8 %=
=
=
fraction
= .
%=
x
=
=
Note that to convert a percent into a fraction, we have multiplied the number
indicating the percent by
and simplify the resulting fraction.
Converting decimals to percents
Let us now see how decimals can be converted to percents.
Example 12: Convert the following decimals to percents:
(i)
0.65
(ii)
0.06
(iii)
3.25
(iv)
10.1
(v)
0.225
Solution:
(i)
0.65 =
= 65%
(ii)
0.06 =
= 6%
(iii)
3.25 =
= 325%
(iv)
10.1 =
=
(v)
0.225 =
percent
=
=
x
= 1010%
= (22.5) x
= 22.5%
We can also write
14
0.225 = 0.225 x 100% = (0.225 x 100)%
= 22.5%
Thus, to convert a decimal into percent
.
we shift the decimal point two places to the right and insert ( ) symbol.
Converting percents to decimals
Example 13: Convert the following percents into decimals:
(i)
76%
(ii)
25.3%
(iii)
(iv)
2.98%
(v)
0.125%
340.8%
Solution :
(i)
76% =
= 0.76 (decimal)
(ii)
25.3% = 25.3 x
(iii)
340.8% = 340.8 x
(iv)
2.98% = 2.98 x
(v)
0.125% = 0.125 x
=
x
=
=
=
x
x
=
= 0.253
=
= 3.408
=
x
= 0.0298
=
= 0.00125
Note that to convert a percent into a decimal, we remove the percent symbol and then
shift the decimal points two places to the left.
7. Applications
Percentages have various applications in daily life. Some of these are discussed
below:
Finding percent of a quantity
Many times, we need to find a given percent of a given quantity. Let us
explain it through some examples.
Example 14: Find:
(i)
25% of ` 125
(ii)
150% of 4 kg
15
(iii)
35% of a litre
(iv)
15% of 10 m
Solution:
25% of ` 125 =
(i)
of ` 125
=`
x 125
=`
= ` 31.25
(ii)
150% of 4kg =
=
of 4 kg
x 4 kg
= 6 kg
(iii)
35% of a litre =
=
of a litre
x 1 litre
= 0.35 litre
(iv)
15% of 10m =
=
of 10 m
x 10 m
= 1.5 m
Example 15: In a school, there are 600 students, of which 48% are boys. Find the number
of boys and girls.
Solution: Number of boys = 48% of 600
=
x 600
= 288
So, the number of girls
= 600 – 288 = 312
Note: You can also find the number of girls as (100 48%) of 600, i.e., 52% of 600.
Ratios as percents
16
Example 16: Ratio of apple trees to other type of trees in an orchard is 5 : 3. Find the
percentage of apple trees.
Solution: Number of apple trees : Number of other trees = 5 : 3
It means if there are 5+3 (=8) trees in all, then 5 are apple trees.
This means part is apple trees and part is other trees.
So, percentage of apple trees = x 100%
=
%
= 62 %
Example 17: Sonia obtained 366 marks out of a maximum of 600 marks.
Find the percentage of marks obtained by her.
Solution: Required percentage of marks =
x 100
= 61
Thus, Sonia obtained 61% marks.
Increase/decrease percents
Example 18: Monthly expenditure on ration of a family is ` 15000. If the prices of the
items are increased by 75%, find the increase in expenditure of the family. Also, find the
new monthly expenditure.
= `15000
Solution: Monthly expenditure on ration
Percentage increase in expenditure
= 7.5
Increase in expenditure
= 7.5% of `15000
=`
=`
x 15000
x 15000
= ` 1125
So, new expenditure = `1500 + `1125 = `16125
17
Example 19: The number of illiterates in a city decreased from 45 lakhs to 32 lakhs in a
year. What is the percentage decrease of illitrates?
Solution: Decrease in the number of illiterates
= 45 lakhs – 32 lakhs
= 13 lakhs
So, percentage decrease
=
x 100
= 28.9% approximately.
Thus, decrease of number of illitrates of the city = 28.9% (approx.)
Example 20: To cut down the expenditure, a company decides to reduce its
consumption of petrol by 18%. If the reduction in the monthly consumption of petrol is
288 litres, find the original monthly consumption of petrol.
Solution: Let the original monthly consumption of petrol = 100 litres.
Reduction
= 18%
Savings
= 18% of 100 litres
= 18 litres
If saving is 18 litres, then original consumption = 100 litres
If saving is 1 litre, then original consumption =
litres
If saving is 288 litres, then original consumption =
x 288 litres
= 1600 litres
Alternatively, we can proceed as follows:
Let the original monthly consumption of petrol
= x litres
Reduction
= 18% of x
=
x
So, saving =
Thus,
= 288
18
or,
18x = 28800
or,
x=
= 1600
Petrol saved = 1600 litres
Comparing quantities using percents
Example 21: David obtained 32 marks out of 50 and Rina obtained 48 marks out of 80.
Who has performed better?
Solution:
Just by observing the marks, you may say that Rina‟s performance is better because she
has obtained 48 marks while David obtained 32 marks. But this may not be a correct
way of comparing their performances, because David has obtained 32 marks out of 50
and Rina has obtained 48 out marks of 80
We can compare their performances by first converting their marks into percentages.
Percentage of marks obtained by David =
x 100 = 64
Percentage of marks obtained by Rina =
x 100 = 60
Thus, David has performed better.
Example 22: In a Board‟s examination 160 students of school A appeared. Out of these,
118 passed. In the same examination, 200 students of school B appeared and 150
students passed. Which school has performed better in the examination?
Solution: Pass percentage of school A =
Pass percentage of school B =
x 100 = 73
x 100 = 75
So, performance of school B is better as 75 > 73 .
Profit and Loss
Savita bought a watch for `500 and sold it for `520.
The buying price of an item is known as its cost price (abbreviated as CP).
The price at which it is sold in known as its selling price (abbreviated as SP).
So, CP of watch is `500 and SP of watch is `520.
Note that `520 > `500. That is, Savita has earn (` 520
19
` 500) on this transaction.
Thus, we can say that if SP > CP, then there is a profit or gain.
In this case, profit = SP
CP
= ` 520
` 500 = ` 20
In other words, if SP > CP, then
there is profit and
Profit = SP
CP
Suppose I purchase a doll for ` 80 and sell it ` 75.
In this case, SP < CP.
We say that there is loss of (` 80
` 75), i.e., ` 5 on this transaction.
Thus, if SP < CP, then
Loss = CP
SP
Profit (or Loss) percent
Profit (or loss) percent is always calculated on the CP.
In the case of watch,
Profit percent =
=
x 100
x 100 = 4
Thus, profit of Savita on the watch is 4%.
Similarly, loss percent on doll
=
x 100
=
x 100 = 6
Thus, loss on the doll is 6 %.
We now consider some more examples related to profit and loss.
Example 23: A shopkeeper bought a table for `4500 and sold it for `4725. Find gain or
loss percent
Solution: Here, CP of table = ` 4500.
20
SP of the table = ` 4725
As SP > CP, so there is a profit
Profit = SP – CP
`4500 = `225
= `4725
Profit percent =
x 100
=
x 100 = 5
Thus, shopkeeper had a gain of 5%.
Example 24: Anvi bought a flat for `4700000. She had to sell it for `460600. Find her
profit or loss percent.
Solution: Here, CP = `4700000
and SP = `4606000
As SP < CP, so there is loss.
Loss = CP – SP
= `4700000
`4606000
= `94000
Loss percent =
=
x 100
x 100 = 2
Thus, Anvi suffers a loss of 2%.
Example 25: An article was bought for ` 2250 and was sold at a gain of 5%. Find the SP
of the article.
Solution: CP = ` 2250
Gain = 5%
so, gain = 5% of ` 2250
=
x ` 2250
= ` 112.50
Therefore, SP = CP + profit (gain)
21
= ` 2250 + ` 112.50
= ` 2362.50
Example 26: By selling an article for `4600, Neetu losses 8%. Find the cost price of the
article.
Solution: Let CP = `100
Loss = 8%
So, loss on `100 = 8
or
SP of the article = ` 100
`8
= ` 92
If SP is ` 92, CP `100
So, if SP is ` 4600, CP = `
x 4600
= ` 5000
Alternate Method.
Let CP be ` x.
Loss = 80%
or
of ` x
Loss =
=`
=`
Now, SP = CP
=`x
Loss
`
=`
=`
But the SP = ` 4600
So,
or x =
= 4600
= 5000
Thus, cost price of the article was ` 5000.
22
Example 27: Wasim bought eggs at the rate of ` 36 per dozen and Sold them at 4 eggs
for `10. Find his gain or loss percent.
Solution: Let the number of eggs purchased = 48
CP of eggs at ` 36 per dozen = `
SP at 4 for ` 10 = `
x 48 = ` 144
x 48 = `120
Since SP < CP, there is a loss.
Loss = ` 144 - ` 120 = ` 24
Loss percent =
=
x 100
x 100 =
= 16
Thus, there is loss of 16 %.
Simple Interest
When we borrow money from a Bank or a Financial Company or an Individual
(called a Lender) , for a certain time, we have to pay at the end of the time period,
some extra money to the lender besides the borrowed money. This extra is called
the interest.
The money borrowed is called the principal.
The money returned after the entire time period is called amount.
Thus, Amount = Principal + Interest
Interest is paid according to some agreement which is in the form of a rate
percent per annum.
If the principal remains the same for the entire loan period, then the interest is
known as simple interest. Now, we take some examples to illustrate the
calculation of simple interest (or interest).
Example 28: Find the interest on ` 5000 for 3 years at the rate of 5% per annum.
Solution: Interest on `100 for 1 year = ` 5
So, interest on `1 for 1 year = `
So, interest on `1500 for 1 year = `
23
x 5000
So, interest for 5000 for 3 years = `
x 5000 x 3 = ` 750
From the expression at (A) above, we find that
Interest =
` 5000 5 3
100
i.e., I =
, when I is the interest, P is the principal, R% is the rate per
annum and T is the number of years (Time period). Clearly
Amount (A) = P + I
we can directly use the formula
I=
to calculate interest.
In this formula, four quantities I, P, R and T are involved. If any three of these are given,
we can find the fourth.
We now illustrate this through some examples.
Example 28: Find simple interest on ` 8000 at 10% per annum for 4 years.
Solution: Here, P = 8000, R = 10, T = 3 years.
I=
=`
= ` 2400
Example 29: Determine the amount to be paid for a sum of money ` 7500 at 8 per
annum simple interest at the end of 5 years.
Solution : Here, P = ` 7500, R = 8 , T = 5 years.
I=
=`
=`
x
x5
= ` 3187.50
So, Amount = P + I
24
= ` 7500 + ` 3187.50 = ` 10687.50
Example 30: Pravin pays an interest of ` 13500 for 4 years on a sum of ` 45000. Find the
rate of interest.
Solution: Here, I = ` 13500, P = ` 45000,
T = 4 years, R = ?
Using the formula
I=
13500 =
13500 = 1800 R
So,
R=
=7
Hence, rate of interest = 7.5% per annum.
Example 31: On a certain sum, the interest paid after 3 years is ` 5940 at 9% per annum.
Find the sum and the amount.
Solution: Here, I = ` 5940, R = 9, T = 3 years.
P = ?, A = ?
We know that
I=
5940 =
or
5940 =
or
P=`
P
= ` 22000
Thus, the sum = ` 22000
Amount = ` 22000 + ` 5940 = ` 27940
Example 32: In how many years will a sum of ` 40000 yield an interest of `11200 at 14%
per annum?
Solution: Here, P = ` 40000, I = ` 11200, R = 14, T = ?
25
I=
11200 =
or,
11200 = 5600 T
T=
=2
Thus, the required time = 2 years.
Example 33: A sum of money becomes double of itself in 10 years. In how many years,
will it be four times of itself at the same rate of simple interest?
Solution: Let the sum be ` 100
So, amount after 10 years = `200
Interest = ` 200
` 100 = ` 100
Using the formula,
I=
100 =
R = 10
Now, new amount = `400
= `400
So, interest
`100 = `300
I=
300 =
T = 30
Hence, the sum will of four times in 30 years.
26
STUDENT’S
SUPPORT
MATERIAL
27
STUDENT’S WORKSHEET – 1
Recall Ratios
WARM UP Worksheet W1
Name of the student ______________________
Date ______
Activity: There are two bags, one containing ‘smart fractions’ and the other ‘charming
ratios’. Each fraction has to find the right match for itself. Help them to form pairs and
complete the table below.
28
• RATIO
FRACTION
29
30
STUDENT’S WORKSHEET – 2
Comparison of Ratios
WARM UP Worksheet W2
Name of the student ______________________
Date ______
Given below are a few rings with ratios written in the center. Express each ratio in the
simplest form and write it in the star diamond.
31
Activity 2 – Compare the following fractions and hence conclude the relationship
between their ratios. (First one is done for you)
i)
3
3
and
4
5
15 > 12
20
Hence
3/4 > 3/5
3:4 > 3: 5
Now write three equivalent ratios of each of the ratios
ii)
15
5
and
21
7
32
Hence
Now write three equivalent ratios of each of the ratios
iii)
2
5
and
7
11
Hence
33
Now write three equivalent ratios of each of the ratios
iv)
15
4
and
20
5
Hence
Now write three equivalent ratios of each of the ratios
34
v)
4
8
and
3
15
Hence
Now write three equivalent ratios of each of the ratios
35
STUDENT’S WORKSHEET – 3
Proportions
PRE CONTENT WORKSHEET P 1
Name of the student ______________________
Date ______
Activity1 – Reaching Proportion
Compare the ratios.
6:9
2:5
12:16
and
and
and
14:21
3:2
9:12
3:15
5:20
and
and
6:20
6:24
Now write the ratios which are equivalent :
The terms in the above answer are said to be in ……………………………………..
Activity2 – Check whether the ratios are proportional:
i)
14 : 14 and 2 : 1
ii)
18 : 48 and 12 : 32
iii)
10 : 3 and 9 : 2
iv)
4 : 5 and 28 : 35
v)
18 : 48 and 3 : 8
36
STUDENT’S WORKSHEET – 4
More Proportions
PRE CONTENT WORKSHEET P 2
Name of the student ______________________
Date ______
Activity 1- Proportion Maniac
Create a proportion from each set of numbers:
i)
9,1.3,27
ii)
12,3,11,44
iii)
36,9,8,32
iv)
16,10,8,5
Activity 2 – Determine if the given ratios form a proportion. Also, write the middle
terms and extreme terms where the ratios form a proportion:
i)
27 litres: 135 litres and9 cans: 45 cans
ii)
2kg: 60kg and 25g: 125g
iii)
20cm: 1 meter and 35cm: 175cm
iv)
3g : 4g and 5g: 6g
v)
25cm: 5cm and 16m: 80m
37
STUDENT’S WORKSHEET – 5
Mean Proportion
Content Worksheet C1
Name of the student ______________________
Activity1 – Finding x
i)
6 : 15 : : x : 35
ii)
30 : 25 : : 42 : x
iii)
9 : x : : 6 : 10
iv)
x : 1.5 : : 6.3 : 4.5
v)
4 : x : : 8 : 14
Activity 2- Find the fourth proportion to:
i)
2,6,1
ii)
30,32,.42
iii)
18,54,6
iv)
4,3,6
v)
6,10,14
38
Date ______
Activity 3- Find the third proportion to :
i)
5 and 10
ii)
2 and 8
iii)
12 and 36
iv)
7 and 35
v)
3 and 18
Activity 4- Find the mean proportion to :
i)
6 and 54
ii)
9 and 16
iii)
8 and 32
iv)
3 and 14
39
STUDENT’S WORKSHEET – 6
Unitary Method
Content Worksheet C2
Name of the student ______________________
Date ______
Activity1 – Fruit Bonanza
John goes to a fruit shop. He bought 2kg apples, 5 oranges,10 bananas, 1/2kg
strawberries, 1kg grapes and 2 pineapples. Give a bill to John by filling the missing
details:
$250 per kg
$350 per kg
$120 per dozen
$100 per dozen
$150 per kg
$25 per piece
40
Fruit Bonanza
S.No.
Item
Rate
1.
APPLES
2.
ORANGES
3.
BANANAS
4.
STRAWBERRY
5.
GRAPES
6.
PINEAPPLES
Quantity
TOTAL
41
Cost
STUDENT’S WORKSHEET –7
Percentage Fractions and Decimals
CONTENT Worksheet C3
Name of the student ______________________
Date ______
Activity - Dropping Drill
% is ‘per hundred’
x% means ‘x-hundredth’
A) Convert percent into fraction and ratio by dropping the % sign and dividing by
100.
Convert percent into decimal by dropping the % sign and shift the decimal 2 places
to the left.
30% =
=
45% =
34% =
32% =
56%
97%
105%
48%
34%
3%
61%
100%
= 3: 10
= 0.3
42
B) Solve the % given below and drop the equivalent ratios, decimals and fractions
from the clue box (find as many as possible).
30%
125%
40%
5%
92%
300%
2/5
3:1
23:25 8/20
0.92
6/2
5/4
1/20
0.4
46/50 2/5
3:10
3/1
0.3
4/10
1.25
0.05
3
0.92
4/80
25/20 90:30 9:30
46/50
0.30
5/100 0.40
43
920/1000
STUDENT’S WORKSHEET – 8
Fractions and Decimals into Percentage
CONTENT Worksheet C4
Name of the student ______________________
Date ______
Activity - Movers on Move
Percentage Packers and Movers are on a move and they are converting fractions, ratios
and decimals to percentage so that it is easy to pack them. Help them complete their
work.
1. 11/50
7. 6/15
2. 0.23
8. 2.45
3. 7/8
9. 1/5
4. 18/60
10. 18 :60
5. 7/20
11. 0.3
6. 5 : 4
12. 3/5
44
STUDENT’S WORKSHEET – 9
Percentage of a Given Quantity
CONTENT Worksheet C5
Name of the student ______________________
Date ______
Activity - Picture Percentile
A) Use the given colour grid of 10 by 14 which is made up of tiles of different colours
and write the percent and the simplest fraction for different colour tiles in the space
provided(do complete working).
%
Fraction
%
Fraction
%
Fraction
45
%
Fraction
%
Fraction
B) Carefully observe the given box having tiles of different colours and size and answer
the following questions.
1. How many ‘b’ parts equal ‘a’ part?
2. How many ‘c’ parts equal ‘a’ part?
3. How many ‘e’ parts equal one ‘a’ part?
4. How many ‘d’ parts equal one ‘b’ part?
5. If ‘a’ contains 50% of the given box then
a. ‘b’ contains _____% of the given box.
46
b. ‘c’ contains _____% of the given box.
c. ‘d’ contains _____% of the given box.
6. How many ‘b’ boxes will make up 75% of the whole box?
7. How many ‘c’ boxes will make up 75% of the whole box?
8. How many ‘d’ boxes will make up 25% of the whole box?
9. How many ‘e’ boxes will make up 75% of the whole box?
10. What percentage of the whole box are ‘c’ and ‘d’ put together?
C)
Solve the following:
1. 45% of 250
2. 35% of 500m
3. 120% of 40
4. 30% of $800
5. 51% of 900 marks
47
STUDENT’S WORKSHEET -10
Application of Percentage
CONTENT WORKSHEET C6
Name of the student ______________________
Date ______
Activity – Cricket Mania
Obtain the score card of an IPL Cricket match played between your favorite teams
and find the following details of the five top scorers of both teams
___________________________ V/S ____________________________
Match Format __________________ Won by ___________________ Played on
______
Name of
Batsman
Runs
Balls Strike
scored Faced rate
A
B
Name of
Batsman
Runs
Balls Strike
scored Faced rate
A
A/B 100
48
B
A/B 100
Now announce the following award
Runs Scored
__________
_
Strike Rate
BEST STRIKE RATE – TEAM 1
________
Name:
Runs Scored
__________
_
Strike Rate
BEST STRIKE RATE – TEAM 2
________
Name:
Runs Scored
__________
_
BEST STRIKE RATE OF THE MATCH
Strike Rate
________
Name:
49
STUDENT’S WORKSHEET -11
Percentage Drill
CONTENT WORKSHEET C7
Name of the student ______________________
Activity – Independent Practice
1. What percent is each of the following?
a) $15 of $600
b) 250 gram of 4kg
c) 800 m of 20 km
d) 450 ml of 4.5 liters
2. Find the number whose 16% is 32.
3. Find the number whose 12 % is 64.
4. Find the number whose 6 % is 12.
5. Find the total distance covered by a bus whose 3 % is 1.5 km.
50
Date ______
6. 72% of 250 students are good in mathematics. How many are not good in
mathematics?
7. Timford High School football team won 10 matches out of the total
number of matches they played. If their win percentage was 40, then
how many matches did they play in all?
8. If Maria had $ 600 left after spending 75% of her money, how much did she have in
the beginning?
9. In a toy shop, Sports items, toy cars and Barbie dolls are in the
ratio of 3 : 5 : 2. What is the percentage of toy cars in the shop?
10. A man got a 10% increase in his salary. If his new salary is $15400, find his original
salary.
11. In an examination 94% of the candidates passed and 105 failed. How many
candidates appeared for the examination?
51
12. On last Monday, 25% of the boys were absent in class VII and 18 were present. How
many boys are on roll in class VII?
13. The composition of gunpowder is as under:
Nitre 75%, Charcoal 15% and Sulphur . How many grams of each are there in 1.5 kg
of gunpowder?
14. A salesman gets 9.5% commission on his sales. Find his commission for a laptop
sold for $520.
15. Find
a) 15 % more than 600 m.
b) 18 % less than $500.
52
STUDENT’S WORKSHEET -12
Increase and Decrease as Percent
CONTENT WORKSHEET C8
Name of the student ______________________
Date ______
Activity- Essential Information
A) Obtain the daily maximum temperature for one week from newspaper daily or
from the website of metreological department.
Day of Week
Max. Temp.(0C)
Monday
Tuesday
Wednesday
Thursday
Friday
Saturday
Sunday
Find the Increase or decrease percentage in the temperature of the following:
1. Monday to Wednesday
2. Tuesday to Wednesday
3. Wednesday to Friday
4. Tuesday to Saturday
53
5. Thursday to Sunday
C) Obtain the price of Gold for one week from newspaper daily or from a reliable
source.
Day of Week
Price of Gold (per 10 gm)
Monday
Tuesday
Wednesday
Thursday
Friday
Saturday
Sunday
Find the Increase or decrease percentage in the price of the Gold on daily basis.
1. Monday to Tuesday
2. Tuesday to Wednesday
3. Wednesday to Thursday
4. Thursday to Friday
54
5. Friday to Saturday
6. Saturday to Sunday
B) Solve the following
1. The size of a bag that could hold 5 kg of weight has now been increased to hold
7 kg. What is the percentage increase in size?
2. A computer that usually sells for $856 goes on sale for $630. What is the percent
decrease in the selling price of computer?
3. In ten years the population of a town increased from 358,000 to 416,000. Find the
percentage increase.
55
Student’s WORKSHEET – 13
Profit and Loss
CONTENT Worksheet C9
Name of the student ______________________
Date ______
Activity - Profit Loss Race
PROFIT LOSS CARDS
$800
0
$550
$500
$ 670
$ 900
$600
$250
$50
$ 700
$ 980
$650
$960
$250
$500
$ 670
$ 980
$600
$800
$550
$50
$ 700
$ 900
$650
$960
Materials Required: 1 die,
1 Board per group
2-3 counters to move depending upon the number of players
To Play:
Note: Number on the cards represents the C.P.
1. Shuffle the Profit and Loss cards, and then place them on the table, face down.
2. Player 1 rolls the die and simultaneously picks up a card. Green cards represent
profit to be calculated on the given C.P whereas Red cards are for loss.
3. Player 1 then finds the profit or the loss on the C.P. as per the color of the card.
56
4. If player 1 solves the problem correctly then he or she moves the counter same
number of steps as the number obtained on dice forward on the board. If the answer
is incorrect then the player moves the same number of steps backwards. However,
they cannot move further backwards from the start position.
5. If the players use all of the cards in the deck, they reshuffle the deck and start again.
6. The player who reaches Finish first wins.
PLAY BOARD
57
Student’s WORKSHEET – 14
Profit and Loss as Percentage
CONTENT Worksheet C10
Name of the student ______________________
Date ______
Activity - Club Percentage
A) Find the profit% or loss% as the case may be and put the following Q.No. in the
appropriate % Box.
1. C.P. = $700 , Profit = $35
2. C.P. = $5000 , S.P. = $5100
3. S.P. = $630, C.P. = $700
4. P = $10, C.P. = $150 , Overhead Expenses = $50
5. C.P. = $1400 , S.P. = $1428
6. S.P. = $720 , C.P. = $800
7. C.P. = $1400 , S.P. = $1372
8. C.P. = $800 , S.P. = $960
9. C.P. = $8100 , S.P. = $8019
10. C.P. = $80 , S.P. = $96
11. C.P. = $360 , Overhead Expenses = $40, Loss = $16
58
B) Solve the following problems:
1. The cost of an article was $15,500 and $450 was spent on its repairs. If it is sold
for a profit of 15%, find the selling price of the article.
2. A VCR and TV were bought for $ 8000 each. The shopkeeper made a loss of 4%
on the VCR and a profit of 8% on the TV. Find the gain or loss percent on the
whole transaction.
3. A shopkeeper buys 80 articles for $ 2400 and sells them for a profit of 16%. Find
the selling price of one article.
4. If selling price is doubled, the profit triples. Find the profit percent.
5. Alfred buys an old scooter for $ 4700 and spends $ 800 on its repairs. If he sells
the scooter for $ 5800, his gain as a percentage of cost price.
59
6. A milkman sold two of his buffaloes for Rs 20,000 each. On one he made a gain
of 5% and on the other a loss of 10%. Find his overall gain or loss.
7. A bookseller sells books at a profit of 10%. If he buys a book from a distributor at
$ 200, how much does he sell it for?
8. Richard sold a painting from his collection, making a profit of 20%. How much
did the painting cost him?
9. By selling a camera for $ 240, a shopkeeper makes a profit of 20%. What is his
C.P.? What would his profit percent be if he sold the article for $ 275?
10. Find the gain or loss percent if C.P. = $332 and S.P. = $350.
60
STUDENT’S WORKSHEET – 15
Simple Interest
CONTENT Worksheet C11
Name of the student ______________________
Date ______
Activity - Interesting Interest
I want to get a loan to buy a car? What
should I know about the loan or amount
of my monthly payments?
1. If P= $5000 , R = 12.5%p.a. , T = 5 years then A = ____________
2. If P= $8500 , R = 5%p.a. , T = 3 years then S.I. = ____________
3. If P= $2000 , R = 11%p.a. , T = 2 years 6 months then A = ____________
4. Vanya deposited $14600 at a rate of 10%per annum for 175 days. Find the interest and
the amount she got back after 175 days.
61
5. Sara deposited $1400 for 3 years and 9 months at the rate of 5% p.a. Find the amount
she may get at maturity.
6. Maria borrowed $17000 at the rate of 6% p.a. for 8 years and 5 months. Find the
amount she paid back.
7. Mr. Mark opened a saving account that earns 3.5% annual interest. He wants to earn
at least $315 in interest after 3 years. How much money should he invest in order to
earn $315 as interest?
62
STUDENT’S WORKSHEET – 16
Finding Rate
CONTENT Worksheet C12
Name of the student ______________________
Activity - My Choice EMI
A)
Date ______
All banks offered various schemes.
Earn $ 400
interest on $3000
in 3 years.
Earn $ 1600
interest on
$10000 in 5 years.
Earn $ 900 interest
on $6000 in 3
years.
Earn $ 350
interest on $2100
in 2 years.
Help Ronald select the best scheme. (Hint: offering the best rate of interest).Do the
working in the space provided and answer accordingly.
Ronald’s choice = ___________________
Give reason to support his choice. _________________________________
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B)
Solve the following problems:
1. If Ronald wants a yearly interest of $750 on a deposit of $15000 then which of the
above scheme he would select?
Earn $__________
interest on $______ in
________ years
2. Find Rate when, Principal is $ 6000, Interest is $ 900 and Time is 7 years.
3. Ronald took another scheme and deposited $5400 and got back an amount of
$6000 after 2 years. Find Ronald's interest rate.
4. Marcus borrowed $3000 for 4 years at simple interest to pay for his car. If Marcus
repaid a total of $3750, at what interest rate did he borrow the money?
5. Richard deposits 5400 and got back an amount of 6000 after 2 years. Find
Richard's interest rate.
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STUDENT’S WORKSHEET –17
Finding Time and Principal
CONTENT Worksheet C13
Name of the student ______________________
Activity - Time Machine
Date ______
1. Find Time when, Principal = $ 1500; Interest = $ 150; Rate = 5%p.a.
2. Find Time when, Principal = $ 4000; Interest = $ 800; Rate = 4% p.a.
3. Find Time when, Principal = $ 2000; Interest = $ 60; Rate = 3% p.a.
4. Find in what time will $ 200 amount to $ 270 at the rate of 5%p.a. simple interest?
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5. Find Time when, Principal is $ 3200, Interest is $ 160, Rate is 2% p.a.
6. Find Principal when Time = 3 years, Interest = $ 600; Rate = 4% p.a.
7. Find Principal when Time = 5 years, Interest = $ 500; Rate = 5% p.a.
8. Find Principal when Time = 4 years, Interest = $ 400; Rate = 5% p.a.
9. Find the Principal which gives an Amount of $ 4116 at the rate of 8% for 5 years.
10. Find the Principal which gives an amount of $ 4116 in 5 years at the rate of 8%per
annum.
66
STUDENT’S WORKSHEET –18
Assess Ratios and proportion
CONTENT Worksheet PC1
Name of the student ______________________
Date ______
Activity- Assessment of the chapter
Q1. Choose the correct answer:
1. The ratio of 2 minutes to 1 hour is
(a) 1 : 30
(b) 2 : 1
(c) 1 : 2
(d) 30 : 1
2. Percent means a fraction with denominator
(a) 1
(b) 10
(c) 100
(d) 1000
3. The present population of a town is 50,000. If the population increases by 5%,
then population after an year will be
(a) 55,000
(b) 52,500
(c) 57.500
(d) 5,200
(c) 200
(d) 5000
4. If 25% of x is 500, then x is
(a) 20,000
(b) 2000
5. x% of y is
(a)
xy
100
(b)
x
100
(c)
y
100
(d)
100
xy
6. If a certain amount doubles after ten years then the rate of interest is
(a) 1%
(b) 10%
(c) 20%
(d) 25%
7. What value of x will make the number 40 : 35 :: 16 : x in proportion
(a) 4
(b) 87
(c) 17
(d) 14
8. Anil bought a shirt for Rs.250 and sold it at a gain of 25%. He sold it for
(a) Rs.320.5 (b) Rs.312.5 (c) Rs.3250
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(d) 300
9. 0.075 is equal to
(a) 75 percent (b)
10.
7
percent
5
1
as percent is equal to
3
1
1
(a) 33 %
(b) 30 %
3
3
(c) 7.5%
(c) 30%
(d) 60%
11. Which bottle of moisture is the best buy?
A. 8 oz for $1.49
B. 12 oz for $2.29
C. 15 oz for $2.75
D. 20 oz for $3.89
Q2. Write the following ratios in lowest terms:
ii)
1
3 to2
2
18 to 57
iii)
75paise to Rs.5
iv)
3m5cm to 35cm
v)
3dozen to 2 score
i)
Q3. Check if the following ratios are equivalent?
i)
2 : 3 and 6 : 9
ii)
1 : 4 and 2 : 3
iii)
7 : 9 and 9 : 7
iv)
5 : 8 and 15 : 24
v)
12 : 16 and 21 : 28
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(d) none
Q4. Solve for x:
i)
3/7 = x/28
ii)
3/5 =15/x
iii)
x/18 = 12/72
iv)
32/x = 4/17
v)
24/x = 6/3
Q5. Find three equal ratios for each of the following:
i)
5 to 8
ii)
60/40
iii)
9 : 21
iv)
25 to 33
v)
4:9
1
3
Q6. A vase contains 16 roses, 10 carnations, and 14 daisies. Write each ratio in lowest
terms using to.
a. roses to daisies
b. carnations to all flowers
c. daisies to roses and carnations
Q7. At a certain speed Jack travels 320 miles in 4.5 hours. At the same speed, how far
would he travel in 7 hours?
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Q8. If a restaurant likes to keep a ratio of 3 waiters per 42 seated guests, how many
waiters should the restaurant have on staff for a night where crowds of 112 seated
guests are expected?
Q9. A baker usually bakes 4 cakes per 90 guests. How many cakes should the caterer
bake for a party with 225 guests?
Q10. A recipe calls for 2 cups of sugar for every 5 cups of white flour. If 12 cups of flour
are used, how many cups of sugar are needed?
Q11. A truck can travel 232 miles on 14.5 gallons of diesel. How many gallons would
the truck need to travel 400 miles?
Q12. A sum of money is to be distributed among A, B, C, D in the proportion of
5 : 2 : 4 : 3. If C gets $1000 more than D, then what is B's share?
Q13. Seats for Mathematics, Physics and Biology in a school are in the ratio 5 : 7 : 8.
There is a proposal to increase these seats by 40%, 50% and 75% respectively. What
will be the ratio of increased seats?
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Q14.Two numbers are respectively 20% and 50% more than a third number. Then find
the ratio of the two numbers.
4
2
of Rose's amount is equal to
of
5
5
Jasmine's amount, how much amount does Jasmine have?
Q15. Rose and Jasmine together have $1210. If
Q16. To make 3 glasses of mango squash you need 600ml of water. Calculate how much
water you need to make :
i)
5 glasses of squash
ii)
7 glasses of squash
iii)
1 glass of squash
Q17. If 10 litres of petrol cost £8.20, calculate the cost of:
i)
4 litres
ii)
12 litres
iii)
30 litres
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Q18. Peter buys 21 football stickers for 84p.Calculate the cost of:
i)
7 stickers
ii)
50 stickers
iii)
15 stickers
Q19. The first, third and fourth term of a proportion are 6, 9 and 36. Find the second
term.
Q20. If $500 is to be divided amongst Raj, Rack and Mack , so that Raj get two parts,
Rack three parts and Mack five parts. How much money will each get?
Q21. Express the following as percentage:
a) 3: 15
b) 4: 5
c) 15:40
d) 3.64: 5.6
Q22. Jery got 50% marks in Hindi, 75% in English and 90 marks in Maths. The
maximum marks in each subject were 100,140 and 160 respectively. Find his
aggregate marks.
Q23.By selling a jacket for $729, Sherlyn lost 10%. What was the cost price?
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Q24. Two chairs were sold at$990 each, one at a profit of 10% and the other at loss of
10%.What are the cost prices of each of the tables? In this sale, did the seller make a
profit or loss? What is the percentage of profit or loss?
Q25.Kartik and Sameer borrowed $6000 each at the rate of 12% p.a. for a period of 2
years and 3 years respectively. Find the difference of interest paid by them.
73
STUDENT’S WORKSHEET –19
Test Your Progress
CONTENT Worksheet PC2
Name of the student ______________________
Date ______
a) Find the ratio of the following in the lowest form.
a) 75 paise to Rs.5
b) 7kg to 700grams
c) 3 dozen to 2 scores
d) 3m 5cm to 35cm
e) 48 mins to 2 hours 50 min
1. Peter earns $.25,000 per month and saves $.5,000 per month. Find the ratio of:
a) earnings to savings
b) earning to expenditure
2. Length and breadth of rectangular fields are 20m and 15m respectively. Find the
ratio of the length to the breadth of the field
3. Compare the following ratios. Find the smaller ratio
a) 7:6 and 24:9
b)4:7 and 5:8
4. Two numbers are in the ratio 8:7. Their sum is 60. Find the two numbers.
5. The sum of two angles is 90 degrees. The angles are in the ratio 2:3. Find the
measure of each angle
6. The sum of the angles of a triangle is 180 degrees and the angles are in the ratio
1:2:3. Find the measure of each angle
7. Vicky divides a sum of $1200 between his two children in the ratio 5:3. How
much does each child get?
8. Divide $1500 between John and George in the ratio 5:7. How much will each of
them get?
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9. State which of the following are in proportion?
a) 7:3::63:27
b) 60g:50g::180l:150l
c) 10 minute:1 hour::6 hours:36 hours
d) 20 days:1 year::60 days:2 years
10. Find x in each of the following proportion.
a) x:12::14:24
b) 15:7::60:x
c) x:6::55:11
d) 7:14=15:x
e) 18:x=27:3
11. Shekar drives his car at a constant speed. If he travels 8 km. in 10 minutes, how
long will he take to travel 36 km?
12. Sally sells 60 tickets for the school raffle and receives a commission of $5. Sony
sells 96 tickets of the raffle. What is her commission?
13. Last month Ronnie sold Rs.2,500 worth of toys and got a commission of $.1,050.
If he sells $4,200 worth toys this month, what will be his commission?
14. The interest on $.3,000 for one year is $210. What is the interest on $.4,500 for a
year? If one wants to get an interest of $700, what should be the Principal?
15. Express 3:4 and 1:8 as percent.
16. Convert 25% and 150% to fraction, decimal, as well as ratio.
17. Henry earns $12,000 per month. He spends 30% of his monthly income on house
rent, 40% on household expenses, 5% on travel, and 5% on entertainment for the
family. The remaining is saved in a bank. Find out how much Henry spends on
each item and how much does he save.
18. Express 45cm as a percentage of 3m.
19. Express 84g as a percentage of 3.36kg.
75
20. The strength of a high school is 2.500. Out of this, 600 are in pre-primary section,
800 in primary section, 700 in middle school, and the remaining are in high
school. Find the percentage of students in each section.}
21. If 30% of a sum of money is $270, what is the total sum of money?
22. If 25% of a journey is 750km, how long is the whole journey?
23. Find the percentage decreases in the price of a shirt when it decreased from $80
to $60.
24. Estimate the following:
a) 40% of 390
b) 48% of 510
25. Percy buys a cycle for $1,700 and sells it at $1,870. What is his profit percentage?
26. Jones buys a radio for $2,200 and sells it for a profit of 20%. What is the selling
price of the radio?
27. Raj sells a TV for $8,400, making a profit of 20%. What was the C.P. of the TV?
28. Rashmi buys a camera for $12,000 and since he was in need of money, he had to
sell it for $8,400. What is his loss percent?
29. A music system was sold at a profit of 7%. If it had been sold at profit of 10%, the
profit would have been $780 more. What is its cost price?
30. Boston borrowed $7,500 from Sam and after one year returned $8,250 to his
friend. Calculate the simple interest and rate.
31. In how many years will a sum of money double itself at 10% simple interest per
annum?
1
32. If $750 yields $885 in three years at simple interest, what will $1,200 yield in 3
2
years?
33. If a certain sum amounts to $44,000 at 5% p.a. over 2 years, how much will it
amount to in 3 years at the same rate?
34. Jazz invested $15,000 at 15% p.a. interest from 17 June 2006 to 10 November 2011.
Calculate the interest.
76
Suggested Video links
Name
Video Clip 1
Title/Link
Revise Ratios and Proportion
www.bgfl.org/bgfl/.../Ratio_n_Proportion_FINAL_PUPILS_F3.ppt
Video Clip 2 Ratios and Proportion
www.klvx.org/DocumentView.asp?DID=144
Video Clip 3
What is a Ratio
www.grossmont.edu/carylee/Ma125/lectures/ratio.ppt
Video Clip 4 Proportions
www.primaryresources.co.uk/.../powerpoint/ratio_and_proportion.ppt
Video Clip 5
Percentages
http://bit.ly/Alb3kS
Video Clip 6
Simple Interest
sreagles.wikispaces.com/file/view/Simple+Interest.ppt
Video Clip 7
Fractions and Percentages
www.maths4gcse.co.uk/Percentages.ppt
Weblink1
http://www.youtube.com/watch?v=D6uYQYgSVd8
Weblink 2
Introduction to Ratios
http://www.youtube.com/watch?v=UsPmg_Ne1po
Weblink 3
Equivalent Ratios
http://www.youtube.com/watch?v=jwpmEiYJrS8&feature=related
Weblink 4
Proportions
http://www.youtube.com/watch?v=8v40pLjmbEM&feature=related
Weblink 5
http://www.arcademicskillbuilders.com/games/dirt-bikeproportions/dirt-bike-proportions.html
Weblink 6
http://www.arcademicskillbuilders.com/games/ratio-blaster/ratioblaster.html
Weblink 7
http://www.arcademicskillbuilders.com/games/ratio-stadium/ratiostadium.html
77
CENTRAL BOARD OF SECONDARY EDUCATION
Shiksha Kendra, 2, Community Centre, Preet Vihar,
Delhi-110 092 India