Table of Contents
Exercise 2.1
Question: 1
Compare the following fractions by using the symbol > or < or =;
![](https://files.askiitians.com/cdn/images/20181113-19052101-3156-equation-1.png)
Solution:
(i) We have,
![](https://files.askiitians.com/cdn/images/20181113-1910539-5404-equation-3.png)
Taking the LCM of 9 and 13, we get,
9 x 13 = 117
Now, we convert the given fractions to equivalent fractions by making the denominators 117,
![](https://files.askiitians.com/cdn/images/20181113-1915586-2210-equation-4.png)
As we know, 91 > 72
![](https://files.askiitians.com/cdn/images/20181113-19110102-9713-equation-5.png)
Hence, 79 > 813
(ii) We have,
![](https://files.askiitians.com/cdn/images/20181113-19113961-5330-equation-6.png)
The given fractions are equivalent fractions as the denominators are equal,
And we know that, 11 > 5
Therefore, 119 > 59
(iii) We have,
![](https://files.askiitians.com/cdn/images/20181113-19118305-4261-equation-7.png)
Taking the LCM of 41 and 30, we get,
41 x 30 = 1230
Now, we convert the given fractions to equivalent fractions by making the denominators 1230,
![](https://files.askiitians.com/cdn/images/20181113-19121852-4687-equation-8.png)
Now, we clearly know 1110 > 779
![](https://files.askiitians.com/cdn/images/20181113-19125446-653-equation-9.png)
Hence, 3741 > 1930
(iv) We have,
![](https://files.askiitians.com/cdn/images/20181113-19129868-2300-equation-10.png)
Taking the LCM of 15 and 105, we get,
5 x 3 x 7 = 105
Now, we convert the given fractions to equivalent fractions by making the denominators 105,
![](https://files.askiitians.com/cdn/images/20181113-19134133-8964-equation-11.png)
Question: 2
Arrange the following fractions in ascending order:
![](https://files.askiitians.com/cdn/images/20181113-19142180-3305-equation-13.png)
Solution:
(i) We have,
![](https://files.askiitians.com/cdn/images/20181113-19146227-5857-equation-14.png)
Taking the LCM of 8, 6, 8, 4 and 3, we get,
2 x 4 x 3 = 24
Now, we convert the given fractions to equivalent fractions by making the denominators 24,
![](https://files.askiitians.com/cdn/images/20181113-19149524-220-equation-15.png)
We know that, 8 < 9 < 12 < 18 < 20
![](https://files.askiitians.com/cdn/images/20181113-19153212-7266-equation-16.png)
Hence, 13 < 38 < 24 < 68 < 56
(ii) We have,
![](https://files.askiitians.com/cdn/images/20181113-19157133-6786-equation-17.png)
Taking the LCM of 6, 8, 12 and 16, we get,
2 x 2 x 2 x 2 x 3 = 48
Now, we convert the given fractions to equivalent fractions by making the denominators 48,
![](https://files.askiitians.com/cdn/images/20181113-192156-1083-equation-18.png)
We know that, 12 < 15 < 18 < 32
![](https://files.askiitians.com/cdn/images/20181113-192571-1684-equation-19.png)
Question: 3
Arrange the following fractions in descending order:
![](https://files.askiitians.com/cdn/images/20181113-19217728-9831-equation-20.png)
Solution:
(i) We have,
![](https://files.askiitians.com/cdn/images/20181113-1922256-399-equation-21.png)
Taking the LCM of 5, 10, 15 and 20, we get,
5 x 2 x 2 x 3 = 60
Now, we convert the given fractions to equivalent fractions by making the denominators 48
![](https://files.askiitians.com/cdn/images/20181113-19225416-3889-equation-22.png)
As we know 51 > 48 > 44 > 42
![](https://files.askiitians.com/cdn/images/20181113-19229244-2330-equation-23.png)
(ii) We have,
![](https://files.askiitians.com/cdn/images/20181113-19234353-3040-equation-24.png)
Taking the LCM of 7, 35, 14 and 28, we get,
7 x 5 x 2 x 2 = 140
Now, we convert the given fractions to equivalent fractions by making the denominators 140
![](https://files.askiitians.com/cdn/images/20181113-19238150-9529-equation-25.png)
As we know 40 > 44 > 65 > 90
![](https://files.askiitians.com/cdn/images/20181113-19242509-7273-equation-26.png)
Question: 4
Write the equivalent fractions of 3/5
Solution:
Multiplying or dividing both the numerator and denominator by the same number,
so that the fraction keeps its value.
So the equivalent fractions of 3/5 are
![](https://files.askiitians.com/cdn/images/20181113-19246150-3554-equation-27.png)
are the five equivalent fractions of 3/5
Question: 5
Find the sum:
![](https://files.askiitians.com/cdn/images/20181113-19249447-7917-equation-28.png)
Solution:
(i) We have,
![](https://files.askiitians.com/cdn/images/20181113-19253806-436-equation-29.png)
Taking the LCM of 8 and 10, we get,
2 x 4 x 5 = 40
Now, we convert the given fractions to equivalent fractions by making the denominators 40
![](https://files.askiitians.com/cdn/images/20181113-19257213-9467-equation-30.png)
(ii) We have,
![](https://files.askiitians.com/cdn/images/20181113-1932354-2128-equation-31.png)
Taking out the LCM of 4 and 5, we get,
4 x 5 = 20
Now, we convert the given fractions to equivalent fractions by making the denominators 20
![](https://files.askiitians.com/cdn/images/20181113-1937573-7505-equation-32.png)
(iii) We have,
![](https://files.askiitians.com/cdn/images/20181113-1931358-7309-equation-33.png)
Taking out the LCM of 6 and 4, we get,
2 x 2 x 3 = 12
Now, we convert the given fractions to equivalent fractions by making the denominators 12
![](https://files.askiitians.com/cdn/images/20181113-191411769-3751-equation-34.png)
(iv) We have,
![](https://files.askiitians.com/cdn/images/20181113-19323870-7158-equation-35.png)
Taking out the LCM of 5, 10 and 15, we get,
5 x 2 x 3 = 30
Now, we convert the given fractions to equivalent fractions by making the denominators 30
![](https://files.askiitians.com/cdn/images/20181113-19328355-2709-equation-36.png)
Question: 6
Find the difference of
![](https://files.askiitians.com/cdn/images/20181113-19333136-5088-equation-37.png)
Solution:
(i) We have,
![](https://files.askiitians.com/cdn/images/20181113-19337871-6702-equation-38.png)
Taking out the LCM of 24 and 16, we get,
2 x 2 x 2 x 2 x 3 = 48
Now, we convert the given fractions to equivalent fractions by making the denominators 48
![](https://files.askiitians.com/cdn/images/20181113-19341652-9603-equation-39.png)
(ii) We have, 6 and 23/3
The difference between 6 and 23/3
![](https://files.askiitians.com/cdn/images/20181113-19345230-1981-equation-40.png)
(iii) We have,
![](https://files.askiitians.com/cdn/images/20181113-191645400-8003-equation-41.png)
Taking out the LCM of 25 and 20, we get,
5 x 5 x 4 = 100
Now, we convert the given fractions to equivalent fractions by making the denominators 100
![](https://files.askiitians.com/cdn/images/20181113-191656354-9248-equation-42.png)
The difference between both the fractions are
![](https://files.askiitians.com/cdn/images/20181113-19171900-8180-equation-43.png)
(iv) We have,
![](https://files.askiitians.com/cdn/images/20181113-19176666-6971-equation-44.png)
Taking out the LCM of 10 and 15, we get,
2 x 3 x 5 = 30
Now, we convert the given fractions to equivalent fractions by making the denominators 30
![](https://files.askiitians.com/cdn/images/20181113-191714729-4900-equation-45.png)
The difference between both the fractions are
![](https://files.askiitians.com/cdn/images/20181113-191724260-2481-equation-46.png)
Question: 7
Find the difference:
![](https://files.askiitians.com/cdn/images/20181113-191736792-7597-equation-47.png)
Solution:
(i) We have,
![](https://files.askiitians.com/cdn/images/20181113-191745323-5700-equation-48.png)
Taking out the LCM of 7 and 11, we get,
7 x 11 = 77
Now, we convert the given fractions to equivalent fractions by making the denominators 77
![](https://files.askiitians.com/cdn/images/20181113-191758245-3499-equation-49.png)
The difference between both the fractions are
![](https://files.askiitians.com/cdn/images/20181113-19184902-6128-equation-50.png)
(ii) We have,
![](https://files.askiitians.com/cdn/images/20181113-19181190-3807-equation-51.png)
(iii) We have,
![](https://files.askiitians.com/cdn/images/20181113-191819422-1387-equation-52.png)
(iv) We have,
![](https://files.askiitians.com/cdn/images/20181113-191828173-3602-equation-53.png)
Taking out the LCM of 10 and 15, we get,
2 x 3 x 5 = 30
Now, we convert the given fractions to equivalent fractions by making the denominators 30
![](https://files.askiitians.com/cdn/images/20181113-191845954-6047-equation-55.png)
Question: 8
Simplify:
![](https://files.askiitians.com/cdn/images/20181113-191852611-8676-equation-56.png)
Solution:
(i) We have,
![](https://files.askiitians.com/cdn/images/20181113-191858736-7226-equation-57.png)
Taking out the LCM of 3, 6 and 9, we get,
3 x 3 x 2 = 18
Now, we convert the given fractions to equivalent fractions by making the denominators 18, we get,
![](https://files.askiitians.com/cdn/images/20181113-192736654-2862-equation-58.png)
(ii) We have,
![](https://files.askiitians.com/cdn/images/20181113-19194705-793-equation-59.png)
(iii) We have,
![](https://files.askiitians.com/cdn/images/20181113-19191248-7708-equation-60.png)
Taking out the LCM of 6, 8 and 12, we get,
2 x 2 x 2 x 3 = 48
Now, we convert the given fractions to equivalent fractions by making the denominators 48, we get,
![](https://files.askiitians.com/cdn/images/20181113-191917236-5909-equation-61.png)
Question: 9
What should be added to to get 12?
Solution:
We have,
![](https://files.askiitians.com/cdn/images/20181113-191926815-3806-equation-63.png)
Let x be the number added to 38/7 to get 12
Therefore,
x + 387 = 12
![](https://files.askiitians.com/cdn/images/20181113-191932159-2215-equation-64.png)
Question: 10
What should be added to?
Solution:
![](https://files.askiitians.com/cdn/images/20181113-191941455-2096-equation-66.png)
Let x be the number added to 79/15 to get 63/5
![](https://files.askiitians.com/cdn/images/20181113-191946377-5870-equation-67.png)
Taking out the LCM of 5 and 15, we get,
3 x 5 = 15
Now, we convert the given fractions to equivalent fractions by making the denominators 15, we get,
![](https://files.askiitians.com/cdn/images/20181113-191953361-2054-equation-68.png)
Question: 11
Suman studies forhours daily. She devotes
hours of her time for science and mathematics. How much time does she devote for other subjects?
Solution:
Given,
Suman studies forhours daily
She devoteshours of her time for science and mathematics. Let x be time she devotes for other subjects.
![](https://files.askiitians.com/cdn/images/20181113-192022914-6650-equation-72.png)
Taking out the LCM of 3 and 5, we get,
3 x 5 = 15
Now, we convert the given fractions to equivalent fractions by making the denominators 48, we get,
![](https://files.askiitians.com/cdn/images/20181113-192033273-4688-equation-73.png)
Question: 12
A piece of wire of lengthIf it is cut into two pieces in such a way that the length of one piece is
what is the length of the other piece?
Solution:
A piece of wire of lengthone piece is
![](https://files.askiitians.com/cdn/images/20181113-192049242-787-equation-76.png)
Let the length of other piece be x m.
![](https://files.askiitians.com/cdn/images/20181113-192054805-8083-equation-77.png)
Question: 13
A rectangular piece of paper islong and
wide. Find its perimeter?
Solution:
Given,
A rectangular piece of paper islong and
wide
![](https://files.askiitians.com/cdn/images/20181113-192112946-1259-equation-80.png)
Perimeter = 2(length + width)
![](https://files.askiitians.com/cdn/images/20181113-192118305-8031-equation-81.png)
Question: 14
In a “magic square”, the sum of numbers in each row, in each column and along the diagonal is same. Is this a “magic square”?
4/11 | 9/11 | 2/11 |
3/11 | 5/11 | 7/11 |
8/11 | 1/11 | 6/11 |
![](https://www.askiitians.com/resources/images/fourm-img/advertisement8-8.png)
Solution:
Given,
4/11 | 9/11 | 2/11 |
3/11 | 5/11 | 7/11 |
8/11 | 1/11 | 6/11 |
![](https://www.askiitians.com/resources/images/fourm-img/advertisement8-8.png)
![](https://files.askiitians.com/cdn/images/20181113-192122852-2710-equation-82.png)
Therefore, the sum of numbers in each row, in each column and along the diagonal is same and the sum is 15/11.
Question: 15
The cost of Mathematics book is Rsand that of science book is
Which costs more and by how much?
Solution:
Given,
The cost of Mathematics book is Rsand that of science book is
We need to compare the cost of mathematics and science book,
![](https://files.askiitians.com/cdn/images/20181113-192144464-8821-equation-85.png)
Taking out the LCM of 4 and 2, we get,
2 x 2 = 4
Now, we convert the given fractions to equivalent fractions by making the denominators 4, we get
= 103/4
![](https://files.askiitians.com/cdn/images/20181113-192149339-7055-equation-86.png)
As we know, 103 > 82
![](https://files.askiitians.com/cdn/images/20181113-192155229-7906-equation-87.png)
Hence, the cost of mathematics book is more than that of the cost of the science book.
Question: 16
Provide the number in the box [] and also give its simplest form in each of the following:
![](https://files.askiitians.com/cdn/images/20181113-19225292-9565-equation-88.png)
Solution:
![](https://files.askiitians.com/cdn/images/20181113-192210855-6861-equation-89.png)
Exercise 2.2
Question: 1
Multiply
![](https://files.askiitians.com/cdn/images/20181113-194820925-1786-equation-1.png)
Solution:
(i) We have,
![](https://files.askiitians.com/cdn/images/20181113-194825456-8101-equation-2.png)
(ii) We have,
![](https://files.askiitians.com/cdn/images/20181113-19482965-2431-equation-3.png)
(iii) We have,
![](https://files.askiitians.com/cdn/images/20181113-194833161-523-equation-4.png)
(iv) We have,
![](https://files.askiitians.com/cdn/images/20181113-194836614-9869-equation-5.png)
Question: 2
Find the product:
![](https://files.askiitians.com/cdn/images/20181113-194840661-2421-equation-6.png)
Solution:
(i) We have,
![](https://files.askiitians.com/cdn/images/20181113-194845130-9609-equation-7.png)
(ii) We have,
![](https://files.askiitians.com/cdn/images/20181113-195547535-5481-equation-38.png)
(iv) We have,
![](https://files.askiitians.com/cdn/images/20181113-194855287-7124-equation-9.png)
Question: 3
Simplify:
![](https://files.askiitians.com/cdn/images/20181113-19485952-6437-equation-10.png)
Solution:
(i) We have,
![](https://files.askiitians.com/cdn/images/20181113-19492552-1323-equation-11.png)
(ii) We have,
![](https://files.askiitians.com/cdn/images/20181113-194912322-1379-equation-12.png)
(iii) We have,
![](https://files.askiitians.com/cdn/images/20181113-194916900-8010-equation-13.png)
Question: 4
Find:
![](https://files.askiitians.com/cdn/images/20181113-194920728-6451-equation-14.png)
Solution:
(i) We have,
![](https://files.askiitians.com/cdn/images/20181113-194925275-1130-equation-15.png)
(ii) We have,
![](https://files.askiitians.com/cdn/images/20181113-194929681-9189-equation-16.png)
(iii) We have,
![](https://files.askiitians.com/cdn/images/20181113-1949339-728-equation-17.png)
Question: 5
Which is greater?
![](https://files.askiitians.com/cdn/images/20181113-194936587-3106-equation-18.png)
Solution:
![](https://files.askiitians.com/cdn/images/20181113-19494087-7992-equation-19.png)
While comparing two fractions, when the numerators of both the fractions are same, then the denominator having higher value shows the fraction has lower value.
So, 6/14 is greater.
Therefore, 1/2 of 6/7 is greater.
Question: 6
(i) 7/11 of 330
(ii) 5/9 of 108 meters
(iii) 3/7 of 42 litres
(iv) 1/12 of an hour
(v) 5/6 of an year
(vi) 3/20 of a Kg
(vii) 7/20 of a litres
(viii) 5/6 of a day
(ix) 2/7 of a week
Solution:
(i) We have,
![](https://files.askiitians.com/cdn/images/20181113-194943915-6433-equation-20.png)
(ii) We have,
5/9 of 108 meters
= 5/9 × 108 meters
= 5 × 12 meters
60 meters
(iii) We have,
3/7 of 42 litres
= 3/7× 42 litres
= 3 × 6 litres
=18 litres
(iv) We have,
1/12 of an hour
An hour = 60 minutes
Therefore, 1/12 × 60 minutes
= 5 minutes
(v) We have,
5/6 of an year
I Year = 12 months
Therefore,
5/6 × 12 months
= 5 × 2 months
= 10 months
(vi) We have,
3/20 of a kg
1 Kg = 1000 gms
Therefore,
3/20 × 1000 gms
= 3 × 50 gms
= 150 gms
(vii) We have,
7/20 of a litre
1 litre = 1000 ml
Therefore,
7/20 × 1000 ml
= 7 × 50 ml
= 350 ml
(viii) We have,
5/6 of a day
I day = 24 hours
Therefore,
5/6 × 24 hours
= 5 × 4 hours
= 20 hours
(ix) We have,
2/7 of a week
I week = 7 days
Therefore,
2/7 × 7 days = 2 days
Question: 7
Shikha plans 5 saplings in a row in her garden. The distance between two adjacent saplings is 3/4 m. Find the distance between first and last sapling.
Solution:
There are 4 adjacent spacing for 5 saplings.
Given, the distance between two adjacent saplings is 3/4 m.
4 adjacent spacing for 5 saplings = 3/4 × 4 = 3 m
Therefore, the distance between first and last sapling is 3 m.
Question: 8
Ravish reads 1/3 part of a book in one hour. How much part of the book will be read in 2(1/5) hours?
Solution:
Let x be the full part of book.
Given, Ravish reads 1/3 part of a book in one hour
1 hour = (1/3) x
Part of the book will he read in 2(1/5) hours
![](https://files.askiitians.com/cdn/images/20181113-194951400-4743-equation-22.png)
Question: 9
Lipika reads a book for 1(3/4) hours every day. She reads the entire book in 6 days. How many hours in all were required by her to read the book?
Solution:
Given,
Time taken by Lipika to read a book per day = 1(3/4) = 7/4 hours
Time taken by Lipika to read a book for 6 days
Question: 10
Find the area of a rectangular park which is 41(2/3)m long and 18(3/5)m broad.
Solution:
![](https://files.askiitians.com/cdn/images/20181113-19509978-918-equation-26.png)
Area of a rectangular park = (length x breadth)
![](https://files.askiitians.com/cdn/images/20181113-195013665-7964-equation-27.png)
Question: 11
If milk is available at Rs 17(3/4) per litre, find the cost of 7(2/5) litres of milk.
Solution:
Given,
![](https://files.askiitians.com/cdn/images/20181113-195027181-3745-equation-30.png)
Question: 12
Sharda can walk 8(1/3) km in one hour. How much distance will she cover in 2(2/5) hours.
Solution:
Given,
![](https://files.askiitians.com/cdn/images/20181113-20844901-5680-equation-39.png)
Distance covered by Sharda in one hour = 25/3 km
![](https://files.askiitians.com/cdn/images/20181113-20849417-8407-equation-40.png)
Question: 12
A sugar bag contains 30 kg of sugar. After consuming 2/3 of it, how much sugar is left in the bag
Solution:
Given, A sugar bag contains 30 kg of sugar
After consuming 23 of it, the amount of sugar left in the bag
![](https://files.askiitians.com/cdn/images/20181113-195048197-6649-equation-34.png)
Question: 14
Each side of a square is 6 (2/3)m long. Find its area.
Solution:
Given,
![](https://files.askiitians.com/cdn/images/20181113-195055998-151-equation-36.png)
Question: 15
There are 45 students in a class and 3/5 of them are boys. How many girls are. there in the class?
Solution:
Given,
There are 45 students in a class,
And 3/5 of them are boys.
Therefore, no of girls in the class = 45 –3/5 × 45
= 45 – 27
= 18
Exercise 2.3
Question: 1
Find the reciprocal of each of the following fractions and classify them as proper, improper and whole numbers
(i) 3/7
(ii) 5/8
(iii) 9/7
(iv) 6/5
(v) 12/7
(vi) 1/8
Solution:
(i) 3/7
7/3 = improper number
(ii) 5/8
8/5 = improper number
(iii) 9/7
7/9 = proper number
(iv) 6/5
5/6 = proper number
(v) 12/7
7/12 = proper number
(vi) 1/8
8 = whole number
Question: 2
Divide:
![](https://files.askiitians.com/cdn/images/20181113-201233657-7174-equation-1.png)
Solution:
![](https://files.askiitians.com/cdn/images/20181113-201237907-250-equation-2.png)
![](https://files.askiitians.com/cdn/images/20181113-201242297-4721-equation-3.png)
![](https://files.askiitians.com/cdn/images/20181113-201246110-4798-equation-4.png)
Question: 3
Divide:
![](https://files.askiitians.com/cdn/images/20181113-201250688-1429-equation-5.png)
Solution:
![](https://files.askiitians.com/cdn/images/20181113-20125578-1125-equation-6.png)
![](https://files.askiitians.com/cdn/images/20181113-201259141-2041-equation-7.png)
Question: 4
Simplify:
![](https://files.askiitians.com/cdn/images/20181113-20133704-308-equation-8.png)
Solution:
![](https://files.askiitians.com/cdn/images/20181113-20137766-1224-equation-9.png)
![](https://files.askiitians.com/cdn/images/20181113-201311547-4125-equation-10.png)
![](https://files.askiitians.com/cdn/images/20181113-201315969-5773-equation-11.png)
Question: 5
A wire of length 12(1/2)m is cut into 10 pieces of equal length. Find the length of each piece.
Solution:
![](https://files.askiitians.com/cdn/images/20181113-201321329-7770-equation-13.png)
Question: 6
The length of a rectangular plot of area 65(1/3) m2 is 12(1/4) m. What is the width of the plot?
Solution:
Given,
The length of a rectangular plot of area 65(1/3) m2 is 12(1/4) m.
![](https://files.askiitians.com/cdn/images/20181113-201335659-869-equation-15.png)
Question: 7
By what number 6(2/9) be multiplied to get 4(4/9)?
Solution:
Given,
![](https://files.askiitians.com/cdn/images/20181113-201343956-6497-equation-17.png)
Let x be the number which needs to be multiplied by 56/9,
Now,
![](https://files.askiitians.com/cdn/images/20181113-201348112-8493-equation-18.png)
Question: 8
The product of two numbers is 25(5/6). If one of the numbers is 6(2/3), find the other?
Solution:
Given,
The product of two numbers is 25(5/6). If one of the numbers is 6(2/3),
![](https://files.askiitians.com/cdn/images/20181113-201352817-8156-equation-21.png)
Let the other number be x.
![](https://files.askiitians.com/cdn/images/20181113-201357489-5866-equation-22.png)
Question: 9
The cost of 6(1/4) kg of apples is Rs 400. At what rate per kg are the apples being sold?
Solution:
Given,
![](https://files.askiitians.com/cdn/images/20181113-20141551-6782-equation-24.png)
Question: 10
By selling oranges at the rate of Rs 5(1/4) per orange, a fruit seller get Rs 630. How many dozens of oranges does he sell?
Solution:
Given,
By selling oranges at the rate of Rs 5(1/4) per orange, a fruit seller get Rs 630.
5(1/4) = 2(1/4)
![](https://files.askiitians.com/cdn/images/20181113-20147270-4286-equation-27.png)
12 apples =1 dozen
Therefore, 120 apples = 10 dozen
Question: 11
In mid-day meal scheme 3/10 litre of milk is given to each student of a primary school. If 30 litres of milk is distributed every-day in the school, how many students are there in the school?
Solution:
Given,
3/10 litre of milk is given to each student of a primary school.
30 litres of milk is distributed everyday in the school
Number of students given 3/10 litres of milk = 1
Number of students given 1 litre of milk = 10/3
Number of students given 30 litres of milk = 10/3 × 30 = 100 Students
Question: 12
In a charity show Rs 6496 were collected by selling some tickets. If the price of each ticket was Rs 50(3/4), how many tickets were sold?
Solution:
Given,
Rs 6496 were collected by selling some tickets. If the price of each ticket was Rs 50(3/4),
![](https://files.askiitians.com/cdn/images/20181113-201410895-2204-equation-29.png)