NCERT Class 7 Maths Integers
The chapter 1 begins with an introduction to Integers which forms the bigger collection of numbers. Then the representation of integers on a number line which helps us to visualize integers is dealt and the first exercise has problems on the same. Then next section of the chapter deals with properties of addition and subtraction of integers such as Closure Property, Commutative Property and Associative Property. Then the multiplication of integers and different cases involved in it are dealt in the later part of the chapter.This is followed by properties of multiplication of integers such as Closure Property,Commutative Property, Associative Property and Distributive Property.Division of integers and properties associated with them are dealt in the last section of the chapter.
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Exercise 1.1
Chapter 1 Ex.1.1 Question 1
Following number line shows the temperature in degree Celsius (\(^\circ \rm C\)) at different places on a particular day.
(a) Observe this number line and write the temperature of the places marked on it.
(b) What is the temperature difference between the hottest and the coldest places among the above?
(c) What is the temperature difference between Lahulspiti and Srinagar?
(d) Can we say temperature of Srinagar and Shimla taken together is less than the temperature at Shimla? Is it also less than the temperature at Srinagar?
Solution
(a)
What is known?
Temperature in degree Celsius (\(^\circ \rm C\)) at different places on a particular day marked on the number line.
What is unknown?
Temperature at different places marked on the number line.
Reasoning:
Identify the integer on the given number line and match it with the place which is written on the top (and connected to the point by an arrow). That integer is the temperature of the place in degree Celsius (\(^\circ \rm C\)).
Steps:
Place |
Temperature |
Lahulspiti |
\(-8\) \(^\circ \rm C\) |
Srinagar |
\(–2\) \(^\circ \rm C\) |
Shimla |
\(5\) \(^\circ \rm C\) |
Ooty |
\(14\) \(^\circ \rm C\) |
Bengaluru |
\(22\) \(^\circ \rm C\) |
(b)
What is known?
Different places and their respective temperatures.
What is unknown?
The temperature difference between the hottest and the coldest places.
Reasoning:
Identify the hottest place – highest temperature (or the highest positive value of the integer) and the coldest place – lowest temperature (or the lowest negative value of the integer). Find the difference between the two values.
Steps:
Hottest place is Bengaluru at \(22 ^\circ \rm C\).
Coldest place is Lahulspiti at \(-8 ^\circ \rm C\)
The difference between the temperature of the two cities:
\[\begin{align} &= {22^{\rm{o}}}{\rm{C}} - \left( { - {8^{\rm{o}}}{\rm{C}}} \right) = {22^{\rm{o}}}{\rm{C}} +{8^{\rm{o}}}{\rm{C}}\\&= {30^{\rm{o}}}{\rm{C}}\end{align}\]
Remember that for subtraction, we add the additive inverse of the integer that is being subtracted, to the other integer. The additive inverse of \(-8\) is \(8\) and therefore, we added \(22\) and \(8\) to get the answer.
(c)
What is known?
Different places and their respective temperatures.
What is unknown?
The temperature difference between Lahulspiti and Srinagar.
Reasoning:
Use the table and find the temperature of Lahulspiti and Srinagar. Then subtract the two values. But remember one thing – subtract the er value from the higher one.
Steps:
Temperature of Srinagar: \(–2^\circ \rm C\)
Temperature of Lahulspiti: \(-8\)\(^\circ \rm C\)
Although the absolute value of the temperature of Lahulspiti is higher than that of Srinagar, it is a negative integer and thus, it is lower. Therefore, the difference between the temperature is:
\[\begin{align} {-2^{\rm{o}}}{\rm{C}} - \left( { - {8^{\rm{o}}}{\rm{C}}} \right) &= {-2^{\rm{o}}}{\rm{C}} +{8^{\rm{o}}}{\rm{C}}\\&= {8^{\rm{o}}-2^{\rm{o}}}{\rm{C}}\\ &= 6^\rm {o} C \end{align}\]
(d)
What is known?
Different places and their respective temperatures.
What is unknown?
The temperature difference between Lahulspiti and Srinagar.
Reasoning:
Use the table and find the temperature values for Lahulspiti and Srinagar. Subtract the lower value from the higher values.
Steps:
Temperature of Srinagar: \(–2\)\(^\circ \rm C\)
Temperature of Shimla: \(5\)\(^\circ \rm C\)
Temperature of Srinagar and Shimla together
\[\begin{align}&= - 2^\circ {\rm{C}} + {\rm{ }}5^\circ {\rm{C}}\\&= {\rm{ }}3^\circ {C}\end{align}\]
So, we can say that temperature of Srinagar and Shimla, taken together is less than the temperature of Shimla.
\[\begin{align}3^\circ {\rm{C}} < 5^\circ {\rm{C}}\end{align}\]
But the temperature of Srinagar and Shimla taken together is not less than the temperature of Srinagar.
\[\begin{align}-2^\circ {\rm{C}} < 3^\circ {\rm{C}}\end{align}\]
Chapter 1 Ex.1.1 Question 2
In a quiz, positive marks are given for correct answers and negative marks are given for incorrect answers. If Jack’s score in five successive rounds were \(\rm{}25, –5, –10,15, 10.\) What was his total in the end?
Solution
What is known?
Jack’s score in five successive rounds.
What is unknown?
Jack’s total score in five rounds.
Reasoning:
Just add the scores in five rounds to get the total score.
Steps:
Jack’s scores in the five successive rounds are \(25, –5, –10, 10\) and \(15\)
Total marks obtained by jack
\[\begin{align}& = 25 + \left( { - 5} \right) + \left( { - 10} \right) + 10 + 15\\& = 25-5-10 + 10 + 15\\& = 35\end{align}\]
Thus, jack obtained \(35\) marks in the quiz.
Chapter 1 Ex.1.1 Question 3
In Srinagar, temperature was \(–5^\circ \rm C\) on Monday and then it was dropped \(2^\circ \rm C\) on Tuesday. What was the temperature of Srinagar on Tuesday? On Wednesday it rose up by \(4^\circ \rm C.\) What was the temperature on this day?
Solution
What is known?
Different days and their respective temperatures.
What is the unknown?
The temperature of Srinagar on Tuesday and Wednesday.
Reasoning:
Steps:
Temperature of Srinagar on Monday\(= \,–5^\circ \rm C\)
On Tuesday it was dropped by \(2^\circ \rm C\)
Temperature on Tuesday
\[\begin{align}&= -5^\circ \rm C - 2^\circ \rm C \\ &={{\rm -7^\circ C}}\end{align}\]
On Wednesday it rose up by \(4^\circ \rm C\)
Temperature on Wednesday
\[\begin{align}&= -7^\circ \rm C\,{\rm{ + }}\,{\rm{4^\circ C}}\\&={\rm{ -3^\circ C}}\end{align}\]
Thus, temperature on Tuesday was \(–7^\circ \rm C\) and on Wednesday was \(–3^\circ \rm C.\)
Chapter 1 Ex.1.1 Question 4
A plane is flying \(5000\rm\,m\) above the sea level. At a particular point, it is exactly above a submarine floating \(1200\rm\,m \) below the sea level. What is the vertical distance between them?
Solution
What is known?
Distance of plane and submarine from the sea level
What is unknown?
The vertical distance between plane and submarine.
Steps:
Height of the plane above the sea level \(= 5000 \rm\,m\)
Depth of submarine below the sea level \(= \,–1200 \rm\,m\)
Distance between plane and submarine
\[\begin{align}& = 5000\rm{}\,\, m -\left( \rm{}-1200\, m \right)\\\rm{} &= 5000 + 1200\\& = 6200\rm{} m\\{\rm{}}\end{align}\]
Hence, the vertical distance between \(\rm{}= 6200m.\)
Chapter 1 Ex.1.1 Question 5
Mohan deposits \(₹\,2000\) in his bank account and withdraws \(₹\,1,642\) from it the next day. If withdrawal of amount from the account is represented by a negative integer, then how will you represent the amount deposited? Find the balance in Mohan’s account after the withdrawal?
Solution
What is known?
The amount deposited and the amount withdrawal.
What is the unknown?
Balance after the withdrawal.
Reasoning:
To find the balance you have to subtract the amount deposited and the amount withdrawal.
Steps:
Amount deposited by Mohan \(= ₹\, 2000\) (amount deposited can be represented by a positive integer)
Amount withdrawn by Mohan \(= ₹\,1642\) (amount deposited can be represented by negative integer)
Balance in Mohan’s account \(=\) amount deposited – amount withdrawal
\[\begin{align}\\&= ₹ (2000–1642)\\&= ₹\,358\end{align}\]
Hence,the balance in Mohan’s account after withdrawal is \(₹\,358.\)
Chapter 1 Ex.1.1 Question 6
Rita goes \(20\rm\, km\) towards east from a point \(A\) to the point \(B.\) From \(B\) she moves \(30\rm\, km\) towards west along the same road. If the distance towards east is represented by a positive integer,then how will you represent the distance travelled towards west? By which integer will you represent her final position from \(A?\)
Solution
What is known?
Distance travelled by Rita towards east from point \(A\) and towards west from point \(B\).
What is the unknown?
Final position of Rita from \(A\)
Reasoning:
Distance travelled towards east from point \(A\) (or the highest positive value of the integer) and Distance travelled towards west from point \(B\) (or the lowest negative value of the integer). Find the difference between the two values.
Steps:
Distance travelled by Rita towards east \(= 20\rm\,km\)
(distance travelled towards east is represented by positive integer)
Distance travelled by Rita towards west \(=\, – \,30\rm\,km\)
(distance travelled towards west is represented by negative integer)
Final position of Rita from A
\[\begin{align} &=[20 +(-30)]\rm{\,km}\\&= (20-30)\rm{km}\\&= -10\rm{\,km}\end{align}\]
Therefore, we will represent her final position from \(A\) by negative integer i.e. \(–10\rm\,km.\)
Chapter 1 Ex.1.1 Question 7
In a magic square, each row, column and the diagonal has the same sum. Check which of the following is a magic square?
\(5\) |
\(–1\) |
\(–4\) |
\(–5\) |
\(–2\) |
\(–7\) |
\( 0\) |
\(3\) |
\(–3\) |
1 |
\(–10\) |
\(0\) |
\(–4\) | \(–3\) | \(–2\) |
\(–6\) |
\(4\) |
\(–7\) |
Solution
What is known?
Value of each row, column and the diagonal.
What is unknown?
To check which row, columnand the diagonal have same sum.
Reasoning:
Use the table and find the sum of the row, column and the diagonal.
Steps:
In square (i)
(a) Taking rows
\[\begin{align} {R_1} &= 5 + \left( {-1} \right) + \left( {-4} \right) = 0\\{R_2}& = -5 + \left( {-2} \right) + 7 = 0\\{R_3} &= \,\,\,0 + 3 + \left( {-3} \right) = 0\end{align}\]
(b) Taking columns
\[\begin{align}{C_1} &= {\rm{5 + }}\left( {{\rm{-5}}} \right){\rm{ + 0}}\,{\rm{ = }}\,{\rm{0}}\\ {C_2} &=\rm{-1 + }\left( \rm{-2} \right){\rm{ + }}\left( {{\rm{-3}}} \right){\rm{ = }}\,{\rm{0}}\\{C_3} &= \rm{-}\,{\rm{4 + 7 + }}\left( {{\rm{-3}}} \right){\rm{ = }}\,\,{\rm{0}}\end{align}\]
(c) Taking diagonals
\[\begin{align}{d_1} &= 5 + \left( {-2} \right) + \left( {-3} \right) = 0\\{d_2} &= -\,4 + \left( {-2} \right) + 0 = -\,6\end{align}\]
This square is not a magic square because the sum of one of its diagonal is not equal to the sum of its other diagonal,
In square (ii)
(a) Taking rows
\[\begin{align}{R_1} &= 1 + \left( {-10} \right) + 0 = -9\\{R_2} &= -4 + \left( {-3} \right) + \left( {-2} \right) = -9\\{R_3} &= \,-6 + 4 + \left( {-7} \right) = -9\end{align}\]
(b) Taking columns
\[\begin{align}{C_1} &= 1 + \left( {-4} \right) + \left( {-6} \right) = -9\\{C_2} &= -10 + \left( {-3} \right) + 4 = -9\\{C_3} &= 0 + \left( {-2} \right) + \left( {-7} \right) = -9\end{align}\]
(c) Taking diagonals
\[\begin{align}{d_1} &= 1 + \left( {-3} \right) + \left( {-7} \right) = -9\\{d_2} &= 0 + \left( {-3} \right) + \left( {-6} \right) = -9\end{align}\]
This square box is a magic square because the sum of its rows, columns and diagoanls are equal.
Hence, (ii) is a magic square.
Chapter 1 Ex.1.1 Question 8
Verify \(a– (–b) = a + b\) for the following values of \(a\) and \(b\)
(i) \(a=21, \quad b=18\)
(ii) \(a= 118, \quad b =125\)
(iii) \(a =75, \quad b=84\)
(iv) \(a= 28, \quad b =11\)
Solution
What is known?
Different value of \(a\) and \(b\).
What is unknown?
To verify: \(a-(-b)=a+b.\)
Steps:
Let, \(a-(-b)=a+b. \) (equation 1)
i) \(a = 21, b = 18\)
put the values of \(a\) and \(b\) in equation (1).
\[\begin{align}a-( -b)&= a + b\\ & =21-( -18)\\& = 21 + 18\\&= 21 + 18\\&= 21 + 18\\ 39 &= 39\\\rm{LHS}&= \rm{RHS}\,\,\text{(Hence Verified.)}\end{align}\]
ii) \(a= 118, b = 125\)
put the value of \(a\) & \(b\) in equation (1).
\[\begin{align}a-(-b)&= a + b\\ & =118-(-125)\\&=118 +125\\&= {118+ 125}\\&=118 + 125\\ 243 &=243\\\rm{LHS}&= \rm{RHS}\,\,\text{(Hence Verified.)}\end{align}\]
iii) \(a= 75, b= 84\)
put the value of \(a\) & \(b\) in equation (1).
\[\begin{align}a-(-b)&= a + b\\ & =75-(-84) \\&=75 + 84\\&=75+ 84\\&=75 + 84\\ 159 &= 159\\\rm{LHS}&= \rm{RHS}\,\,\text{(Hence Verified.)}\end{align}\]
iv) \(a=28 ,b=11\)
put the values of \(a\) & \(b\) in equation (1).
\[\begin{align}a-(-b)&= a + b\\ & =28-(-11)\\&=28 + 11\\&= 28+ 11\\&=28 + 11\\ 39 &= 39\\\rm{LHS}&= \rm{RHS}\,\,\text{(Hence Verified.)}\end{align}\]
Chapter 1 Ex.1.1 Question 9
Use the sign of \(<, > \) or \(=\) to make the statement true?
\(\begin{align} {\rm a.}\, –8+(–4) \;&\boxed{\;\;}\; –8–(–4)\quad \\-8-4\quad&\boxed{\;\;} \quad -8+4\quad\\–12 \quad&\boxed{\lt} \quad –4 \end{align}\)
\[\begin{align}&A\left( {3,2} \right), \qquad\;\;\;\;\;\;\;\;\;\;\;\;B\left( { - 2,3} \right), \qquad\;\;\;\;\;\; C\left( {0,4} \right)\\&D\left( { - \frac{1}{2}, - \frac{3}{2}} \right), \qquad E\left( {\sqrt 2 , - \sqrt 2 } \right), \qquad F\left( {\pi ,0} \right) & \end{align}\]
a.\( (-8) + (–4)\) |
\(\quad\boxed{\;\;}\) |
\( (–8) – (–4)\) |
b. \((–3)+7–(19)\) |
\(\quad\boxed{\;\;}\) |
\(15–8+(–9)\) |
c. \(23–41+11\) |
\(\quad\boxed{\;\;}\) |
\(23–41–11\) |
d. \(39 + (–24) –(15)\) |
\(\quad\boxed{\;\;}\) |
\( 36+ (– 52) – (–36)\) |
e. \(–231+79+51\) |
\(\quad\boxed{\;\;}\) |
\( –399+159+81\) |
Solution
What is known?
The statement.
What is unknown?
To find out which is greater than, smaller than or equal to.
Reasoning:
Add or Subtract the two values and after check the statement is <,>,or =
Steps:
\(\begin{align} {\rm a.}\, –8+(–4) \;&\boxed{\;\;}\; –8–(–4)\quad \\-8-4\quad&\boxed{\;\;} \quad -8+4\quad\\–12 \quad&\boxed{\lt} \quad –4 \end{align}\)
\(\begin{align}{\rm b.}\; –3+7–19\;&\boxed{\;\;}\; 15 –8 + (–9)\\–3+7–19\quad&\boxed{\;\;} \quad 15 –8 –9 \\4–19 \quad&\boxed{\;\;} \quad 7–9\\ –15 \quad&\boxed{\lt} \quad –2\\ \end{align}\)
\(\begin{align}{\rm c.}\;23–41+11 \;&\boxed{\;\;}\; 23 –41 –11 \\ –18+11\quad&\boxed{\;\;} \quad –18 –11 \\ -7\quad&\boxed{\gt} \quad -29 \\ \end{align}\)
\(\begin{align}{\rm d.}\; 39+(–24)–15 \;&\boxed{\;\;}\; 36 + (–52) – (–36) \\39–24–15\quad&\boxed{\;\;} \quad 36–52+36 \\ 15-15\quad&\boxed{\;\;} \quad -16 + 36\\0\quad&\boxed{\lt} \quad 20 \\ \end{align}\)
\(\begin{align} {\rm e.}\; –23 + 79 + 51 \;&\boxed{\;\;}\; –399 + 159 + 81\quad \\ –152 + 51\quad&\boxed{\;\;} \quad –240+81\quad\\–101 \quad&\boxed{\gt} \quad –159 \end{align}\)
Chapter 1 Ex.1.1 Question 10
A water tank has steps inside it. A monkey is sitting on the topmost step \(C\) (i.e; the first step).The water level is on the ninth step.
i) He jumps \(3\) steps down and then jumps back \(2\) up. In how many jumps will he reach the water level?
ii) After drinking water he wants to go back. For this he jumps \(4\) steps up and then jumps back \(2\) steps down in every move. In how many jumps will he reach back the top step?
iii) If the number of steps moved down is represented by negative integers and the number of steps moved up by positive integers, represent his move in part(i) & (ii) by completing the following
\({\rm{a)\,\,-3 + 2 -}}...{\rm{ = -8}}\)
\({\rm{b)\,\,4-2 + \ldots }}{\rm{. = 8}}{\rm{.}}\)
In (a) the sum \((-8)\) represents going down by \(8\) steps. So, what will the sum \(8\) in (b) represent?
Solution
What is the known?
Number of steps going up and down.
What is unknown?
In how many jumps will he reach the water level and In how many jumps will he reach back the top step?
Steps:
i) Monkey jumps \(3\) steps down and then back \(2\) steps up. The jumps of monkey can be represented as:–
First jump \(=1+3 = 4\) steps
Second jump \(=4–2 = 2\) steps
Third jump \(=2+3 = 5\) steps
Fourth jump \(=5–2 = 3\) steps
Fifth jump \(=3+3 =6\) steps
Sixth jump \(=6–2 =4\) steps
Seventh jump \(=4+3 =7\) steps
Eighth jump \(=7–2 =5\) steps
Ninth jump \(=5+3 =8\) steps
Tenth jump \(=8–2 =6\) steps
Eleventh jump \(=6+3 =9\) steps
Monkey will reach ninth step in \(11\) jumps.
ii) Monkey jumps \(4\) steps up and then jumps back \(2\) steps down in every ove. The jumps of monkey can be represented as follows;–
First jump \(=9–4 =5\) steps
Second jump \(=5+2 =7\) steps
Third jump \(=7–4 =3\) steps
Fourth jump \(=3+2 =5\) steps
Fifth jump \(=5–4 =1\) step
The monkey will reach back the top step after \(5\) jumps.
iii) Moves in part (i)
\begin{align*} \left[\begin{array}{l} {-3+2-3+2-3} \\ {+2-3+2-3+2=-8} \end{array}\right] \end{align*}
which represents the monkey goes down by \(\rm{}8\) steps
Moves in part (ii)
\(\begin{align}4–2+4–2+4=8,\end{align}\) the sum \(8\) in (b) represents the monkey goes up by \(8\) steps.