GSEB Class 7 Maths Solutions Chapter 1 Integers InText Questions

Try These (Page 2)

 

Question 1. A number line representing integers is given below.
- 3 and – 2 are marked by E and F respectively. Which integers are marked by B, D, H, J, M and O?
Answer: Let us complete the given number line so that integers marked by different alphabets are shown. Therefore, the number line becomes:

A B C D E F G H J K L M N O
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

The integer marked by B is \( -6 \).
The integer marked by D is \( -4 \).
The integer marked by H is \( 0 \).
The integer marked by J is \( 2 \).
The integer marked by M is \( 5 \).
The integer marked by O is \( 7 \).
In simple words: First, fill in all the numbers on the number line. Then, just read the numbers that are under each letter given in the question.

Exam Tip: For number line questions, draw the complete line clearly and precisely identify each point before listing the answers.

 

Question 2. Arrange 7, – 5, 4, 0 and -4 in ascending order and then mark them on a number line to check your answer.
Answer: Since every positive integer is greater than \( 0 \), and every negative integer is less than \( 0 \).
Thus, \( -5 < (-4) < 0 < 4 < 7 \)
\( \implies \) The required ascending order is: \( -5, -4, 0, 4, 7 \).
Because the integer occurring to the right on a number line is greater than the one on the left, and the integer on the left is smaller than the one on the right. Now, representing \( 7, -5, 4, 0 \) and \( -4 \) on a number line, we get:

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
  ^  ^        ^       ^     ^
  -5 -4       0       4     7

In simple words: Arrange the numbers from smallest to largest. Then, put them on a number line. Make sure the numbers go from left to right as they get bigger.

Exam Tip: Remember that negative numbers are smaller when they are further from zero, and positive numbers are larger when further from zero. Zero is always between them.

 

Question 3. State whether the following statements are correct or incorrect. Correct those which are wrong:
(i) When two positive integers are added we get a positive integer.
(ii) When two negative integers are added we get a positive integer.
(iii) When a positive integer and a negative integer are added, we always get a negative integer.
(iv) Additive inverse of an integer 8 is (-8) and additive inverse of (-8) is 8.
(v) For subtraction, we add the additive inverse of the integer that is being subtracted, to
(vi) (-10) + 3 = 10-3
(vii) 8 + (-7) – (-4) = 8 + 7-4
Construct five examples in support of your answer.
Answer:
(i) Correct statement:
Examples:
(a) \( 55 + 52 = 107 \)
(b) \( 45 + 71 = 116 \)
(c) \( 110 + 71 = 181 \)
(d) \( 145 + 171 = 316 \)
(e) \( 210 + 103 = 313 \). The results \( 107, 116, 181, 316 \) and \( 313 \) are all positive integers.
(ii) Incorrect statement – The correct statement is: When two negative integers are added, we always get a negative integer.
Examples:
(a) \( (-72) + (-57) = -129 \)
(b) \( (-114) + (-25) = -139 \)
(c) \( (-110) + (-81) = -191 \)
(d) \( (-39) + (-31) = -70 \)
(e) \( (-201) + (-211) = -412 \). The results \( -129, -139, -191, -70 \) and \( -412 \) are all negative integers.
(iii) Incorrect statement – The correct statement is: When a positive integer and a negative integer are added, we find the difference between their numerical values and assign the sign of the integer with the larger numerical value.
Examples:
(a) \( (-52) + 75 = 23 \)
(b) \( 121 + (-89) = 32 \)
(c) \( (-225) + 105 = -120 \)
(d) \( 712 + (-908) = -196 \)
(e) \( 369 + (-693) = -324 \)
(iv) Correct statement:
Examples:
(a) The additive inverse of \( -38 \) is \( 38 \).
(b) The additive inverse of \( 29 \) is \( -29 \).
(c) The additive inverse of \( -63 \) is \( 63 \).
(d) The additive inverse of \( 99 \) is \( -99 \).
(e) The additive inverse of \( -45 \) is \( 45 \).
(v) Correct statement:
Examples:
(a) \( 58 - 76 \) means \( 58 + (-76) = -18 \)
(b) \( 58 - (-176) \) means \( 58 + (+176) = 234 \)
(c) \( (-81) - (45) \) means \( (-81) + (-45) = -126 \)
(d) \( -81 - (-45) \) means \( (-81) + (+45) = -36 \)
(e) \( (-110) - (-152) \) means \( (-110) + (+152) = 42 \)
(vi) Incorrect statement:
Since, R.H.S. \( = 10 - 3 = 7 \)
And L.H.S. \( = (-10) + 3 = -10 + 3 = -7 \)
i.e., L.H.S. \( \neq \) R.H.S.
\( \implies \) The correct statement is: \( (-10) + 3 = -7 \).
(vii) Incorrect statement.
Since, L.H.S. \( = 8 + (-7) - (-4) = 8 - 7 + 4 = 1 + 4 = 5 \).
And R.H.S. \( = 8 + 7 - 4 = 15 - 4 = 11 \).
i.e., L.H.S. \( \neq \) R.H.S.
\( \implies \) The correct statement is: \( 8 + (-7) - (-4) = 5 \).
In simple words: Read each statement carefully. Decide if it is true or false. If it's false, write down the right statement and show examples to prove it. Remember rules for adding and subtracting positive and negative numbers.

Exam Tip: Always provide clear examples to support your corrections or statements, especially for rules involving positive and negative integers.

 

Try These (Page 3)

 

Question 1. We have done various patterns with numbers in our previous class. Can you find a pattern for each of the following? If yes, complete them:
(a) 7, 3, -1, -5, ______, ______, ______.
(b) -2, -4, -6, -8, ______, ______, ______.
(c) 15, 10, 5, 0, ______, ______, ______.
(d) -11, -8, -5, -2, ______, ______, ______.
Make some more such patterns and ask your friends to complete them.
Answer:
(a) Since \( 7 - 4 = 3 \), \( 3 - 4 = -1 \), \( -1 - 4 = -5 \).
So, \( -5 - 4 = -9 \), \( -9 - 4 = -13 \), \( -13 - 4 = -17 \).
Thus, we have: \( 7, 3, -1, -5, -9, -13, -17 \).
(b) Since, \( -2 + (-2) = -4 \), \( -4 + (-2) = -6 \), \( -6 + (-2) = -8 \).
So, \( -8 + (-2) = -10 \), \( -10 + (-2) = -12 \), \( -12 + (-2) = -14 \).
Thus, we have: \( -2, -4, -6, -8, -10, -12, -14 \).
(c) Since, \( 15 + (-5) = 10 \), \( 10 + (-5) = 5 \), \( 5 + (-5) = 0 \).
So, \( 0 + (-5) = -5 \), \( -5 + (-5) = -10 \), \( -10 + (-5) = -15 \).
Thus, we have: \( 15, 10, 5, 0, -5, -10, -15 \).
(d) Since \( -11 + 3 = -8 \), \( -8 + 3 = -5 \), \( -5 + 3 = -2 \).
So, \( -2 + 3 = 1 \), \( 1 + 3 = 4 \), \( 4 + 3 = 7 \).
Thus, we have: \( -11, -8, -5, -2, 1, 4, 7 \).
In simple words: Look closely at the numbers and figure out what is being added or subtracted each time. Then, keep doing that same step to fill in the missing numbers in the pattern.

Exam Tip: To identify a pattern, always find the difference or ratio between consecutive terms. This helps you predict the next numbers accurately.

 

Try These (Page 8)

 

Question 1. Write a pair of integers whose sum gives
(a) a negative integer.
(b) zero.
(c) an integer smaller than both the integers.
(d) an integer smaller than only one of the integers.
(e) an integer greater than both the integers.
Answer:
(a) A negative integer:
Let's take \( -15 \) and \( 9 \).
Sum: \( (-15) + 9 = -6 \).
[\( -6 \) is a negative integer].
(b) Zero:
Let's take \( -18 \) and \( 18 \).
Sum: \( (-18) + 18 = 0 \).
(c) An integer smaller than both integers:
Let's take \( -6 \) and \( -4 \).
Sum: \( (-6) + (-4) = -10 \).
[\( -10 \) is smaller than \( -6 \) and \( -4 \)].
(d) An integer smaller than only one of the integers:
Let's take \( 4 \) and \( -6 \).
Sum: \( 4 + (-6) = -2 \).
[\( -2 \) is smaller than \( 4 \) only].
(e) An integer greater than both integers:
Let's take \( 19 \) and \( 21 \).
Sum: \( 19 + 21 = 40 \).
[\( 40 \) is greater than \( 19 \) and \( 21 \)].
In simple words: For each part, think of two numbers that, when added together, give the type of answer asked for. Use positive and negative numbers as needed to meet the conditions.

Exam Tip: When dealing with sums of integers, remember that adding two negative integers results in a more negative integer, while adding a positive and negative integer can result in either positive, negative, or zero depending on their absolute values.

 

Question 2. Write a pair of integers whose difference gives
(a) a negative integer.
(b) zero.
(c) an integer smaller than both the integers.
(d) an integer greater than only one of the integers.
(e) an integer greater than both the integers.
Answer:
(a) A negative integer:
Let's take \( 3 \) and \( 8 \).
Difference: \( 3 - 8 = -5 \) or \( 8 - 3 = 5 \). If we take \( -8 \) and \( 3 \), then \( -8 - 3 = -11 \).
[\( -11 \) is a negative integer]. Example: \( 3 - 8 = -5 \).
(b) Zero:
Let's take \( -3 \) and \( -3 \).
Difference: \( (-3) - (-3) = (-3) + 3 = 0 \).
(c) An integer smaller than both integers:
Let's take \( 5 \) and \( 9 \).
Difference: \( 5 - 9 = -4 \).
[\( -4 \) is smaller than \( 5 \) and \( 9 \)].
(d) An integer greater than only one of the integers:
Let's take \( 16 \) and \( 5 \).
Difference: \( 16 - 5 = 11 \).
[\( 11 \) is greater than \( 5 \) only].
(e) An integer greater than both integers:
Let's take \( 15 \) and \( -6 \).
Difference: \( 15 - (-6) = 15 + 6 = 21 \).
[\( 21 \) is greater than \( 15 \) as well as \( -6 \)].
In simple words: For each part, find two numbers whose subtraction gives the result specified. Remember that subtracting a negative number is like adding a positive number.

Exam Tip: Be careful with the order when calculating differences. \( a - b \) is generally not the same as \( b - a \), and subtracting a negative number changes the sign to addition.

 

Try These (Page 10)

 

Question 1. Using number line, find:
(i) 4 x (-8)
(ii) 8 x (-2)
(iii) 3 x (-7)
(iv) 10 x (-1)
Answer:
(i) \( 4 \times (-8) \): Starting from 0, make 4 jumps of \( -8 \) units each to the left.

-40 -32 -24 -16 -8 0 8
  ^---^---^---^
  4 jumps of -8

From the number line, we have:
\( (-8) + (-8) + (-8) + (-8) = -32 \).
\( \implies 4 \times (-8) = -32 \).
(ii) \( 8 \times (-2) \): Starting from 0, make 8 jumps of \( -2 \) units each to the left.

-16 -14 -12 -10 -8 -6 -4 -2 0 1
^--^--^--^--^--^--^--^
8 jumps of -2

From the number line, we have:
\( (-2) + (-2) + (-2) + (-2) + (-2) + (-2) + (-2) + (-2) = -16 \).
\( \implies 8 \times (-2) = -16 \).
(iii) \( 3 \times (-7) \): Starting from 0, make 3 jumps of \( -7 \) units each to the left.

-28 -21 -14 -7 0
    ^-----^-----^
    3 jumps of -7

From the number line, we have:
\( (-7) + (-7) + (-7) = -21 \).
\( \implies 3 \times (-7) = -21 \).
(iv) \( 10 \times (-1) \): Starting from 0, make 10 jumps of \( -1 \) unit each to the left.

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
^--^--^--^--^--^--^--^--^--^
10 jumps of -1

From the number line, we have:
\( (-1) + (-1) + (-1) + (-1) + (-1) + (-1) + (-1) + (-1) + (-1) + (-1) = -10 \).
\( \implies 10 \times (-1) = -10 \).
In simple words: To multiply on a number line, start at zero. For \( A \times B \), make \( A \) jumps, and each jump should be \( B \) units long. If \( B \) is negative, jump to the left. If \( B \) is positive, jump to the right. The spot you land on is the answer.

Exam Tip: When using a number line for multiplication, ensure your jumps are consistent in size and direction. The starting point is always zero.

 

Try These (Page 10)

 

Question 1. Find:
(i) 6 x (-19)
(ii) 12 x (-32)
(iii) 7 x (-22)
Answer:
(i) \( 6 \times (-19) = -[6 \times 19] = -[114] = -114 \).
(ii) \( 12 \times (-32) = -[12 \times 32] = -[384] = -384 \).
(iii) \( 7 \times (-22) = -[7 \times 22] = -[154] = -154 \).
In simple words: When you multiply a positive number by a negative number, first multiply them like regular numbers. Then, always put a minus sign in front of your answer.

Exam Tip: Remember the rule: Positive times Negative always equals Negative. Multiply the absolute values and then apply the negative sign.

 

Try These (Page 11)

 

Question 1. Find:
(a) 15 x (-16)
(b) 21 x (-32)
(c) (-42) x 12
(d) -55 x 15
Answer: While multiplying a positive integer and a negative integer, we multiply them as whole numbers and put a minus sign \( (-) \) before the product, i.e., the product of a positive integer and a negative integer is a negative integer. Therefore,
(a) \( 15 \times (-16) = -[15 \times 16] = -[240] = -240 \).
(b) \( 21 \times (-32) = -[21 \times 32] = -[672] = -672 \).
(c) \( (-42) \times 12 = -[42 \times 12] = -[504] = -504 \).
(d) \( (-55) \times 15 = -[55 \times 15] = -[825] = -825 \).
In simple words: When one number is positive and the other is negative, you multiply them normally. Then, your final answer will always have a minus sign.

Exam Tip: Be consistent with the sign rule: if there is one negative number in the multiplication, the result is negative. If there are two negative numbers, the result is positive.

 

Question 2. Check if
(a) 25 x (-21) = (-25) x 21.
(b) (-23) x 20 = 23 x (-20).
Write five more such examples.
Answer:
(a) For \( 25 \times (-21) = (-25) \times 21 \):
L.H.S. \( = 25 \times (-21) = -[25 \times 21] = -[525] = -525 \).
R.H.S. \( = (-25) \times 21 = -[25 \times 21] = -[525] = -525 \).
Thus, L.H.S. \( = \) R.H.S.
\( \implies 25 \times (-21) = (-25) \times 21 \). (This statement is correct).
(b) For \( (-23) \times 20 = 23 \times (-20) \):
L.H.S. \( = (-23) \times 20 = -[23 \times 20] = -[460] = -460 \).
R.H.S. \( = 23 \times (-20) = -[23 \times 20] = -[460] = -460 \).
Thus, L.H.S. \( = \) R.H.S.
\( \implies (-23) \times 20 = 23 \times (-20) \). (This statement is also correct).
Other examples:
(i) \( (-12) \times 19 = 12 \times (-19) \)
(ii) \( 15 \times (-17) = (-15) \times 17 \)
(iii) \( 51 \times (-40) = (-51) \times 40 \)
(iv) \( (-20) \times 25 = 20 \times (-25) \)
(v) \( 16 \times (-15) = (-16) \times 15 \)
In simple words: Check if swapping the negative sign from one number to the other gives the same answer when multiplying. Then, write five more pairs that show the same thing.

Exam Tip: This property shows that \( a \times (-b) = (-a) \times b = -(a \times b) \). It's an important concept for understanding integer multiplication.

 

Try These (Page 12)

 

Question 1. (i) Starting from (-5) x 4, find (-5) x (-6).
(ii) Starting from (-6) x 3, find (-6) x (-7).
Answer:
(i) Look at the following pattern:
\( (-5) \times 4 = -[5 \times 4] = -20 \)
\( (-5) \times 3 = -[5 \times 3] = -15 = -20 + 5 \)
\( (-5) \times 2 = -[5 \times 2] = -10 = -15 + 5 \)
\( (-5) \times 1 = -[5 \times 1] = -5 = -10 + 5 \)
\( (-5) \times 0 = -[5 \times 0] = 0 = -5 + 5 \)
From this pattern, we have:
\( (-5) \times (-1) = 0 + 5 = 5 \)
\( (-5) \times (-2) = 5 + 5 = 10 \)
\( (-5) \times (-3) = 10 + 5 = 15 \)
\( (-5) \times (-4) = 15 + 5 = 20 \)
\( (-5) \times (-5) = 20 + 5 = 25 \)
\( (-5) \times (-6) = 25 + 5 = 30 \)
Thus, \( (-5) \times (-6) = 30 \).
(ii) Look at the following pattern:
\( -6 \times 3 = -[6 \times 3] = -18 \)
\( -6 \times 2 = -12 = -18 + 6 \)
\( -6 \times 1 = -6 = -12 + 6 \)
\( -6 \times 0 = 0 = -6 + 6 \)
From this pattern, we have:
\( -6 \times (-1) = 0 + 6 = 6 \)
\( -6 \times (-2) = 6 + 6 = 12 \)
\( -6 \times (-3) = 12 + 6 = 18 \)
\( -6 \times (-4) = 18 + 6 = 24 \)
\( -6 \times (-5) = 24 + 6 = 30 \)
\( -6 \times (-6) = 30 + 6 = 36 \)
\( -6 \times (-7) = 36 + 6 = 42 \)
Thus, \( (-6) \times (-7) = 42 \).
In simple words: Create a sequence of multiplications. Start with multiplying a negative number by a positive number, then gradually decrease the positive number to zero, and then to negative numbers. Observe how the product changes by adding the first number's opposite each time.

Exam Tip: This method helps visualize why the product of two negative integers is positive. Each step in the pattern adds the first integer to the previous result.

 

Try These (Page 12)

 

Question 1. Find: (-31) x (-100), (-25) x (-72), (-83) x (-28)
Answer: We multiply the two negative integers as whole numbers and put the positive sign \( (+) \) before the product.
\( (-31) \times (-100) = +[31 \times 100] = +[3100] = 3100 \).
\( (-25) \times (-72) = +[25 \times 72] = +[1800] = 1800 \).
\( (-83) \times (-28) = +[83 \times 28] = +[2324] = 2324 \).
In simple words: When you multiply two negative numbers together, the answer is always a positive number. Just multiply the numbers as if they were positive, and the result will be positive.

Exam Tip: A key rule in integer multiplication is that a negative multiplied by a negative always yields a positive result. Always double-check your signs.

 

Think, Discuss and Write (Page 14)

 

Question 1. (i) The product (-9) x (-5) x (-6) x (-3) is positive whereas the product (-9) x (-5) x 6 x (-3) is negative. Why?
(ii) What will be the sign of the product if we multiply together:
(a) 8 negative integers and 3 positive integers?
(b) 5 negative integers and 4 positive integers?
(c) (-1), twelve times?
(d) (-1), 2 m times, m is a natural number?
Answer: We know that, if the number of negative integers in a product is even, then the product is a positive integer; if the number of negative integers in a product is odd, then the product is a negative integer.
(i) The product \( (-9) \times (-5) \times (-6) \times (-3) \) is positive because an even number (four) of negative integers are multiplied.
The product \( (-9) \times (-5) \times 6 \times (-3) \) is negative because an odd number (three) of negative integers are multiplied.
(ii) The sign of the product will be:
(a) Positive [: Product of 8 negative integers is positive] (since 8 is an even number).
(b) Negative [: Product of 5 negative integers is negative] (since 5 is an odd number).
(c) Positive [: 12 is an even number, and the product of an even number of negative integers is positive].
(d) Positive [: \( 2m \) is an even number, so the product will be positive].
In simple words: The sign of a product depends on how many negative numbers you are multiplying. If there's an even count of negative numbers, the answer is positive. If there's an odd count, the answer is negative. Positive numbers don't change the final sign.

Exam Tip: Count only the negative factors to determine the sign of the product. The number of positive factors does not influence the final sign.

 

Try These (Page 18)

 

Question 1. (i) Is 10 x [(6 + (-2)] = 10 x 6 + 10 x (-2)?
(ii) Is (-15) x [(-7) + (-1)] = (-15) x (-7) + (-15) x (-1)?
Answer:
(i) Yes, [:: \( a \times (b + c) = a \times b + a \times c \)]
\( 10 \times [6 + (-2)] = 10 \times [6 - 2] = 10 \times 4 = 40 \).
And, \( 10 \times 6 + 10 \times (-2) = 60 - 20 = 40 \).
Thus, \( 10 \times [6 + (-2)] = 10 \times 6 + 10 \times (-2) \).
(ii) Yes, [:: \( a \times (b + c) = a \times b + a \times c \)]
\( (-15) \times [(-7) + (-1)] = (-15) \times [-7 - 1] = (-15) \times (-8) = (+) (15 \times 8) = 120 \).
And, \( (-15) \times (-7) + (-15) \times (-1) = (+) (15 \times 7) + (+) (15 \times 1) = 105 + 15 = 120 \).
Thus, \( (-15) \times [(-7) + (-1)] = (-15) \times (-7) + (-15) \times (-1) \).
In simple words: Check if the distributive property works for these integer multiplications. This means multiplying the number outside the bracket by each number inside, and then adding the results, should give the same answer as calculating inside the bracket first and then multiplying.

Exam Tip: The distributive property \( a \times (b + c) = a \times b + a \times c \) holds true for all integers, making calculations simpler by breaking them into smaller parts.

 

Try These (Page 18)

 

Question 1. (i) Is 10 x [6 – (-2)] = 10 x 6 – 10 x (-2)?
(ii) Is (-15) x [(-7) – (-1)] = (-15) x (-7) – (-15) x (-1)?
Answer:
(i) Yes, [:: \( a \times (b - c) = a \times b - a \times c \)]
\( 10 \times [6 - (-2)] = 10 \times [6 + 2] = 10 \times 8 = 80 \).
And, \( 10 \times 6 - 10 \times (-2) = 60 - (-20) = 60 + 20 = 80 \).
Thus, \( 10 \times [6 - (-2)] = 10 \times 6 - 10 \times (-2) \).
(ii) Yes, [:: \( a \times (b - c) = a \times b - a \times c \)]
\( (-15) \times [(-7) - (-1)] = (-15) \times [-7 + 1] = (-15) \times (-6) = (+) (15 \times 6) = 90 \).
And, \( (-15) \times (-7) - (-15) \times (-1) = (+) (15 \times 7) - (+) (15 \times 1) = 105 - 15 = 90 \).
Thus, \( (-15) \times [(-7) - (-1)] = (-15) \times (-7) - (-15) \times (-1) \).
In simple words: This question checks if the distributive property works for subtraction as well. You can multiply the outer number by each number inside the bracket, then subtract the results, and it should be the same as solving the bracket first, then multiplying.

Exam Tip: The distributive property applies to both addition and subtraction. Always remember that \( a - (-b) \) becomes \( a + b \).

 

Try These (Page 19)

 

Question 1. By using distributive property, find: (-49) x 18; (-25) x (-31); 70 x (-19) + (-1) x 70
Answer:
(i) \( (-49) \times 18 \):
\( 18 = 10 + 8 \)
\( (-49) \times 18 = (-49) \times [10 + 8] \)
\( = (-49) \times 10 + (-49) \times 8 \) [using distributivity]
\( = -490 + (-392) = -882 \).
(iii) \( (-25) \times (-31) \):
\( -31 = (-30) + (-1) \)
\( (-25) \times (-31) = (-25) \times [(-30) + (-1)] \)
\( = (-25) \times (-30) + (-25) \times (-1) \) [using distributivity]
\( = +(25 \times 30) + (25 \times 1) \)
\( = 750 + 25 = 775 \).
(iv) \( 70 \times (-19) + (-1) \times 70 \):
[:: \( a \times b + a \times c = a \times (b + c) \)]
\( 70 \times (-19) + (-1) \times 70 = 70 \times [(-19) + (-1)] \)
\( = 70 \times [-20] = -[70 \times 20] = -1400 \).
In simple words: To use the distributive property, break one of the numbers into parts (like \( 18 = 10+8 \)) or factor out a common number (like \( 70 \)). Then, multiply or add the parts as shown to get the final answer.

Exam Tip: The distributive property helps simplify complex multiplications by breaking them into easier steps. Remember that \( a \times (b+c) = ab + ac \) and \( ab + ac = a \times (b+c) \).

 

Try These (Page 22)

 

Question 1. Find:
(a) 100 ÷ 5
(b) (-81) ÷ 9
(c) (-75) ÷ 5
(d) (-32) ÷ 2
Answer: We know that to divide a negative integer by a positive integer, we divide them as whole numbers and then put a minus sign \( (-) \) before the quotient.
(a) \( 100 \div 5 = 20 \).
(b) \( (-81) \div 9 = -9 \).
(c) \( (-75) \div 5 = -15 \).
(d) \( (-32) \div 2 = -16 \).
In simple words: When a negative number is divided by a positive number, the answer is always negative. Just divide the numbers normally and put a minus sign in front of the result. If both are positive, the answer is positive.

Exam Tip: The rule for division signs is similar to multiplication: if the signs are different, the result is negative. If the signs are the same, the result is positive.

 

Try These (Page 23)

 

Question 1. Find:
(a) 125 ÷ (-25)
(b) 80 ÷ (-5)
(c) 64 ÷ (-16)
Answer: We know that to divide a positive integer by a negative integer, we first divide them as whole numbers and then put a minus sign \( (-) \) before the quotient.
(a) \( 125 \div (-25) = -5 \).
(b) \( 80 \div (-5) = -16 \).
(c) \( 64 \div (-16) = -4 \).
In simple words: To divide a positive number by a negative number, perform the division as you would with two positive numbers. Then, place a negative sign in front of the final answer.

Exam Tip: Any time you divide numbers with different signs (one positive, one negative), the outcome will always be negative.

 

Try These (Page 23)

 

Question 1. Find:
(a) (-36) ÷ (-4)
(b) (-201) ÷ (-3)
(c) (-325) ÷ (-13)
Answer: To divide a negative integer by a negative integer, we first divide them as whole numbers and put a positive sign \( (+) \) before the quotient.
(a) \( (-36) \div (-4) = 9 \).
(b) \( (-201) \div (-3) = 67 \).
(c) \( (-325) \div (-13) = 25 \).
In simple words: When you divide a negative number by another negative number, the answer will always be positive. Just divide the numbers as if they were positive.

Exam Tip: A negative divided by a negative always results in a positive. This is a fundamental rule for integer division.

 

Try These (Page 24)

 

Question 1. Is (i) 1 ÷ a = 1 and (ii) a ÷ (-1) = –a for any integer a? Take different values of a and check.
Answer:
(i) Let us take \( a = -1, 1, 2, 3, ... \).
For \( a = -1 \):
L.H.S. \( = 1 \div (-1) = -1 \).
R.H.S. \( = 1 \).
i.e., L.H.S. \( \neq \) R.H.S. (So, \( 1 \div a = 1 \) is false for \( a = -1 \)).
For \( a = 1 \):
L.H.S. \( = 1 \div 1 = 1 \).
R.H.S. \( = 1 \).
i.e., L.H.S. \( = \) R.H.S. (So, \( 1 \div a = 1 \) is true for \( a = 1 \)).
For \( a = 2 \):
L.H.S. \( = 1 \div 2 = \frac{1}{2} \).
R.H.S. \( = 1 \).
i.e., L.H.S. \( \neq \) R.H.S. (So, \( 1 \div a = 1 \) is false for \( a = 2 \)).
For \( a = 3 \):
L.H.S. \( = 1 \div 3 = \frac{1}{3} \).
R.H.S. \( = 1 \).
i.e., L.H.S. \( \neq \) R.H.S. (So, \( 1 \div a = 1 \) is false for \( a = 3 \)).
Thus, \( 1 \div a = 1 \) is true only for \( a = 1 \). It is not true for *any* integer \( a \).
(ii) Let us take \( a = 1, 2, 3, ... \).
For \( a = 1 \):
L.H.S. \( = a \div (-1) = 1 \div (-1) = -1 \).
R.H.S. \( = -a = -1 \).
i.e., L.H.S. \( = \) R.H.S.
For \( a = 2 \):
L.H.S. \( = a \div (-1) = 2 \div (-1) = -2 \).
R.H.S. \( = -a = -2 \).
i.e., L.H.S. \( = \) R.H.S.
For \( a = 3 \):
L.H.S. \( = a \div (-1) = 3 \div (-1) = -3 \).
R.H.S. \( = -a = -3 \).
i.e., L.H.S. \( = \) R.H.S.
For \( a = 7 \):
L.H.S. \( = a \div (-1) = 7 \div (-1) = -7 \).
R.H.S. \( = -a = -7 \).
i.e., L.H.S. \( = \) R.H.S.
Thus, for every integer, we have \( a \div (-1) = -a \). (This statement is correct for any integer \( a \)).
In simple words: For the first part, test different numbers to see if \( 1 \) divided by that number is always \( 1 \). You'll find it's only true for \( 1 \). For the second part, test if any number divided by \( -1 \) always equals the negative of that number. This rule holds true for all integers.

Exam Tip: Be cautious with universal statements like "for any integer." Always test with positive, negative, and zero values to confirm their validity.

Try These (Page 24)

 

Question 1. Is (i) \( 1 \div a = 1 \) and (ii) \( a \div (- 1) = -a \) for any integer \( a \)? Take different values of \( a \) and check.
Answer:
(i) Let's consider different integer values for \( a \).
For \( a = -1 \):
L.H.S. \( = 1 \div (-1) = -1 \)
R.H.S. \( = 1 \)
Since L.H.S. \( \neq \) R.H.S.
For \( a = 1 \):
L.H.S. \( = 1 \div 1 = 1 \)
R.H.S. \( = 1 \)
Since L.H.S. \( = \) R.H.S.
For \( a = 2 \):
L.H.S. \( = 1 \div 2 = \frac{1}{2} \)
R.H.S. \( = 1 \)
Since L.H.S. \( \neq \) R.H.S.
For \( a = 3 \):
L.H.S. \( = 1 \div 3 = \frac{1}{3} \)
R.H.S. \( = 1 \)
Since L.H.S. \( \neq \) R.H.S.
Therefore, \( 1 \div a = 1 \) is only true for \( a = 1 \).

(ii) Let's choose various integer values for \( a \).
For \( a = 1 \):
L.H.S. \( = a \div (-1) = 1 \div (-1) = -1 \)
R.H.S. \( = -a = -1 \)
Since L.H.S. \( = \) R.H.S.
For \( a = 2 \):
L.H.S. \( = a \div (-1) = 2 \div (-1) = -2 \)
R.H.S. \( = -a = -2 \)
Since L.H.S. \( = \) R.H.S.
For \( a = 3 \):
L.H.S. \( = a \div (-1) = 3 \div (-1) = -3 \)
R.H.S. \( = -a = -3 \)
Since L.H.S. \( = \) R.H.S.
For \( a = 7 \):
L.H.S. \( = a \div (-1) = 7 \div (-1) = -7 \)
R.H.S. \( = -a = -7 \)
Since L.H.S. \( = \) R.H.S.
Thus, for every integer, we find that \( a \div (-1) = -a \).
In simple words: We tested several integers for \( a \). For the first part, \( 1 \div a = 1 \) only works when \( a \) is exactly 1. For the second part, \( a \div (-1) = -a \) holds true for all integers we checked.

Exam Tip: Always test with both positive and negative integers, including 0 and 1, to thoroughly check mathematical properties for any integer \( a \).