GSEB Class 6 Maths Solutions Chapter 6 Integers Exercise 6.2

Gujarat Board Textbook Solutions Class 6 Maths
Chapter 6 Integers Ex 6.2

Question 1. Using the number line write the integer which is:
(a) 3 more than 5
(b) 5 more than – 5
(c) 6 less than 2
(d) 3 less than – 2
Answer:
(a) 3 more than 5:

To get the integer that is 3 more than 5, we begin at 5 and move 3 steps to the right. This leads us to 8, as shown in the diagram. So, 3 more than 5 is 8.
In simple words: To find 3 more than 5 on the number line, start at 5 and jump 3 places to the right. You will land on 8.

Exam Tip: When finding "more than" a number on a number line, always move to the right. The number of units to move is the second number given.

(b) 5 more than – 5:

To find 5 more than \( -5 \), we begin at \( -5 \) and move 5 units to its right. We arrive at 0, as depicted on the number line. Thus, 5 more than \( -5 \) is 0.
In simple words: Start at \( -5 \) on the number line. Move 5 steps to the right. You will end up at 0.

Exam Tip: Remember that moving right on a number line increases the value, while moving left decreases it. Always mark your starting and ending points clearly.

(c) 6 less than 2:

To find the integer 6 less than 2, we begin at 2 and move 6 steps, each of 1 unit, to the left. We then arrive at \( -4 \), as illustrated in the figure. So, 6 less than 2 is \( -4 \).
In simple words: Start at 2 on the number line. Move 6 steps to the left. You will land on \( -4 \).

Exam Tip: When finding "less than" a number, always move to the left on the number line. The number of steps you take is crucial for accuracy.

(d) 3 less than – 2:

To get 3 less than \( -2 \), we begin at \( -2 \) and move 3 units to the left. We then arrive at \( -5 \), as shown in the diagram. Therefore, 3 less than \( -2 \) is \( -5 \).
In simple words: Find \( -2 \) on the number line. Move 3 places to the left. You will reach \( -5 \).

Exam Tip: Pay close attention to negative starting numbers when subtracting; moving left continues into more negative values.

Question 2. Use number line and add the following integers:
(a) \( 9 + (-6) \)
(b) \( 5 + (-11) \)
(c) \( (- 1) + (-7) \)
(d) \( (-5) + 10 \)
(e) \( (- 1) + (- 2) + (- 3) \)
(f) \( (- 2) + 8 + (-4) \)
Answer:
(a) \( 9 + (-6) \)

On the number line, we first move 9 units to the right from 0 and arrive at 9. After that, we move 6 steps to the left from 9 and reach 3, as shown in the diagram. So, \( 9 + (-6) = +3 \).
In simple words: Start at 0, jump right 9 steps to reach 9. Then, from 9, jump left 6 steps. You will end up at 3.

Exam Tip: Adding a negative number means moving to the left on the number line, just like subtraction. Always visualize the movement from the current position.

(b) \( 5 + (-11) \)

On the number line, we first move 5 units to the right from 0 to arrive at 5. After that, we move 11 units to the left from 5 and reach at \( -6 \), as shown in the diagram. So, \( 5 + (-11) = -6 \).
In simple words: Start at 0, move right 5 steps to reach 5. Then, from 5, move left 11 steps. You will end up at \( -6 \).

Exam Tip: When the negative number is larger than the positive number in an addition, the result will always be negative. This is clearly seen when moving left more steps than you moved right.

(c) \( (- 1) + (-7) \)

First, we move 1 unit to the left from 0 to reach \( -1 \). Then, we move 7 units further left from \( -1 \) and arrive at \( -8 \), as shown in the illustration. Thus, \( (-1) + (-7) = -8 \).
In simple words: Start at 0, move left 1 step to \( -1 \). From \( -1 \), move left another 7 steps. You will reach \( -8 \).

Exam Tip: When adding two negative integers, the movement on the number line is always to the left from the starting point, resulting in a more negative sum.

(d) \( (-5) + 10 \)

First, we move 5 units to the left from 0 to arrive at \( -5 \). Then, we move 10 units to the right from \( -5 \) and reach \( +5 \), as depicted in the figure. So, \( (-5) + 10 = +5 \).
In simple words: Start at 0, move left 5 steps to \( -5 \). From \( -5 \), move right 10 steps. You will end up at \( +5 \).

Exam Tip: When adding a positive and a negative number, the direction of movement depends on which number has a larger absolute value. The final position determines the sign of the answer.

(e) \( (- 1) + (- 2) + (- 3) \)

First, we move 1 unit to the left from 0 and arrive at \( -1 \). Then, we move 2 units further left from \( -1 \) and arrive at \( -3 \). Following this, we move 3 more units to the left from \( -3 \) and reach \( -6 \), as illustrated in the figure. So, \( (-1) + (-2) + (-3) = -6 \).
In simple words: Start at 0. Go left 1 step to \( -1 \). Then go left 2 more steps to \( -3 \). Finally, go left 3 more steps. You will end up at \( -6 \).

Exam Tip: When adding multiple negative integers, each addition means moving further to the left on the number line, accumulating a larger negative value.

(f) \( (- 2) + 8 + (-4) \)

First, we move 2 units to the left from 0 to arrive at \( -2 \). Then, we move 8 units to the right from \( -2 \) and reach 6. Following this, from 6, we move 4 units to the left and arrive at 2, as shown in the diagram. So, \( (-2) + 8 + (-4) = 2 \).
In simple words: Start at 0. Go left 2 steps to \( -2 \). From there, go right 8 steps to reach 6. Finally, from 6, go left 4 steps. You will finish at 2.

Exam Tip: For problems with three or more integers, combine them step-by-step on the number line. Always identify your current position before making the next move.

Question 3. Add without using number line:
(a) \( 11 + (-7) \)
(b) \( (-13) + (+ 18) \)
(c) \( (-10) + (+ 19) \)
(d) \( (-250) + (+ 150) \)
(e) \( (-380) + (-270) \)
(f) \( (-217) + (-100) \)
Answer:
(a) \( 11 + (-7) \):
Since, \( 11 = 7 + 4 \)
\( 11 + (-7) = 7 + 4 + (-7) = 7 + (-7) + 4 \)
\( = 0 + 4 \)
\( [ (- 7) + (7) = 0 ] \)
\( = + 4 \)
Thus, \( 11 + (-7) = 4 \)
In simple words: To add 11 and -7, we can think of 11 as 7 + 4. So, we add 7 and -7, which gives 0. Then we add the remaining 4, making the total 4.

Exam Tip: When adding integers with different signs, subtract the smaller absolute value from the larger absolute value, and use the sign of the number with the larger absolute value.

(b) \( (-13) + (+ 18) \):
Since, \( (+18) = (+13) + (+5) \)
\( (-13) + (+18) = (-13) + (+13) + (+5) \)
\( = 0 + (+5) \)
\( [ (-13) + (+13) = 0 ] \)
\( = (+5) \)
Thus, \( (-13) + (+18) = 5 \)
In simple words: To add -13 and 18, we can split 18 into 13 + 5. When -13 and +13 are added, they cancel out to 0. So, only 5 is left.

Exam Tip: Breaking down one of the numbers into parts that can cancel out with the other number is a useful technique for mental calculation.

(c) \( (-10) + (+ 19) \):
Since, \( (+19) = (+10) + (+9) \)
\( (-10) + (+19) = (-10) + (+10) + (+9) \)
\( = 0 + (+9) \)
\( [ (-10) + (+10) = 0 ] \)
\( = +9 \)
Thus, \( (-10) + (+19) = 9 \)
In simple words: To add -10 and 19, we split 19 into 10 + 9. The -10 and +10 cancel out to 0, leaving 9 as the answer.

Exam Tip: This method of finding additive inverses (numbers that sum to zero) is effective for simplifying integer addition problems.

(d) \( (-250) + (+ 150) \):
Since, \( (-250) = (-150) + (-100) \)
\( (-250) + (+150) = (-150) + (-100) + (+150) \)
\( = (-150) + (+150) + (-100) \)
\( = 0 + (-100) \)
\( [ (-150) + (+150) = 0 ] \)
\( = -100 \)
Thus, \( (-250) + (+150) = -100 \)
In simple words: To add -250 and 150, we break -250 into -150 and -100. The -150 cancels out with +150 to make 0, leaving -100.

Exam Tip: Always look for ways to create zero pairs (a number and its opposite) to simplify calculations when adding integers.

(e) \( (-380) + (-270) \):
Since, the given integers have the same sign,
\( (-380) + (-270) = -[380 + 270] \)
\( = -650 \)
Thus, \( (-380) + (-270) = -650 \)
In simple words: When adding two negative numbers, simply add their absolute values together and keep the negative sign. Here, 380 + 270 equals 650, so the answer is -650.

Exam Tip: Remember, adding two negative numbers always results in a larger negative number. Think of it as moving left on the number line twice.

(f) \( (-217) + (-100) \):
Since, the given integers have the same sign,
\( (-217) + (-100) = -(217 + 100) \)
\( = -317 \)
Thus, \( (-217) + (-100) = -317 \)
In simple words: Add the two numbers 217 and 100, which gives 317. Since both numbers were negative, the final answer also stays negative, making it -317.

Exam Tip: Combining negative numbers is similar to combining positive numbers in terms of magnitude, but the result retains the negative sign.

Question 4. Find the sum of:
(a) 137 and -354
(b) - 52 and 52
(c) - 312, 39 and 192
(d) - 50,- 200 and 300
Answer:
(a) \( 137 + (-354) \):
Since, \( (-354) = (-137) + (-217) \)
\( 137 + (-354) = 137 + (-137) + (-217) \)
\( = 0 + (-217) \)
\( [ 137 + (-137) = 0 ] \)
\( = -217 \)
Thus, \( 137 + (-354) = -217 \)
In simple words: Break -354 into -137 and -217. The 137 and -137 cancel out to 0, leaving -217 as the total.

Exam Tip: When dealing with a positive and a negative number, consider subtracting the smaller absolute value from the larger one, and keep the sign of the larger absolute value.

(b) \( (-52) + 52 \):
Since, the given integers are opposite of each other, their sum must be equal to 0.
Thus, \( (-52) + 52 = 0 \)
In simple words: When you add a number to its exact opposite (like -52 and +52), they always cancel each other out, giving a total of zero.

Exam Tip: Any number added to its additive inverse (the number with the same magnitude but opposite sign) will always result in zero. This is a fundamental property of integers.

(c) \( (-312) + 39 + 192 \):
First, add the positive numbers:
\( 39 + 192 = 231 \)
Now, add this sum to the negative number:
\( (-312) + 231 \)
Break down \( (-312) = (-231) + (-81) \)
\( (-312) + 231 = (-231) + (-81) + 231 \)
\( = (-231) + 231 + (-81) \)
\( = 0 + (-81) \)
Thus, \( (-312) + 39 + 192 = -81 \)
In simple words: First, add the positive numbers 39 and 192 to get 231. Then, add -312 and 231. Break -312 into -231 and -81. The -231 and +231 cancel, leaving -81.

Exam Tip: When adding multiple integers, it's often easiest to group and add all positive numbers first, then all negative numbers, and finally combine the two sums.

(d) \( (-50) + (-200) + 300 \):
First, add the negative numbers:
\( (-50) + (-200) = (-250) \)
Now, add this sum to the positive number:
\( (-250) + 300 \)
Break down \( 300 = 250 + 50 \)
\( (-250) + 300 = (-250) + 250 + 50 \)
\( = 0 + 50 \)
\( [ (-250) + 250 = 0 ] \)
Thus, \( (-50) + (-200) + 300 = 50 \)
In simple words: Combine the negative numbers first: -50 and -200 make -250. Then add -250 to 300. Since 300 is 250 + 50, the -250 and +250 cancel out, leaving 50.

Exam Tip: For sums with multiple integers, you can combine numbers with the same sign first, and then combine the resulting positive and negative totals.

Question 5. Find the sum:
(a) \( (-7) + (-9) + 4 + 16 \)
(b) \( (37) + (-2) + (-65) + (-8) \)
Answer:
(a) \( (-7) + (-9) + 4 + 16 \):
First, combine the negative integers:
\( (-7) + (-9) = (-16) \)
Next, combine the positive integers:
\( 4 + 16 = 20 \)
Now, add the results:
\( (-16) + 20 \)
We know \( 20 = 16 + 4 \)
\( (-16) + 20 = (-16) + 16 + 4 \)
\( = 0 + 4 \)
\( [ (-16) + 16 = 0 ] \)
\( = 4 \)
Thus, \( (-7) + (-9) + 4 + 16 = 4 \)
In simple words: Add the negative numbers: -7 and -9 make -16. Add the positive numbers: 4 and 16 make 20. Then add -16 and 20. Break 20 into 16 + 4. The -16 and +16 cancel, leaving 4.

Exam Tip: A good strategy for adding multiple integers is to collect all the negative numbers, collect all the positive numbers, sum each group, and then find the sum of the two results.

(b) \( (37) + (-2) + (-65) + (-8) \):
First, combine the positive integer:
\( 37 \)
Next, combine the negative integers:
\( (-2) + (-65) + (-8) = (-75) \)
Now, add the positive total to the negative total:
\( 37 + (-75) \)
We know \( (-75) = (-37) + (-38) \)
\( 37 + (-75) = 37 + (-37) + (-38) \)
\( = 0 + (-38) \)
\( [ 37 + (-37) = 0 ] \)
\( = -38 \)
Thus, \( 37 + (-2) + (-65) + (-8) = -38 \)
In simple words: First, list the positive number: 37. Then, add all the negative numbers: -2, -65, and -8 combine to -75. Now, add 37 and -75. Break -75 into -37 and -38. The 37 and -37 cancel out, leaving -38.

Exam Tip: When combining multiple numbers, rearrange terms to group positives and negatives. This often makes it easier to spot additive inverses and simplify the overall calculation.