Selina Concise Solutions for ICSE Class 6 Mathematics Chapter 7 Number Line

IMPORTANT POINTS

1. Number Line: A Number line is used to represent numbers, such as : fractions, whole numbers, integers, etc.
2. Using A Number Line to Compare Numbers: Out of any two numbers, marked on a number line, the number which is on the right of the other number is greater and the number which is on the left of the other number is lesser (smaller).

EXERCISE 7(A)

Question 1: Fill in the blanks, using the following number line :

(i) An integer, on the given number line, is ………… than every number on its left.
(ii) An integer, on the given number line, is greater than every number to its …………..
(iii) 2 is greater than – 4 implies 2 is to the ………….. of – 4.
(iv) -3 is ………….. than 2 and 3 is ………. than – 2.
(v) – 4 is ………….. than -8 and 4 is …………… than 8.
(vi) 5 is …………. than 2 and -5 is …………… than – 2.
(vii) -6 is …………. than 3 and the opposite of -6 is ………… than opposite of 3.
(viii) 8 is …………. than -5 and -8 is ……….. than -5.

Answer:
(i) An integer, on the given number line, is greater than every number on its left.
(ii) An integer, on the given number line, is greater than every number to its left.
(iii) 2 is greater than - 4 implies 2 is on the right of - 4.
(iv) - 3 is less than 2 and 3 is greater than -2.
(v) - 4 is greater than -8 and 4 is less than 8.
(vi) 5 is greater than 2 and - 5 is less than - 2.
(vii) -6 is less than 3 and the opposite of -6 is greater than opposite of 3.
(viii) 8 is greater than -5 and -8 is less than -5.
A number line helps us visualize the relationship between positive and negative values clearly. As we move from the left toward the right, the value of the numbers continuously increases.
Teacher's Tip: Remember the phrase "Right is Mighty" to remind yourself that numbers on the right side of a number line are always larger.
Exam Tip: When filling in blanks for comparisons, always double-check the signs (+ or -) as they change the entire direction on the number line.

Question 2: In each of the following pairs, state which integer is greater :
(i) -15, -23
(ii) -12, 15
(iii) 0, 8
(iv) 0, -3

Answer:
(i) -15, -23
-15 is greater than -23 as -15 lies on the right side of -23 on the number line
(ii) -12, 15
15 is greater than than -12 as 15 lies on the right side of -12 on the number line
(iii) 0, 8
8 > 0
(iv) 0, -3
0 > - 3
When comparing two negative integers, the one with the smaller absolute numerical value is actually the greater one because it is closer to zero. For example, -15 is "less negative" and further to the right than -23.
Teacher's Tip: Imagine the negative sign as "debt"; having a debt of \15 is better (greater) than having a debt of \23.
Exam Tip: Always mention the position on the number line (left or right) to provide a complete mathematical reason for your answer.

Question 3: In each of the following pairs, which integer is smaller :
(i) o, -6
(ii) 2, -3
(iii) 15, -51
(iv) 13, 0

Answer:
(i) 0, -6
-6 < 0
(ii) 2, -3
-3 < 2
(iii) 15, -51
-51 < 15
(iv) 13, 0
0 < 13
The smaller integer is always the one located further to the left on the number line. Negative numbers are always smaller than zero and any positive number.
Teacher's Tip: Think of "smaller" as being deeper underground or lower in temperature.
Exam Tip: To avoid mistakes, quickly sketch a small number line to see which number falls on the left.

Question 4: In each of the following pairs, replace * with < or > to make the statement true:
(i) 3 * 0
(ii) 0 * -8
(iii) -9 * -3
(iv) 3 * 3
(v) 5 * -1
(vi) -13 * 0
(vii) -8 * -18
(viii) 516 * -316

Answer:
(i) 3 > 0
(ii) 0 > -8
(iii) -9 < -3
(iv) 3 = 3 (Note: textbook lists as 3 < 3 in solution text but logic dictates equality or following specific comparison pairs)
(v) 5 > -1
(vi) -13 < 0
(vii) -8 > -18
(viii) 516 > -316
Using inequality symbols correctly shows the relationship between different points in space. These symbols always "open" their mouths toward the larger value.
Teacher's Tip: The symbol "<" looks like a tilted letter "L" for "Less than".
Exam Tip: Be very careful with large negative numbers; -316 is much smaller than any positive number like 516.

Question 5: In each case, arrange the given integers in ascending order using a number line.
(i) – 8, 0, – 5, 5, 4, – 1
(ii) 3, – 3, 4, – 7, 0, – 6, 2

Answer:
(i) – 8, 0, – 5, 5, 4, – 1
Draw a number line and mark the numbers on it. Arranging in ascending order, as shown -8,-5,-1, 0, 4, 5 as on the number line
(ii) 3, -3, 4, -7, 0, -6, 2
Draw the number line and mark the numbers on it. Arranging in ascending order as shown on the number line. -7, -6, -3, 0, 2, 3, 4
Ascending order means moving from the smallest value to the greatest value. On a number line, this simply means reading the marked points from left to right.
Teacher's Tip: Ascending = Climbing "A" ladder from the bottom to the top.
Exam Tip: When using a number line for ordering, ensure your spacing between marks is relatively even for a neat presentation.

Question 6: In each case, arrange the given integers in descending order using a number line.
(i) -5, -3, 8, 15, 0, -2
(ii) 12, 23, -11, 0, 7, 6

Answer:
(i) -5, -3, 8, 15, 0, -2
Draw the number line and mark these numbers on it. Arranging in descending order 15, 8, 0 -2, -3, -5 as shown on the number line
(ii) 12, 23, -11, 0, 7, 6
Draw a number line and mark these numbers on it. Arranging in descending order. 23, 12, 7, 6, 0, -1 as shown on the number line
Descending order is the opposite of ascending order, starting from the greatest value. To find this on the number line, we read our marked points from right to left.
Teacher's Tip: Descending = "D" for going "Down" from high to low numbers.
Exam Tip: Double check your list after writing to ensure you haven't missed any numbers from the original set.

Question 7: For each of the statements, given below, state whether it is true or false :
(i) The smallest integer is 0.
(ii) The opposite of -17 is 17.
(iii) The opposite of zero is zero.
(iv) Every negative integar is smaller than 0.
(v) 0 is greater than every positive integer.
(vi) Since, zero is neither negative nor positive ; it is not an integer.

Answer:
(i) False
(ii) True
(iii) True
(iv) True
(v) False
(vi) False
Zero acts as the neutral origin point on the number line. While it has no sign, it is still a member of the integer set and occupies a specific position between negatives and positives.
Teacher's Tip: Integers go on forever in both directions, so there is no such thing as a "smallest" integer.
Exam Tip: Remember that the word "integer" includes whole numbers, their negatives, and zero.

EXERCISE 7(B)

Use a number line to evaluate each of the following :

Question 1:
(i) (+ 7) + (+ 4)
(ii) 0 + (+ 6)
(iii) (+ 5) + 0

Answer:
(i) (+ 7) + (+ 4)
For + 7, move 7 units to the right of zero and for + 4 move 4 units to the right of +7
\therefore (+ 7) + (+ 4) = + 11
(ii) 0 + (+ 6)
For 0, No movement for + 6 move 6 units right to zero.
\therefore (0) + (+ 6) = + 6.
(iii) (+ 5) + 0
For + 5 move 5 units to the right of 0, for 0 Nor movement.
\therefore (+ 5) + 0 = + 5.
When we add a positive number, we always move to the right on the number line. Starting from the first number, we "jump" forward by the value of the second number.
Teacher's Tip: Start at zero, do the first jump, then use that spot as your new starting point for the second jump.
Exam Tip: Show your jumps with curved arrows on the number line to get full marks for the "Evaluation" method.

Question 2:
(i) (-4) + (+5)
(ii) 0 + (-2)
(iii) (-1) + (-4)

Answer:
(i) (- 4) + (+ 5)
For (- 4) move 4 units to the left of 0, then for + 5 move 5 units to the right of - 4
\therefore (- 4) + (+ 5) = + 1.
(ii) 0 + (- 2)
For 0 no movement then for - 2 move 2 units to left of 0
\therefore 0 + (- 2) = - 2.
(iii) (- 1) + (+ 4)
For - 1 move 1 unit to the left of 0, then for + 4 move 4 units to the right of - 1
\therefore (- 1) + (+ 4) = + 3.
Adding numbers with different signs involves changing directions on the number line. A negative number moves us left, while a positive number moves us back toward the right.
Teacher's Tip: Think of addition as a series of steps; "minus" means step backward, "plus" means step forward.
Exam Tip: Always start your counting from the end of the previous arrow, not from zero again.

Question 3:
(i) (+ 4) + (-2)
(ii) (+3) + (-6)
(iii) 3 + (-7)

Answer:
(i) (+ 4) + (- 2)
For + 4 move 4 units to the right of 0, then for (- 2) move 2 units to the left of + 4
\therefore (+ 4) + (- 2) = + 2.
(ii) (+3) + (-6)
For (+3), we move 3 unit right of 0 and then For (-6), we move 6 units left of 3, we get -3
\therefore (+3) + (-6) = -3
(iii) 3 + (- 7)
For 3, we move 3 units right of 0 and then, for (- 7) move 7 units to left of 3.
\therefore 3 + (- 7) = - 4.
Even if a positive number comes first, the addition of a negative value pulls the final result toward the left. This can often result in a negative final answer if the negative value being added is larger than the starting positive value.
Teacher's Tip: The "bigger" number (ignoring signs) determines what the final sign of the answer will be.
Exam Tip: Use different colored pens for the "right" jumps and "left" jumps to make your diagram easier to read.

Question 4:
(i) (-1) + (-2)
(ii) (-2) + (-5)
(iii) (-3) + (-4)

Answer:
(i) (- 1) + (- 2)
for - 1, start from zero and move one unit to the left and then again for - 2, move 2 unit to left of - 1.
\therefore (- 1) + (- 2) = - 3.
(ii) (-3) + (- 4)
for - 3, start from zero and move 3 units to the left and then again for - 4, move 4 unit to left of - 3.
\therefore (- 3) + (- 4) = - 7.
(iii) (- 2) + (- 5)
for - 2, start from zero and move 2 units to the left and then again for - 5, move 5 unit to left of - 2.
\therefore (- 2) + (- 5) = - 7.
When adding two negative numbers, we never change direction; we just keep moving further to the left. The result is always a more negative number.
Teacher's Tip: Adding negatives is like digging a hole; adding more negatives just makes the hole deeper.
Exam Tip: Remember that the sum of two negative numbers must always be negative.

Question 5:
(i) (+ 10) – (+2)
(ii) (+8)- (-5)
(iii) (-6) – (+2)
(iv) (-7) – (+5)
(v) (+4) – (-2)
(vi) (-8) – (-4)

Answer:
(i) (+ 10) - (+ 2)
From + 2, to reach the position of number + 10, we find 8 steps to the rights.
\therefore (+ 10) - (+ 2) = + 8
(ii) (+ 8) - (- 5)
Starting from the position of - 5, count the number of steps needed to reach + 8, we find 13 steps towards right.
\therefore (+ 8) - (- 5) = + 13
(iii) (- 6) - (+ 2)
Marking the position of 6, and + 2 on the number line count step from position + 2 to left - 6, there are 8 steps
\therefore (- 6) - (+ 2) = - 8
(iv) (- 7) - (+ 5)
Marking - 7 and + 5, form + 5 position count steps towards left to - 7, there are 12 steps
\therefore (- 7) - (+ 5) = - 12
(v) (+ 4) - (- 2)
Marking (+ 4) and (- 2) from + 4 position count steps toward left to 2. There are 6 steps.
\therefore (+ 4) - (- 2) = + 6
(vi) (- 8) - (- 4)
Draw a number line and mark (- 8) on it. Now mark (- 4) on the same line. Now count from - 8 to - 4, which is - 4 as shown.
Subtraction on a number line can be thought of as finding the "distance" and "direction" between two points. We start at the second number and count how many steps it takes to reach the first number.
Teacher's Tip: Subtraction is like asking, "How do I get from point B to point A?"
Exam Tip: If you move right to reach the target, the answer is positive; if you move left, the answer is negative.

Question 6: Using a number line, find the integer which is :
(i) 3 more than -1
(ii) 5 less than 2
(iii) 5 more than -9
(iv) 4 less than -4
(v) 7 more than 0
(vi) 7 less than -8

Answer:
(i) 3 more than -1
To get 3 more than -1, start from -1 and than move 3 units to the right of -1 to get 2.
\therefore 3 more than -1 is 2
(ii) 5 less than 2
To get 5 less than 2, start from 2 and then move 5 units to the left of 5 to get -3.
\therefore 5 more than 2 is -3
(iii) 5 more than -9
To get 5 more than -9, start from -9 and then move -9 units to the right of -9 to get -4.
\therefore 5 more than -9 is -4
(iv) 4 less than -4
To get 4 less than -4, start from -4 and then move 4 units to the left of 4 to get -8.
\therefore 4 less than -4 is -8
(v) 7 more than 0
To get 7 more than 0, start from 7 and then move 7 and then move 7 units to the right of 7 to get 7.
\therefore 7 more than 0 is 7
(vi) 7 less than -8
To get 7 less than -8, start from -8 and then move 7 units to the left of -8 to get -15.
\therefore 7 less than -8 is -15
"More than" implies adding or moving to the right, while "less than" implies subtracting or moving to the left. The wording tells you the starting point and the direction of the movement.
Teacher's Tip: The number mentioned after "than" is always your starting point on the line.
Exam Tip: Be careful when calculating "less than" a negative number; it makes the number even smaller (more negative).

REVISION EXERCISE

Question 1: Fill in the blanks :

(i) 5 is …………… than -2 and -5 is ………… than 2.
(ii) -3 is ………… than 0 and 3 is …………. than 0.
(iii) on a number line, if x is to the left of y, then x is ………… than y.
(iv) on a number line if x is to the right of y, then y is …………. than x.

Answer:
(i) 5 is greater than -2 and -5 is less than 2.
(ii) -3 is less than 0 and 3 is greater than 0.
(iii) On a number line, if x is to the left of y, then x is less than y.
(iv) On a number line, x is to the right of y, then y is less than x.
Comparing variables x and y on a number line establishes a fixed rule for relative values regardless of the actual numbers. The position relative to another point is the ultimate test of "greater" or "less".
Teacher's Tip: If you can visualize x and y as two kids on a line, the kid on the left always has the "smaller" value.
Exam Tip: Read carefully in part (iv); it asks about y relative to x, which is the inverse of the first half of the sentence.

Question 2: Using a number line, write the numbers -15, 7, 0, -8 and -3 in ascending order of value.

Answer:
On the given number line, we mark the numbers -15, 7, 0, -8 and -3 on it, we see that
We see that -15 < -8 < -3 < 0 < 7
-15, -8, -3, 0, 7 are in ascending order
Ascending order organizes data from the most negative (leftmost) to the most positive (rightmost). This process provides a clear picture of the sequence of integers.
Teacher's Tip: Always place zero first to divide your positives and negatives, then sort each group.
Exam Tip: Use commas to separate the numbers in your final list to make it clear for the grader.

Question 3: Using a number line, write the numbers 8, -6, 2 -12, 0, 15 and -1 in descending order of value.

Answer:
On the given number line, we mark the numbers 8, -6, 2, -12, 0, 15 and -1 on it
We see that
15 > 8 > 2 > 0 > -1 > -6 > -12
15, 8, 2, 0, -1, -6, -12 are in descending order
Descending order starts with the largest positive value and moves toward the most negative value. This is the reverse of how we normally read a number line from left to right.
Teacher's Tip: Descending is like walking "backwards" through the number line.
Exam Tip: Be extra careful not to put -1 before 0; even though 1 is a number, -1 is smaller than zero.

Question 4: Using a number line, evaluate :
(i) (+5) + (+4)
(ii) (+6) + (+8)
(iii) (-3) + (+5)
(iv) (-3) + (+7)
(v) (+6) + (-2)
(vi) (-3) + (+3)
(vii) (-5) + (-5)
(viii) (-7) + (-1)
(ix) (+6) – (+2)
(x) (+5) – (-3)
(xi) (+4) – (-1)
(xii) (-7) – (-2)

Answer:
(i) (+5) + (+4)
First of all, we move 5 units to the right of zero then for (+4), move 4 units right of 5, then we reach at 9, then (+5) + (+4) = +9
(ii) (+6) + (+8)
First of all, we move 6 units to the right of zero then for (+8), we move 8 units to the right of (+6). Then we reach at +14, then (+6) + (+8) = +14
(iii) (-3) + (+5)
First of all for (-3) we move, 3 units to the left of zero, then move (+5) units to the right of -3, then we reach at (+2), then (-3) + (+5) = -3 + 5 = 2
(iv) (-3) + (+7)
First of all, we move for (-3) 3 unit to the left of zero and then for (+7), we move 7 units to the right of (-3) reaching +4 Then (-3) + (+7) = +4
(v) (+6) + (-2)
First of all, we move for (+6), 6 units to the right of zero and then for (-2), move 2 units to the left of 6, then we reach 4 Then (+6) + (-2) = 6 – 2 = 4
(vi) (-3) + (+3)
First of all for (-3), we move 3 units left of zero and then for (+3) we move 3 unit right of (-3) reaching at 0. So, (-3) + (+3) = 0
(vii) (-5) + (-5)
First of all for -5, we move 5 units to left of zero and then for (-5), we move 5 units to left of (-5) reaching at -10. (-5) +(-5) = -10
(viii) (-7) + (-1)
First of all for -7, we move 7 units left of zero and then for (-1) we move 1 unit left of -7 reaching -8. (-7) + (-1) = -8
(ix) (+6) - (+2)
First of all for (+6) we move 6 units right of 0 and then for (+2), we move 2 units left of 6 reaching 4. (+6)-(+2) = 6 – 2 = 4
(x) (+5) - (-3)
Mark the points (+5) and (-3) on the same number line. We see that the position of (-3) is 8 units from (+5) to its right 3. (+5) - (-3) = 5 + 3 = 8
(xi) (+4) - (-1)
Mark the points (+4) and (-1) on the same number line, we see that the position of (-1) is 5 units from (+4) to its right. (+4) - (-1) = 4 + 1 = 5
(xii) (-7) - (-2)
Mark the points (-7) and (-2) on the same number line, we see that (-2) is 5 units on the left (-2). -7 - (-2) = -7 + 2 = -5
Evaluation involves performing the arithmetic step-by-step to arrive at a final integer value. Whether adding or subtracting, the number line provides a visual proof that justifies the calculated sum or difference.
Teacher's Tip: When you subtract a negative, it's the same as adding a positive (Double Negative rule).
Exam Tip: Ensure your final answer has the correct sign (+ or -); a missing sign is a common way to lose points even if the number is correct.