NCERT Solutions for Class 6 Maths Chapter 10: The Other Side of Zero — Free PDF

Question: Express the following using integers: (a) A temperature of $5°$ below zero. (b) A deposit of Rs. $200$. (c) A withdrawal of Rs. $150$. (d) $3$ floors below ground level.

Solution:

(a) $5°$ below zero $= -5°$C

(b) Deposit (money in) $= +200$ or simply $200$

(c) Withdrawal (money out) $= -150$

(d) $3$ floors below ground $= -3$

Answer: Negative numbers represent values below a reference point (zero). Positive numbers represent values above it.

Convention: Positive values often represent: above sea level, profit, deposit, temperature above zero. Negative values represent the opposite.

Question: Represent $-4, -1, 0, 2, 5$ on a number line. Which is the smallest? Which is the largest?

Solution:

On the number line:

Numbers increase as we move to the right and decrease as we move to the left.

From left to right: $-4, -1, 0, 2, 5$

Answer: Smallest $= -4$ (farthest left), Largest $= 5$ (farthest right).

Key rule: On the number line, every number to the right is greater than every number to the left. So $-1 > -4$ and $0 > -1$.

Question: Fill in with $<$, $>$, or $=$: (a) $-3 \enspace\square\enspace 2$ (b) $-5 \enspace\square\enspace -2$ (c) $0 \enspace\square\enspace -7$ (d) $-3 \enspace\square\enspace -3$

Solution:

(a) $-3 < 2$ — any negative number is less than any positive number.

(b) $-5 < -2$ — on the number line, $-5$ is to the left of $-2$. Among negative numbers, the one with the larger absolute value is smaller.

(c) $0 > -7$ — zero is greater than any negative number.

(d) $-3 = -3$ — a number equals itself.

Answer: (a) $<$, (b) $<$, (c) $>$, (d) $=$.

Important insight: $-5 < -2$ even though $5 > 2$. For negative numbers, "more negative" means smaller.

Question: Compute: (a) $(+5) + (+3)$ (b) $(-4) + (-6)$

Solution:

(a) Both positive:

(b) Both negative:

Rule for same signs: Add the absolute values and give the result the common sign.

Answer: (a) $8$, (b) $-10$.

Question: Compute: (a) $(+7) + (-3)$ (b) $(-8) + (+5)$ (c) $(-6) + (+6)$

Solution:

(a) $7 + (-3)$: Subtract the smaller absolute value from the larger. $7 - 3 = 4$. The number with the larger absolute value ($7$) is positive, so the answer is $+4$.

(b) $(-8) + 5$: $8 - 5 = 3$. The number with the larger absolute value ($8$) is negative, so the answer is $-3$.

(c) $(-6) + 6 = 0$. A number and its opposite (additive inverse) always sum to $0$.

Rule for different signs: Subtract the smaller absolute value from the larger. Give the result the sign of the number with the larger absolute value.

Answer: (a) $4$, (b) $-3$, (c) $0$.

Question: Use a number line to compute $(-3) + 5$.

Solution:

Step 1: Start at $-3$ on the number line.

Step 2: Adding $5$ means moving $5$ steps to the right.

We land on $2$.

Answer: $(-3) + 5 = 2$.

Rules for the number line:
- Adding a positive number: move right
- Adding a negative number: move left

Question: Compute: $3 - 7$.

Solution:

Now apply the rule for different signs: $|7| - |3| = 4$, and the larger absolute value ($7$) is negative.

Answer: $3 - 7 = -4$.

On the number line: start at $3$, move $7$ steps left, land on $-4$.

Question: Compute: (a) $5 - (-3)$ (b) $(-4) - (-6)$

Solution:

The key rule: Subtracting a negative number is the same as adding its positive.

(a) $5 - (-3) = 5 + 3 = 8$

(b) $(-4) - (-6) = -4 + 6 = 2$

Answer: (a) $8$, (b) $2$.

Why does this work? Subtraction means "finding the difference." Moving $-3$ in the negative direction is the same as adding $3$ in the positive direction. Think of it as "two negatives make a positive."

Question: The temperature at noon was $8°$C. By midnight, it dropped by $13°$C. What is the temperature at midnight?

Solution:

The temperature dropped below zero.

Answer: The temperature at midnight is $-5°$C.

Another example: The temperature at the top of a mountain is $-12°$C and at the base is $6°$C. The difference is: