NCERT Solutions for Class 9th Maths Chapter 3 .3 Integers: Expanding the Horizon — Exercise Set 3.2
Book page 463 Updated on2026-09-08
Q1.
The temperature in the high-altitude desert of Ladakh is recorded as 4 °C at noon. By midnight, it drops by 15 °C. What is the midnight temperature?
Answer
The midnight temperature is −11 °C.
midnight = noon temperature − fall
= 4 − 15
= −11 °C
On the number line, start at 4 and move 15 units to the left. You pass 0 after 4 steps and continue 11 more steps into the negative side.
A drop of 15 °C from 4 °C lands 11 units to the left of zero.
Tip: a “drop” always means subtract. A drop of 15 from 4 is written 4 − 15, never 15 − 4.
Q2.
A spice trader takes a loan (debt) of ₹850. The next day, he makes a profit (fortune) of ₹1,200. The following week, he incurs a loss of ₹450. Write this sequence as an equation using integers and calculate his final financial standing.
Answer
Writing a debt as a negative integer and a fortune as a positive one:
(−850) + 1200 + (−450)
= 350 + (−450)
= −100
His final standing is −₹100, that is, he is ₹100 in debt.
Why it happens: Brahmagupta's rules let all three transactions live on one number line. A debt of 850 and a fortune of 1200 partly cancel — 850 of the profit clears the loan and 350 is left over. The later loss of 450 is larger than that 350, so it pushes the trader back across zero by 100.
Check it yourself: the order does not matter. 1200 + (−850) + (−450) = 1200 − 1300 = −100, the same answer, because addition of integers is commutative and associative.
Q3.
Calculate the following using Brahmagupta’s laws: (i) (–12) × 5 (ii) (–8) × (–7) (iii) 0 – (–14) (iv) (–20) ÷ 4
Answer
Part
Brahmagupta's law used
Working
Answer
(i) (−12) × 5
debt × fortune = debt
12 × 5 = 60, sign negative
−60
(ii) (−8) × (−7)
debt × debt = fortune
8 × 7 = 56, sign positive
56
(iii) 0 − (−14)
removing a debt is a fortune
0 + 14
14
(iv) (−20) ÷ 4
debt shared into 4 equal parts
20 ÷ 4 = 5, sign negative
−5
Why it happens: in each case the size of the answer comes from ordinary whole-number arithmetic; only the sign comes from the debt–fortune rule. Part (iv) is a division, and it obeys the same sign rule as multiplication because (−5) × 4 = −20.
Check it yourself: (iii) reads “zero minus a debt of 14”. If you owe nothing and someone cancels a debt of ₹14 on your behalf, you are ₹14 better off — so the answer must be +14.
Q4.
Explain, using a real-world example of debt, why subtracting a negative number is the same as adding a positive number (e.g., 10 – (–5) = 15).
Answer
Suppose you have ₹10 in hand and your account also carries a debt of ₹5 written as −5. If a relative takes that debt away — subtracts the −5 — nothing is removed from your pocket, yet you are ₹5 better off.
10 − (−5)
= 10 + 5
= 15
The algebraic reason is that subtracting means adding the additive inverse, and the additive inverse of −5 is +5:
a − b = a + (−b)
10 − (−5) = 10 + [−(−5)] = 10 + 5 = 15
Why it happens: −(−5) must equal 5, because −5 and 5 add to 0 and a number has only one additive inverse. So the two minus signs undo one another. In the debt picture: a debt is a negative amount, and cancelling a negative amount is a gain.
Tip: read “−(−5)” as “remove a debt of 5”, not as “two minus signs somehow become plus”. The meaning makes the rule impossible to forget.