Exercise Set : 3.1
1. A merchant in the port city of Lothal is exchanging bags of spices for copper ingots. He receives 15 ingots for every 2 bags of spices. If he brings 12 bags of spices to the market, how many copper ingots will he leave with?
Given:
- 2 bags of spices → 15 copper ingots
- 12 bags of spices → ? copper ingots
First find how many groups of 2 bags are present in 12 bags:
12 ÷ 2 = 6
So, there are 6 groups of 2 bags.
Each group gives 15 copper ingots.
Therefore,
6 × 15 = 90
Answer: 90 copper ingots
2. Look at the sequence of numbers on one column of the Ishango bone: 11, 13, 17, 19. What do these numbers have in common? List the next three numbers that ft this pattern.
The numbers are:
11, 13, 17, 19
All four numbers are prime numbers.
A prime number has exactly two factors: 1 and itself.
For example:
11 = 1 × 11
13 = 1 × 13
17 = 1 × 17
19 = 1 × 19
So, the pattern is prime numbers greater than 10.
The next three prime numbers are:
Common property: They are prime numbers.
Next three numbers: 23, 29, 31
3. We know that Natural Numbers are closed under addition (the sum of any two natural numbers is always a natural number). Are they closed under subtraction? Provide a couple of examples to justify your answer.
Natural numbers are:
N={1,2,3,4,5,…}
For a set to be closed under subtraction, the difference of any two natural numbers must also be a natural number.
Consider:
5 3 = 2
Here, 2 is a natural number.
But consider:
3 5 = 2
Here, is not a natural number.
Therefore, the difference of two natural numbers is not always a natural number.
Natural Numbers are closed under subtraction.
Subtracting a larger natural number from a smaller one can give a negative number, which is not a natural number.
4. Ancient Indians used the joints of their fngers to count, a practice still seen today. Each fnger has 3 joints, and the thumb is used to count them. How many can you count on one hand? How does this relate to the ancient base-12 counting systems?
There are 4 fingers other than the thumb.
Each finger has 3 joints.
Therefore,
4 × 3 = 12
So, using the thumb to point to the joints, we can count:
1,2,3,……12
on one hand.
This gives a natural way of counting up to 12, which is connected with the idea of a base-12 (duodecimal) counting system.
Answer: 12
Exercise Set : 3.2
1. 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?
Temperature at noon:
= 4 º C
Temperature drops by:
= 15 º C
A drop means that we have to subtract.
Therefore,
Since ,
Hence,
The temperature at midnight is .
2. A spice trader takes a loan (debt) of Rs. 850. The next day, he makes a proft (fortune) of Rs. 1,200. The following week, he incurs a loss of Rs. 450. Write this sequence as an equation using integers and calculate his fnal fnancial standing.
We represent:
- Debt by a negative integer.
- Profit by a positive integer.
- Loss by a negative integer.
Step 1: Represent the debt
The initial debt is Rs 850.
Therefore,
Debt = – 850
Step 2: Add the profit
Profit = Rs. 1200.
Therefore,
−850+1200
Calculate:
1200 – 850 = 350
So, after making the profit, the person’s financial position is:
+ Rs 350
This means the person has Rs 350 remaining after clearing the debt.
Step 3: Subtract the loss
Loss = Rs 450.
Therefore,
350 — 450
= – Rs 100
Thus, after suffering the loss, the final financial position is:
– Rs 100
3. Calculate the following using Brahmagupta’s laws:
(i) (–12) × 5
(ii) (–8) × (–7)
(iii) 0 – (–14)
(iv) (–20) ÷ 4
4. 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).
Think of a negative number as a debt.
Suppose you have a financial balance of:
Rs. 10
Now suppose a debt of ₹5 is removed.
A debt of ₹5 can be represented by:
– Rs. 5
Removing this debt means we are subtracting a negative amount: 10 – (– 5)
When the debt of ₹5 is cancelled, your financial position improves by ₹5.
Therefore,
10 – (– 5) = 10 + 5
= 15
Hence,
10 – (– 5) = 15
Exercise Set : 3.3
