The World of NumbersClass 9 Mathematics NCERT Solutions
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Q1Exercise Set 3.1
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?
Solution
Given:
- Rate of exchange: 15 ingots for 2 bags of spices.
- Quantity of spices brought to market: 12 bags.
To Find:
- The total number of copper ingots the merchant will receive.
Solution:
We can set up a proportion to solve this problem. Let be the number of ingots the merchant will receive for 12 bags of spices.
The ratio of ingots to bags is constant:
To find , we can cross-multiply or solve the equation:
Alternatively, we can find the number of ingots per bag:
Rate = ingots per bag.
For 12 bags, the total ingots = .
Final Answer: The merchant will leave with 90 copper ingots.
Q2Exercise Set 3.1
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 fit this pattern.
Solution
Given:
- The sequence of numbers: 11, 13, 17, 19.
To Find:
- The common property of these numbers.
- The next three numbers in the sequence.
Solution:
The chapter states that the numbers 11, 13, 17, and 19 are the prime numbers between 10 and 20. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
-
Common Property: All the numbers in the sequence (11, 13, 17, 19) are prime numbers.
-
Next three numbers: To find the next three numbers that fit this pattern, we need to find the next three prime numbers after 19.
- The number after 19 is 20 (not prime, divisible by 2).
- 21 is not prime (divisible by 3).
- 22 is not prime (divisible by 2).
- 23 is prime.
- 24, 25, 26, 27, 28 are not prime.
- 29 is prime.
- 30 is not prime.
- 31 is prime.
The next three prime numbers after 19 are 23, 29, and 31.
Final Answer: The numbers have in common that they are all prime numbers. The next three numbers that fit this pattern are 23, 29, and 31.
Q3Exercise Set 3.1
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.
Solution
To Determine:
- Whether the set of Natural Numbers () is closed under subtraction.
Justification:
For a set to be closed under an operation, performing that operation on any two members of the set must result in a member that is also in the set.
Let's test subtraction with a few examples using natural numbers.
Example 1:
Let the natural numbers be 5 and 3.
The result, 2, is a natural number.
Example 2:
Let the natural numbers be 3 and 5.
The result, -2, is a negative integer and is not a natural number.
Conclusion:
Since we found at least one case (Example 2) where subtracting two natural numbers results in a number that is not in the set of natural numbers, the set of Natural Numbers is not closed under subtraction.
Final Answer: No, the Natural Numbers are not closed under subtraction. For example, , and -2 is not a natural number.
Q4Exercise Set 3.1
Ancient Indians used the joints of their fingers to count, a practice still seen today. Each finger 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?
Solution
Given:
- Each finger (excluding the thumb) has 3 joints.
- The thumb is used as a pointer to count these joints.
To Find:
- The total number one can count on a single hand using this method.
- The relation of this method to base-12 counting systems.
Solution:
-
Counting on one hand: A human hand has 4 fingers (index, middle, ring, and pinky) and 1 thumb. The method described uses the thumb to touch the joints of the other four fingers.
- Number of fingers used for counting joints = 4
- Number of joints on each finger = 3
- Total count = (Number of fingers) (Number of joints per finger)
- Total count = So, one can count up to 12 on one hand.
-
Relation to base-12 systems: A base-12 system, also known as a duodecimal system, uses 12 as its base. The ability to easily count to 12 on one hand provides a natural and physical basis for the development of such a system. Many ancient civilizations used base-12 (or its multiple, base-60, which is ) for timekeeping (12 hours in a day/night, 12 months in a year), measurement (12 inches in a foot), and commerce. This finger-counting method is a plausible origin for the prevalence of base-12 in history.
Final Answer: You can count up to 12 on one hand. This method provides a tangible way to represent the numbers 1 through 12, which likely contributed to the development and use of base-12 (duodecimal) counting systems in various ancient cultures.
Q1Exercise Set 3.2
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?
Solution
Given:
- Temperature at noon = 4 °C
- Drop in temperature by midnight = 15 °C
To Find:
- The temperature at midnight.
Solution:
A drop in temperature is represented by subtraction.
Midnight temperature = Noon temperature - Drop in temperature
Final Answer: The midnight temperature is -11 °C.
Q2Exercise Set 3.2
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.
Solution
Given:
- A loan (debt) of ₹850.
- A profit (fortune) of ₹1,200.
- A loss (debt) of ₹450.
To Find:
- An equation representing the sequence of transactions.
- The final financial standing of the trader.
Solution:
We can represent debts and losses as negative integers and profits as positive integers.
- Loan of ₹850 is -850.
- Profit of ₹1,200 is +1200.
- Loss of ₹450 is -450.
Equation:
The sequence of transactions can be written as:
Final Standing =
Calculation:
A final standing of -100 means the trader is in a debt of ₹100.
Final Answer: The equation is . The trader's final financial standing is a debt of ₹100.
Q3Exercise Set 3.2
Calculate the following using Brahmagupta's laws:
(i)
(ii)
(iii)
0 - (-14)
(iv)
Solution
To Calculate:
The results of the given arithmetic operations using Brahmagupta's laws.
Solution:
(i)
Brahmagupta's law states that the product of a debt (negative number) and a fortune (positive number) is a debt.
(ii)
Brahmagupta's law states that the product of two debts (two negative numbers) is a fortune (a positive number).
(iii)
Subtracting a debt is equivalent to adding a fortune. Therefore, subtracting a negative number is the same as adding its positive counterpart.
(iv)
Division can be thought of as the inverse of multiplication. A debt divided by a fortune results in a debt.
Final Answer:
(i)
-60
(ii)
56
(iii)
14
(iv)
-5
Q4Exercise Set 3.2
Explain, using a real-world example of debt, why subtracting a negative number is the same as adding a positive number (e.g., ).
Solution
To Explain:
- Why subtracting a negative number is equivalent to adding a positive number, using a real-world example.
Explanation:
Let's use the concepts of 'fortune' (positive numbers) and 'debt' (negative numbers) as introduced by Brahmagupta.
- A positive number, like
+10, represents having a fortune of ₹10. - A negative number, like
-5, represents having a debt of ₹5.
Consider the expression .
10means you have a fortune of ₹10.- (-5)means you are subtracting a debt of ₹5.
Subtracting or removing a debt makes you richer. If someone forgives or cancels a debt of ₹5 that you owe, your net worth increases by ₹5. It is financially the same as someone giving you ₹5.
So, starting with a fortune of ₹10 and then having a debt of ₹5 removed is equivalent to starting with ₹10 and adding ₹5 to it.
Mathematical Representation:
Starting position: +10 (a fortune of ₹10)
Action: Subtracting a debt of 5, which is .
Final position: .
Your new financial standing is a fortune of ₹15.
Final Answer: Subtracting a negative number is like removing a debt. If you have ₹10 and a debt of ₹5 is removed (subtracted), you are effectively ₹5 richer, making your new total ₹15. This is the same as adding ₹5 to your initial ₹10. Therefore, .
Q1Exercise Set 3.3
Prove that the following rational numbers are equal:
(i)
and
(ii)
and
(iii)
and
(iv)
and 3
Solution
To Prove:
The equality of the given pairs of rational numbers.
Method:
Two rational numbers and are equal if and only if .
Proof:
(i)
and
Here, .
Since , the numbers are equal.
(ii)
and
Here, .
Since , the numbers are equal.
(iii)
and
We can write these as and .
Here, .
Since , the numbers are equal.
(iv)
and 3
We can write 3 as the rational number .
Here, .
Since , the numbers are equal. Alternatively, simplifies to 3.
Q2Exercise Set 3.3
Find the sum:
(i)
(ii)
(iii)
Solution
To Find:
The sum of the given rational numbers.
Method:
To add fractions, we first find a common denominator, then add the numerators.
Solution:
(i)
The least common multiple (LCM) of 5 and 10 is 10.
(ii)
The LCM of 12 and 8 is 24.
(iii)
This can be written as . The LCM of 7 and 14 is 14.
Final Answer:
(i)
(ii)
(iii)
Q3Exercise Set 3.3
Find the difference:
(i)
(ii)
(iii)
Solution
To Find:
The difference of the given rational numbers.
Method:
To subtract fractions, we first find a common denominator, then subtract the numerators.
Solution:
(i)
The least common multiple (LCM) of 6 and 4 is 12.
(ii)
The LCM of 8 and 4 is 8.
(iii)
Subtracting a negative number is the same as adding its positive counterpart.
The LCM of 9 and 3 is 9.
Final Answer:
(i)
(ii)
(iii)
Q4Exercise Set 3.3
Find the product:
(i)
(ii)
(iii)
Solution
To Find:
The product of the given rational numbers.
Method:
To multiply fractions, we multiply the numerators together and the denominators together: .
Solution:
(i)
Simplifying the fraction by dividing the numerator and denominator by their greatest common divisor, which is 6:
(ii)
This fraction is already in its simplest form.
(iii)