Exercise 3.1 Practice
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Overview
This page provides comprehensive Ch 3: Coordinate Geometry - Exercise 3.1 Practice. Practice solving ratio problems, prime number identification, closure properties of number systems, and finger segment counting bases. Free step-by-step NCERT solutions and explanations.
Foundations of Arithmetic & Counting Systems
Q1: Lothal Spice & Copper Exchange
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?
We are given the exchange rate between bags of spices and copper ingots:
$$\text{Ratio} = \frac{15\text{ ingots}}{2\text{ bags of spices}}$$
The merchant brings 12 bags of spices.
Since $12 = 6 \times 2$, we can determine the number of ingots by multiplying the exchange unit: $$\text{Number of ingots} = 6 \times 15 = 90\text{ copper ingots}$$
Since $12 = 6 \times 2$, we can determine the number of ingots by multiplying the exchange unit: $$\text{Number of ingots} = 6 \times 15 = 90\text{ copper ingots}$$
Alternatively, using a proportion:
$$\frac{2}{15} = \frac{12}{x} \implies 2x = 12 \times 15 \implies x = \frac{180}{2} = 90$$
90 copper ingots
Q2: Ishango Bone Sequence
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.
Let's analyze the numbers in the sequence: $11, 13, 17, 19$.
- All of these numbers are positive integers greater than 1.
- Each number has exactly two positive divisors: 1 and itself.
To find the next three numbers in the pattern, we list the next three prime numbers following $19$:
1. The next integer is 20 (not prime, factors: 1, 2, 4, 5, 10, 20).
2. 21 is not prime ($3 \times 7 = 21$).
3. 22 is not prime ($2 \times 11 = 22$).
4. 23 is prime.
5. 24 is not prime.
6. 25, 26, 27, 28 are not prime.
7. 29 is prime.
8. 30 is not prime.
9. 31 is prime.
1. The next integer is 20 (not prime, factors: 1, 2, 4, 5, 10, 20).
2. 21 is not prime ($3 \times 7 = 21$).
3. 22 is not prime ($2 \times 11 = 22$).
4. 23 is prime.
5. 24 is not prime.
6. 25, 26, 27, 28 are not prime.
7. 29 is prime.
8. 30 is not prime.
9. 31 is prime.
Commonality: Prime Numbers; Next three numbers: 23, 29, 31
Q3: Closure Property under Subtraction
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.
Definition of Closure:
A set is closed under an operation if performing that operation on any elements of the set always results in an element that is also in that set.
Natural numbers are $\mathbb{N} = \{1, 2, 3, 4, \dots\}$.
A set is closed under an operation if performing that operation on any elements of the set always results in an element that is also in that set.
Natural numbers are $\mathbb{N} = \{1, 2, 3, 4, \dots\}$.
Closure under Subtraction:
No, natural numbers are **not closed under subtraction**. Subtraction of two natural numbers does not always result in a natural number.
No, natural numbers are **not closed under subtraction**. Subtraction of two natural numbers does not always result in a natural number.
Examples to justify:
1. Let $a = 3$ and $b = 5$ (both are natural numbers). $$a - b = 3 - 5 = -2$$ Since $-2$ is a negative integer and not a natural number, closure is violated.
2. Let $a = 4$ and $b = 4$ (both are natural numbers). $$a - b = 4 - 4 = 0$$ Since $0$ is a whole number but not a natural number, closure is violated.
1. Let $a = 3$ and $b = 5$ (both are natural numbers). $$a - b = 3 - 5 = -2$$ Since $-2$ is a negative integer and not a natural number, closure is violated.
2. Let $a = 4$ and $b = 4$ (both are natural numbers). $$a - b = 4 - 4 = 0$$ Since $0$ is a whole number but not a natural number, closure is violated.
No, not closed under subtraction. Examples: $3 - 5 = -2 \notin \mathbb{N}$ and $4 - 4 = 0 \notin \mathbb{N}$.
Q4: Finger Joints Counting & Base-12
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?
Counting on one hand:
A human hand has 4 fingers (index, middle, ring, little) and 1 thumb.
Each of the 4 fingers has 3 joints (phalanges segments).
Using the thumb of the same hand as a pointer, we count the joints on the other 4 fingers: $$\text{Total Count} = 4 \text{ fingers} \times 3 \text{ joints/finger} = 12 \text{ joints}$$ Therefore, you can count up to **12** on one hand.
A human hand has 4 fingers (index, middle, ring, little) and 1 thumb.
Each of the 4 fingers has 3 joints (phalanges segments).
Using the thumb of the same hand as a pointer, we count the joints on the other 4 fingers: $$\text{Total Count} = 4 \text{ fingers} \times 3 \text{ joints/finger} = 12 \text{ joints}$$ Therefore, you can count up to **12** on one hand.
Relationship to Base-12 Systems:
This physical counting method is a primary origin of the **duodecimal (base-12)** counting system used by several ancient civilisations (like ancient India and Sumeria).
Since 12 is highly divisible (by 1, 2, 3, 4, 6, and 12), counting in dozens and half-dozens made market trades and simple fraction divisions extremely efficient.
This physical counting method is a primary origin of the **duodecimal (base-12)** counting system used by several ancient civilisations (like ancient India and Sumeria).
Since 12 is highly divisible (by 1, 2, 3, 4, 6, and 12), counting in dozens and half-dozens made market trades and simple fraction divisions extremely efficient.
Count on one hand = 12; Originates the base-12 (duodecimal) counting system.