Q1. 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?

Answer:

Solution:

For every 2 bags of spices, the merchant receives 15 ingots.

Number of pairs of spice bags = 12 ÷ 2 = 6 pairs

Total ingots received = 6 × 15 = 90

Answer: 90 copper ingots

Q2. 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.

Answer:

Solution:

These numbers are all prime numbers between 10 and 20. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself.

The next three prime numbers after 19 are: 23, 29, 31.

Answer: They are prime numbers; the next three are 23, 29, and 31.

Q3. 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.

Answer:

Solution:

No, natural numbers are NOT closed under subtraction.

Examples:

• 5 − 2 = 3 (natural number, but this works only when the first number is larger)
• 2 − 5 = −3 (not a natural number, since natural numbers are positive)
• 7 − 7 = 0 (not a natural number, since 0 is not a natural number)

Answer: Natural numbers are not closed under subtraction because subtracting a larger number from a smaller one gives a negative number, and subtracting equal numbers gives zero, neither of which belong to the set of natural numbers.

Q4. 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?

Answer:

Solution:

Each finger has 3 joints, and there are 4 fingers (excluding the thumb). So, on one hand we can count 4 × 3 = 12 joints.

This relates to base-12 counting systems because ancient people could count up to 12 on one hand using the finger joints, making 12 a natural base for counting. Many ancient civilizations used base-12 or base-60 systems.

Answer: 12 on one hand; this shows why base-12 systems were common.