NCERT Solutions for Class 9 Maths Chapter 3 — Exercise Set 3.1#

Natural numbers $\mathbb{N} = \{1, 2, 3, \ldots\}$ are the counting numbers. This exercise uses them for exchange, patterns and counting, and checks whether they are closed under an operation (whether the answer is always a natural number again).

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

Every 2 bags give 15 ingots, so we count how many groups of 2 bags he has:

Number of groups $= \dfrac{12}{2} = 6$

Each group gives 15 ingots:

Number of ingots $= 6 \times 15 = 90$

∴ The merchant leaves with 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 fit this pattern. #

Solution

Each of 11, 13, 17 and 19 has only two factors, 1 and itself:

  • $11 = 1 \times 11$
  • $13 = 1 \times 13$
  • $17 = 1 \times 17$
  • $19 = 1 \times 19$

So they are all prime numbers. In fact, they are all the primes between 10 and 20.

The next primes after 19 are found by skipping the numbers that have other factors ($20, 21 = 3 \times 7, 22, 24, 25 = 5 \times 5, 26, 27 = 3 \times 9, 28, 30$):

∴ The next three numbers are 23, 29 and 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. #

Solution

No, natural numbers are not closed under subtraction. The difference of two natural numbers is not always a natural number.

Examples:

$$3 - 5 = -2$$

$-2$ is not a natural number.

$$4 - 4 = 0$$

$0$ is not a natural number, because $\mathbb{N} = \{1, 2, 3, \ldots\}$.

The difference is a natural number only when the first number is larger, for example $9 - 4 = 5$. One example where the answer is not a natural number is enough to show that the set is not closed.

*4. 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

The thumb touches the joints of the other four fingers, and each of these fingers has 3 joints:

Joints on one hand $= 4 \times 3 = 12$

∴ You can count 12 on one hand.

Link to base-12: counting naturally goes up to 12 on one hand, so 12 became a convenient group, just as 10 did for counting on fingers. This is why base-12 (duodecimal) counting appears in many cultures:

  • a dozen is 12 items, and a gross is $12 \times 12 = 144$
  • there are 12 months in a year and 12 hours on a clock face
  • 12 has many factors ($1, 2, 3, 4, 6, 12$), so it divides evenly into halves, thirds and quarters, which is useful in trade

If the fingers of the other hand keep track of how many twelves have been counted, the two hands together reach $12 \times 5 = 60$. This matches the base-60 system of the Babylonians, which we still use for minutes and seconds.

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