NCERT Solutions for Class 9th Maths Chapter 3 .1 The Dawn of Mathematics: The Human Need to Count — Exercise Set 3.1

Book page 433 Updated on2026-09-08

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

He leaves with 90 copper ingots.

12 bags = 6 lots of 2 bags
each lot of 2 bags → 15 ingots
ingots = 6 × 15 = 90

The same answer through the unit rate:

1 bag → 15 ÷ 2 = 7.5 ingots
12 bags → 12 × 7.5 = 90 ingots

Why it happens: the exchange is a fixed ratio 15 : 2. Multiplying both parts of a ratio by the same number does not change it, and 12 = 6 × 2, so the ingots must be 6 × 15. The unit rate 7.5 is not a whole number, yet the answer is — because 12 is a multiple of 2.

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

All four are prime numbers — in fact they are exactly the primes between 10 and 20.

11 = 1 × 11 only
13 = 1 × 13 only
17 = 1 × 17 only
19 = 1 × 19 only

Continuing past 19, we test each number for a factor other than 1 and itself:

Number	Factors	Prime?
20, 21, 22	2·10, 3·7, 2·11	no
23	none but 1 and 23	yes
24 … 28	all composite	no
29	none but 1 and 29	yes
30	2·15	no
31	none but 1 and 31	yes

The next three are 23, 29 and 31.

Did you know? To test whether n is prime you need only try prime divisors up to √n. For 31, √31 < 6, so testing 2, 3 and 5 is enough.

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

No — the natural numbers are not closed under subtraction.

3 − 5 = −2, and −2 is not a natural number
7 − 7 = 0, and 0 is not a natural number either

Why it happens: closure means the operation can never take you outside the set. Addition cannot: a + b is always at least as large as a, so it stays in {1, 2, 3, …}. Subtraction can: a − b lands outside whenever b ≥ a. One counter-example is enough to destroy a closure claim; a hundred successful cases prove nothing.

This failure is exactly what pushed mathematics forward. To make subtraction always possible, Brahmagupta added zero and the negative numbers — and the enlarged set, the integers ℤ, is closed under subtraction.

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

On one hand you can count up to 12.

fingers used for counting = 4 (thumb is the pointer)
joints on each finger = 3
total joints = 4 × 3 = 12

The thumb touches each joint in turn — three on the little finger, three on the ring finger, three on the middle finger, three on the index finger — and one full hand is one dozen.

Why it happens: a counting base is simply the size of the natural group you finish before starting again. Here the hand fills up at 12, so 12 becomes the base. Using the other hand to record completed dozens gives 12 × 12 = 144 (a gross), which is why dozens, grosses, 12 inches to a foot and 12 months to a year all survive.

Tip: 12 has factors 1, 2, 3, 4, 6, 12, while 10 has only 1, 2, 5, 10. Halves, thirds and quarters are all whole numbers of a dozen — one practical reason traders liked base 12.

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