NCERT Solutions Ganita Prakash (Part 1) Chapter 3 The Mechanism of Counting — Math Talk
Book page 51 Updated on2026-09-05
Q1.
How do we ensure that all cows have returned safely after grazing?
Answer
Keep one stick for every cow, and match the sticks against the cows as they come back.
In the morning, as each cow leaves, drop one stick into a pot. Nothing is counted and nothing is named — one cow, one stick.
In the evening, as each cow returns, take one stick out of the pot.
If the pot is empty when the last cow is in, every cow is back. If sticks are left over, that leftover collection is the number of missing cows.
Why it happens: the sticks and the cows have been put into a one-to-one mapping — no two cows share a stick and no stick is left unused. Two collections that can be matched one to one have the same size, whether or not anyone can name that size. This is the whole mechanism of counting, and it works without a single number word.
Q2.
Do we have fewer cows than our neighbour?
Answer
Yes/no can be settled without knowing how many either of us has.
Make our pile of sticks — one per cow. Let the neighbour make his.
Now pair the piles off: take one stick from ours and one from his, again and again.
Whichever pile runs out first belongs to the person with fewer cows. If both run out together, the herds are equal.
Why it happens: pairing off tests whether a one-to-one mapping between the two herds is possible. If ours can be matched into his with some of his left over, ours is the smaller herd. Comparing sizes is therefore an easier problem than measuring them — you can answer “fewer?” long before you can answer “how many?”.
Q3.
If there are fewer, how many more cows would we need so that we have the same number of cows as our neighbour?
Answer
The sticks left over in the neighbour’s pile after the pairing are the answer.
our sticks ↔ his sticks (pair them off) ours run out first the sticks still lying in his pile = the number of extra cows we need
Carry that leftover collection of sticks to the market, and buy one cow for each stick. Now the two herds can be matched exactly.
Why it happens: this is subtraction, done entirely by matching. The unmatched part of the bigger collection is the difference between the two collections — the same idea we later write as 15 − 11 = 4, but here the answer is not a word or a numeral, it is a heap of sticks that can be used directly.