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NCERT Solutions Ganita Prakash (Part 1) Chapter 2 –362.5 Did You Ever Wonder? · Linear Growth vs. Exponential Growth — In-text Questions

Book page 35 Updated on2026-09-05

Q1.
How many times can a person circumnavigate (go around the world) the Earth in their lifetime if they walk non-stop? Consider the distance around the Earth as 40,000 km.
Answer

About 18 times, on reasonable assumptions.

Assume walking at 5 km/h for 8 hours a day = 40 km a day
Assume walking for 50 years of a lifetime

Distance in a year = 40 × 365 = 14,600 km
Distance in 50 years = 14,600 × 50 = 7,30,000 km
Number of trips = 7,30,000 ÷ 40,000 ≈ 18 times

In standard form: 7.3 × 105 km walked, against 4 × 104 km per trip.

Why it happens: the answer is startlingly small for a lifetime of walking, and that is the lesson. Walking is linear — the distance grows by a fixed 40 km a day, never faster. Fifty years of it add up to less than twenty trips round the world, while the paper on page 19 passed the Moon in 46 steps. This is the contrast the next page names.
Did you know? Before modern transport, merchants, sages and scholars really did walk thousands of kilometres across deserts, mountains and rivers to reach distant parts of the world.
Q2.
Roxie tells Estu about a science-fiction novel she is reading where they build a ladder to reach the moon, “... I wonder if we actually had a ladder like that, how many steps would it have?”. What do you think? Make an instinctive guess first.
Answer

Guess before calculating. Most first guesses are lakhs; the true answer is about 192 crore steps.

Distance to the Moon = 3,84,400 km
Assume a gap of 20 cm between steps

3,84,400 km = 3,84,400 × 1,00,000 cm = 3.844 × 1010 cm
Number of steps = 3.844 × 1010 ÷ 20
= 1.922 × 109
= 1,92,20,00,000 steps
Why it happens: we judge such questions from ladders we have seen, which have twenty or thirty steps, so a jump to 109 feels absurd. Working in powers of 10 keeps it manageable: cm and km differ by 105, and that single factor is what our intuition drops.
Q3.
Would the number of steps be in thousands, lakhs, crores, or even more?
Answer

In crores — 192 crore and 20 lakh steps, that is 1 billion 922 million steps.

1,92,20,00,000 = 1.922 × 109
1 crore = 107, so this is 192.2 crore
1 arab = 109, so this is about 1.9 arab

A quick way to reach the same conclusion without exact arithmetic: the Moon is roughly 4 × 1010 cm away, and 20 cm is 2 × 101 cm, so the count is about (4 ÷ 2) × 1010–1 = 2 × 109.

Why it happens: this rough check uses only the rules of this chapter — divide the coefficients, subtract the exponents. It gives the right power of 10 in one line, which is exactly what the question asks for. The exact figure adds precision but changes nothing about the answer "crores".
Q4.
We have to find out how many 20 cm make 3,84,400 km.
Answer
3,84,400 km
= 3,84,400 × 1000 m (1 km = 1000 m)
= 3,84,400 × 1000 × 100 cm (1 m = 100 cm)
= 3,84,400 × 105 cm
= 3.844 × 105 × 105 cm = 3.844 × 1010 cm

Number of 20 cm lengths = 3.844 × 1010 ÷ (2 × 101)
= 1.922 × 109
= 1,92,20,00,000
Why it happens: the unit change from km to cm is itself a power of 10 — a factor of 105 — so the whole conversion is one exponent addition, 5 + 5 = 10. Doing it this way removes the commonest mistake in such problems, which is losing or gaining a zero somewhere in the middle.
Q5.
Can you come up with some examples of linear growth and of exponential growth?
Answer

Linear growth adds a fixed amount each step; exponential growth multiplies by a fixed factor each step.

Linear growth (additive)Exponential growth (multiplicative)
Steps up a ladder: + 20 cm each timeFolding paper: × 2 each time
Pocket money saved at ₹50 a weekMoney at compound interest
Filling a tank at a steady rateLotuses in the magical pond: × 2 a day
Distance walked at a steady speedBacteria dividing in two every 20 minutes
Numbering the pages of a bookPasswords as slots are added: × 10 each slot
Height of a stack, one book at a timeDiamonds in "The Stones that Shine": × 3 each level
Linear after n steps: start + n × d
Exponential after n steps: start × rn
Why it happens: the two behave very differently over time. Linear growth crosses any given level once, at a predictable moment. Exponential growth is small for a long while and then overtakes any linear process, however fast, because a fixed multiplier compounds. 46 folds beat 192 crore steps for exactly this reason.
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