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NCERT Solutions Ganita Prakash (Part 1) Chapter 1 –61.2 Land of Tens — Land of Tens

Book page 5 Updated on2026-09-05

Q1.
The Thoughtful Thousands only has a +1000 button. How many times should it be pressed to show: (a) Three thousand? 3 times (b) 10,000? (c) Fifty three thousand? (d) 90,000? (e) One Lakh? (f) ______? 153 times (g) How many thousands are required to make one lakh?
Answer

Every press adds 1000, so the number of presses is simply the number divided by 1000.

TargetWorkingPresses
(a) Three thousand3000 ÷ 10003 times
(b) 10,00010000 ÷ 100010 times
(c) Fifty three thousand53000 ÷ 100053 times
(d) 90,00090000 ÷ 100090 times
(e) One lakh1,00,000 ÷ 1000100 times
(f) 1,53,000153 × 1000153 times

(g) 100 thousands are required to make one lakh.

1 lakh = 1,00,000
1,00,000 ÷ 1,000 = 100
So 100 × 1000 = 1,00,000 ✓
Why it happens: Thoughtful Thousands can only land on multiples of 1000. That is why every answer above is a whole number — and why this calculator can never show a number like 3,700 or 97,600.
Q2.
The Tedious Tens only has a +10 button. How many times should it be pressed to show: (a) Five hundred? (b) 780? (c) 1000? (d) 3700? (e) 10,000? (f) One lakh? (g) ______? 435 times
Answer

Divide each number by 10.

TargetWorkingPresses
(a) Five hundred500 ÷ 1050 times
(b) 780780 ÷ 1078 times
(c) 10001000 ÷ 10100 times
(d) 37003700 ÷ 10370 times
(e) 10,00010000 ÷ 101000 times
(f) One lakh1,00,000 ÷ 1010,000 times
(g) 4,350435 × 10435 times
Why it happens: Dividing by 10 just drops the last zero. That is why the count of presses looks like the original number with one zero removed — 3700 becomes 370, and 1,00,000 becomes 10,000.
Tip: Tedious Tens really is tedious — reaching one lakh needs ten thousand presses. At one press per second that is nearly 3 hours of pressing!
Q3.
The Handy Hundreds only has a +100 button. How many times should it be pressed to show: (a) Four hundred? (b) 3,700? (c) 10,000? (d) Fifty three thousand? (e) 90,000? (f) 97,600? (g) 1,00,000? (h) ______? 582 times (i) How many hundreds are required to make ten thousand? (j) How many hundreds are required to make one lakh? (k) Handy Hundreds says, “There are some numbers which Tedious Tens and Thoughtful Thousands can’t show but I can.” Is this statement true? Think and explore.
Answer

Divide each number by 100.

TargetWorkingPresses
(a) Four hundred400 ÷ 1004 times
(b) 3,7003700 ÷ 10037 times
(c) 10,00010000 ÷ 100100 times
(d) Fifty three thousand53000 ÷ 100530 times
(e) 90,00090000 ÷ 100900 times
(f) 97,60097600 ÷ 100976 times
(g) 1,00,000100000 ÷ 1001000 times
(h) 58,200582 × 100582 times

(i) 10,000 ÷ 100 = 100 hundreds make ten thousand.
(j) 1,00,000 ÷ 100 = 1000 hundreds make one lakh.

(k) No, the statement is not true.

Handy Hundreds can show only multiples of 100.
Every multiple of 100 is also a multiple of 10.
e.g. 3,700 = 37 × 100 = 370 × 10
So Tedious Tens can show every number Handy Hundreds can show.
Why it happens: Handy Hundreds does beat Thoughtful Thousands — 3,700 and 97,600 are impossible for the +1000 button. But it can never beat Tedious Tens, because 100 is itself ten tens. For the claim to be true, a number would have to be a multiple of 100 but not a multiple of 10, and no such number exists.
Q4.
Creative Chitti is a different kind of calculator. It has the following buttons: +1, +10, +100, +1000, +10000, +100000 and +1000000. It always has multiple ways of doing things. To get the number 321, it presses +10 thirty two times and +1 once. Will it get 321? Alternatively, it can press +100 two times and +10 twelve times and +1 once.
Answer

Yes, both ways give exactly 321.

Way 1: (32 × 10) + (1 × 1)
= 320 + 1 = 321 ✓ (33 clicks)

Way 2: (2 × 100) + (12 × 10) + (1 × 1)
= 200 + 120 + 1 = 321 ✓ (15 clicks)
Why it happens: A button press just adds a fixed amount, and addition can be done in any order. So any combination whose total is 321 works. Chitti is showing that a number can be broken up in many different ways — 321 = 320 + 1 = 200 + 120 + 1 = 300 + 21 = …
Try This: Find a third way. (3 × 100) + (2 × 10) + (1 × 1) = 321 uses only 6 clicks — the fewest possible, because 3 + 2 + 1 = 6 is the digit sum.
Q5.
Two of the many different ways to get 5072 are shown below. These two ways can be expressed as: (a) (50 × 100) + (7 × 10) + (2 × 1) = 5072 (b) (3 × 1000) + (20 × 100) + (72 × 1) = 5072. Find a different way to get 5072 and write an expression for the same.
Answer

Here are three fresh ways, each different from the two given.

Way 1: (5 × 1000) + (7 × 10) + (2 × 1)
= 5000 + 70 + 2 = 5072 (14 clicks)

Way 2: (4 × 1000) + (10 × 100) + (7 × 10) + (2 × 1)
= 4000 + 1000 + 70 + 2 = 5072 (23 clicks)

Way 3: (5 × 1000) + (72 × 1)
= 5000 + 72 = 5072 (77 clicks)
Why it happens: Every way is just a different way of splitting 5072 into pieces worth 1, 10, 100, 1000 … Since 1000 = 10 × 100, you can always trade one press of +1000 for ten presses of +100, or one press of +100 for ten presses of +10. That trade never changes the total, only the number of clicks.
Tip: Way 1 uses the digits of 5072 itself (5 thousands, 0 hundreds, 7 tens, 2 ones). That is always the cheapest way — you will prove it on page 7.
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