Large Numbers Around Us | FIO

Question 7

If you notice, the expressions for the least button clicks also give the Indian place value notation of the numbers. Think about why this is so.

Solution

Understand the Question

In a base- $10$ place value system, each digit represents a count of a specific power of $10$ (ones, tens, hundreds, thousands, lakhs, etc.).
To form any number with the least button clicks using buttons of values $1, 10, 100, 1{,}000, \dots$ , we greedily select the largest possible place value denomination first.
This greedy breakdown matches the expanded form of the number, which is precisely how the Indian place value notation names and structures numbers.

Step 1 · Expanded Form of a Number

Our number system is based on place values where each digit's value depends on its position.

For example, writing $543$ in expanded form:

\begin{aligned} 543 &= 5 \times 100 + 4 \times 10 + 3 \times 1 \\ &= 500 + 40 + 3 \end{aligned}

Step 2 · Minimising Button Clicks

To generate a number using denomination buttons ( $1, 10, 100, 1{,}000, \dots$ ) with the fewest clicks, we use the largest available place value units first:

For $543$ :

Number of $100\text{s}$ button clicks $= 5$
Number of $10\text{s}$ button clicks $= 4$
Number of $1\text{s}$ button clicks $= 3$

$\text{Total clicks} = 5 + 4 + 3 = 12 \text{ clicks}$

The expression for clicks is $5 \times 100 + 4 \times 10 + 3 \times 1$ .

Step 3 · Relation to Indian Place Value Notation

The Indian place value system names numbers by breaking them into their period values (ones, thousands, lakhs, crores):

$5{,}43{,}210 = 5 \times 1{,}00{,}000 + 4 \times 10{,}000 + 3 \times 1{,}000 + 2 \times 100 + 1 \times 10 + 0 \times 1$

Since both the least-click method and the Indian place value system decompose a number into the largest possible powers of $10$ , the resulting expressions are identical.

Answer

Both the least button clicks method and the Indian place value notation break down a number into its expanded form (sum of digits multiplied by their respective powers of $10$ ). Minimising button clicks requires using the largest place value denominations first, which corresponds directly to the place value representation of the number.

Common Mistakes

Confusing Face Value with Place Value: Treating a digit only as its face value (e.g., $5$ ) rather than its place value ( $5 \times 100 = 500$ ).
Suboptimal Clicks: Not using the largest denomination first (e.g., clicking the $1$ button $543$ times instead of using $100\text{s}$ , $10\text{s}$ , and $1\text{s}$ ).

More questions in FIO

According to the 2011 Census, the population of the town of Chintamani was about 75,000. How much less than one lakh is 75,000?

The estimated population of Chintamani in the year 2024 is 1,06,000. How much more than one lakh is 1,06,000?

By how much did the population of Chintamani increase from 2011 to 2024?

For each number given below, write expressions for at least two different ways to obtain the number through button clicks. Think like Chitti and be creative.

(a) 8300

(b) 40629

(d) 66666

(e) 367813

For the numbers in the previous exercise, find out how to get each number by making the smallest number of button clicks and write the expression.

Do you see any connection between each number and the corresponding smallest number of button clicks?

If you notice, the expressions for the least button clicks also give the Indian place value notation of the numbers. Think about why this is so.

Read the following numbers in Indian place value notation and write their number names in both the Indian and American systems:

(a) 4050678

(b) 48121620

(d) 246813579

(e) 345000543

(f) 1020304050

Write the following numbers in Indian place value notation:

(a) One crore one lakh one thousand ten

(b) One billion one million one thousand one

(d) Nine billion eighty million seven hundred thousand six hundred

Q10

Compare and write ' $<$ ', ' $>$ ' or ' $=$ ':

(a) 30 thousand ______ 3 lakhs

(b) 500 lakhs ______ 5 million

(d) 640 crore ______ 60 billion

Q11

Find quick ways to calculate these products:

(a) $2 \times 1768 \times 50$

(b) $72 \times 125$ [Hint: $125 = \dfrac{1000}{8}$ ]

Q12

Calculate these products quickly.

(a) $25 \times 12 =$ _______

(b) $25 \times 240 =$ _______

(d) $2500 \times 12 =$ _______

(e) _______ $\times$ _______ $= 120000000$

Q13

Using all digits from $0$ to $9$ exactly once (the first digit cannot be $0$ ) to create a $10$ -digit number, write the —

(a) Largest multiple of $5$

(b) Smallest even number

Q14

The number 10,30,285 in words is Ten lakhs thirty thousand two hundred eighty five, which has 42 letters. Give a 7-digit number name which has the maximum number of letters.

Q15

Write a 9-digit number where exchanging any two digits results in a bigger number. How many such numbers exist?

Q16

Strike out 10 digits from the number 12345123451234512345 so that the remaining number is as large as possible.

Q17

The words 'zero' and 'one' share letters 'e' and 'o'. The words 'one' and 'two' share a letter 'o', and the words 'two' and 'three' also share a letter 't'. How far do you have to count to find two consecutive numbers which do not share an English letter in common?

Q18

Suppose you write down all the numbers $1, 2, 3, 4, \dots, 9, 10, 11, \dots$ The tenth digit you write is '1' and the eleventh digit is '0', as part of the number 10.

(a) What would the 1000th digit be? At which number would it occur?

(b) What number would contain the millionth digit?

Q19

A calculator has only '+10,000' and '+100' buttons. Write an expression describing the number of button clicks to be made for the following numbers:

(a) 20,800

(b) 92,100

(d) 65,30,000

(e) 70,25,700

Q20

How many lakhs make a billion?

Q21

You are given two sets of number cards numbered from $1$ - $9$ . Place a number card in each box below to get the

(a) largest possible sum

(b) smallest possible difference of the two resulting numbers.

Q22

You are given some number cards; 4000, 13000, 3000, 70000, 150000, 20, 5. Using the cards get as close as you can to the numbers below using any operation you want. Each card can be used only once for making a particular number.

(a) 1,10,000: Closest I could make is $4000 \times (20 + 5) + 13000 = 1,13,000$

(b) 2,00,000:

(d) 12,45,000:

(e) 20,90,800:

Q23

Find out how many coins should be stacked to match the height of the Statue of Unity. Assume each coin is 1 mm thick.

Q24

Grey-headed albatrosses have a roughly 7-feet wide wingspan. They are known to migrate across several oceans. Albatrosses can cover about 900 – 1000 km in a day. One of the longest single trips recorded is about 12,000 km. How many days would such a trip take to cross the Pacific Ocean approximately?

Q25

A bar-tailed godwit holds the record for the longest recorded non-stop flight. It travelled 13,560 km from Alaska to Australia without stopping. Its journey started on 13 October 2022 and continued for about 11 days. Find out the approximate distance it covered every day. Find out the approximate distance it covered every hour.

Q26

Bald eagles are known to fly as high as $4500\text{ m}$ – $6000\text{ m}$ above the ground level. Mount Everest is about $8850\text{ m}$ high. Aeroplanes can fly as high as $10{,}000\text{ m}$ – $12{,}800\text{ m}$ . How many times bigger are these heights compared to Somu’s building?