Large Numbers Around Us | IT

Question 17

Creative Chitti has some questions for you —

(a) You have to make exactly 30 button presses. What is the largest 3-digit number you can make? What is the smallest 3-digit number you can make?

(b) 997 can be made using 25 clicks. Can you make 997 with a different number of clicks?

Create questions like these and challenge your classmates.

Solution

We need to find numbers using specific button presses.

Step 1 — Understanding the calculator

The calculator has buttons to add values. We can press +1, +10, or +100. We want to make 3-digit numbers. These numbers are between 100 and 999. The total number of button presses must be 30.

Let $n_1$ be the number of +1 presses. Let $n_{10}$ be the number of +10 presses. Let $n_{100}$ be the number of +100 presses.

The total clicks are $n_1 + n_{10} + n_{100}$ . The total value is $n_1$ plus $10 n_{10}$ plus $100 n_{100}$ .

Diagram 1

Step 2 — Finding the largest 3-digit number

We need to make exactly 30 button presses. We want the largest possible 3-digit number. To make a large number, we use +100 buttons first. Let us try to use $n_{100}$ clicks for +100. Let us try $n_{100} = 9$ . This adds 900 to the total.

The clicks used are 9. The remaining clicks are $30 - 9$ . $30 - 9 = 21$ The value so far is 900. We need to add a value less than 100. We have 21 clicks remaining for +10 and +1 buttons. Let $n_{10}$ be the number of +10 presses. Let $n_1$ be the number of +1 presses. The sum of these clicks is 21. $n_{10} + n_1 = 21$ The value from these clicks is $10 n_{10}$ plus $n_1$ . We want to maximize this value, but keep it below 100. To maximize, we use +10 buttons as much as possible. We know $n_1 = 21 - n_{10}$ . So, the value is $10 \times n_{10} + (21 - n_{10})$ . This simplifies to $9 \times n_{10} + 21$ . This value must be less than 100. $9 \times n_{10} + 21 \le 99$ Subtract 21 from both sides. $9 \times n_{10} \le 99 - 21$ $9 \times n_{10} \le 78$ Divide by 9. $n_{10} \le \frac{78}{9}$ $n_{10} \le 8.66...$ The largest whole number for $n_{10}$ is 8. So, we press +10 eight times. The number of +1 presses is $21 - 8$ . $n_1 = 21 - 8 = 13$ The value from +10 and +1 buttons is $80 + 13$ . $80 + 13 = 93$ The total value is $900 + 93$ . $900 + 93 = 993$ The total clicks are $9 + 8 + 13$ . $9 + 8 + 13 = 30$ This is a 3-digit number. This is the largest 3-digit number.

$\boxed{\text{Largest 3-digit number: } 993}$

Step 3 — Finding the smallest 3-digit number

We need to make exactly 30 button presses. We want the smallest possible 3-digit number. A 3-digit number must be at least 100. So, we must press the +100 button at least once. To make the number smallest, we use +100 buttons least. Let us use $n_{100} = 1$ . This adds 100 to the total.

The clicks used are 1. The remaining clicks are $30 - 1$ . $30 - 1 = 29$ The value so far is 100. We need to add the smallest possible value. We have 29 clicks remaining for +10 and +1 buttons. Let $n_{10}$ be the number of +10 presses. Let $n_1$ be the number of +1 presses. The sum of these clicks is 29. $n_{10} + n_1 = 29$ The value from these clicks is $10 n_{10}$ plus $n_1$ . To minimize this value, we use +1 buttons most. This means we use +10 buttons as little as possible. The smallest possible number for $n_{10}$ is 0. So, we press +10 zero times. The number of +1 presses is $29 - 0$ . $n_1 = 29 - 0 = 29$ The value from +10 and +1 buttons is $0 + 29$ . $0 + 29 = 29$ The total value is $100 + 29$ . $100 + 29 = 129$ The total clicks are $1 + 0 + 29$ . $1 + 0 + 29 = 30$ This is a 3-digit number. This is the smallest 3-digit number.

$\boxed{\text{Smallest 3-digit number: } 129}$

Step 4 — Making 997 with a different number of clicks

We are told that 997 can be made with 25 clicks. Let us verify this first. To make 997, we can use $n_{100} = 9$ . This gives 900. Clicks used = 9. Remaining clicks = $25 - 9 = 16$ . Remaining value needed = $997 - 900 = 97$ . We need to make 97 with 16 clicks. We use +10 and +1 buttons. Let $n_{10}$ be +10 presses and $n_1$ be +1 presses. $n_{10} + n_1 = 16$ $10 \times n_{10} + 1 \times n_1 = 97$ Substitute $n_1 = 16 - n_{10}$ into the value equation. $10 \times n_{10} + (16 - n_{10}) = 97$ $9 \times n_{10} + 16 = 97$ $9 \times n_{10} = 97 - 16$ $9 \times n_{10} = 81$ $n_{10} = \frac{81}{9}$ $n_{10} = 9$ Then $n_1 = 16 - 9$ . $n_1 = 7$ So, $n_{100} = 9$ , $n_{10} = 9$ , $n_1 = 7$ . Total clicks = $9 + 9 + 7$ . $9 + 9 + 7 = 25$ Total value = $9 \times 100 + 9 \times 10 + 7 \times 1$ . $900 + 90 + 7 = 997$ So, 997 can indeed be made with 25 clicks.

Can 997 be made with a different number of clicks? Let us keep $n_{100} = 9$ . We still need to make 97 using +10 and +1 buttons. Let $C_{\text{rem}}$ be the remaining clicks. Let $n_{10}$ be +10 presses and $n_1$ be +1 presses. $n_{10} + n_1 = C_{\text{rem}}$ $10 \times n_{10} + 1 \times n_1 = 97$ Substitute $n_1 = C_{\text{rem}} - n_{10}$ into the value equation. $10 \times n_{10} + (C_{\text{rem}} - n_{10}) = 97$ $9 \times n_{10} + C_{\text{rem}} = 97$ So, $C_{\text{rem}} = 97 - 9 \times n_{10}$ . The total clicks $C = n_{100} + C_{\text{rem}}$ . $C = 9 + (97 - 9 \times n_{10})$ $C = 106 - 9 \times n_{10}$ We need $n_1 \ge 0$ . So, $C_{\text{rem}} - n_{10} \ge 0$ . $97 - 9 \times n_{10} - n_{10} \ge 0$ . $97 - 10 \times n_{10} \ge 0$ . $10 \times n_{10} \le 97$ . $n_{10} \le 9.7$ . So $n_{10}$ can be any integer from 0 to 9. We already found $n_{10} = 9$ gives $C = 25$ clicks. Let us try $n_{10} = 8$ . This means we press +10 eight times. The number of clicks for 997 would be $106 - 9 \times 8$ . $C = 106 - 72$ $C = 34$ This is a different number of clicks than 25. Let us check the values. $n_{100} = 9$ . Value = 900. $n_{10} = 8$ . Value = 80. $n_1 = C_{\text{rem}} - n_{10} = (97 - 9 \times 8) - 8 = 25 - 8 = 17$ . Value = 17. Total value = $900 + 80 + 17 = 997$ . Total clicks = $9 + 8 + 17 = 34$ . Yes, 997 can be made with 34 clicks.

$\boxed{\text{Yes, 997 can be made with a different number of clicks.}}$

Answer

(a) Largest 3-digit number: 993 (a) Smallest 3-digit number: 129 (b) Yes, 997 can be made with a different number of clicks. For example, it can be made with 34 clicks.

More questions in IT

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What do you think? Guess.

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Q10

Write each of the numbers given below in words:

(a) 3,00,600

(b) 5,04,085

(d) 70,53,138

Q11

Write the corresponding number in the Indian place value system for each of the following:

(a) One lakh twenty three thousand four hundred and fifty six

(b) Four lakh seven thousand seven hundred and four

(d) Ten lakhs two hundred and thirty five

Q12

The Thoughtful Thousands only has a $+1000$ button. How many times should it be pressed to show:

(a) $4000$

(b) $7000$

Q13

The Tedious Tens only has a $+10$ button. How many times should it be pressed to show:

(a) Five hundred?

(b) $780$ ?

(d) $3700$ ?

(e) $10,000$ ?

(f) One lakh?

(g) ________ ? $435$ times

Q14

The Handy Hundreds only has a $+100$ button. How many times should it be pressed to show:

(a) Four hundred? __________ times

(b) 3,700? __________

(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.

Q15

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. “How so?”, you might ask. 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.

Q16

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

Q. Find a different way to get 5072 and write an expression for the same.

Q17

Creative Chitti has some questions for you —

(a) You have to make exactly 30 button presses. What is the largest 3-digit number you can make? What is the smallest 3-digit number you can make?

(b) 997 can be made using 25 clicks. Can you make 997 with a different number of clicks?

Create questions like these and challenge your classmates.

Q18

How can we get the numbers (a) 5072, (b) 8300 using as few button clicks as possible?

Find out which buttons should be clicked and how many times to get the desired numbers given in the table. The aim is to click as few buttons as possible.

Here is one way to get the number 5072. This method uses 23 button clicks in total. Is there another way to get 5072 using less than 23 button clicks? Write the expression for the same.

Q19

How many zeros does a thousand lakh have?

Q20

How many zeros does a hundred thousand have?

Q21

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Q22

Think and share situations where it is appropriate to

(a) round up,

(b) round down,

(d) when exact numbers are needed.

Q24

I have a number for which all five nearest neighbours are 5,00,00,000. What could the number be? How many such numbers are there?

Q25

Roxie and Estu are estimating the values of simple expressions.

4,63,128 + 4,19,682,

Roxie: “The sum is near 8,00,000 and is more than 8,00,000.” Estu: “The sum is near 9,00,000 and is less than 9,00,000.”

(a) Are these estimates correct? Whose estimate is closer to the sum?

(b) Will the sum be greater than 8,50,000 or less than 8,50,000? Why do you think so?

(d) Exact value of 4,63,128 + 4,19,682 = ________

Q26

14,63,128 – 4,90,020

Roxie: “The difference is near 10,00,000 and is less than 10,00,000.” Estu: “The difference is near 9,00,000 and is more than 9,00,000.”

(a) Are these estimates correct? Whose estimate is closer to the difference?

(b) Will the difference be greater than 9,50,000 or less than 9,50,000? Why do you think so?

(d) Exact value of 14,63,128 – 4,90,020 = ________

Q27

From the information given in the table, answer the following questions by approximation:

(i) What is your general observation about this data? Share it with the class.

(ii) What is an appropriate title for the above table?

(iii) How much is the population of Pune in 2011? Approximately, by how much has it increased compared to 2001?

(iv) Which city's population increased the most between 2001 and 2011?

(v) Are there cities whose population has almost doubled? Which are they?

(vi) By what number should we multiply Patna's population to get a number/population close to that of Mumbai?

Q33

Using the meaning of multiplication and division, can you explain why multiplying by 5 is the same as dividing by 2 and multiplying by 10?

Q34

How Long is the Product?

In each of the following boxes, the multiplications produce interesting patterns. Evaluate them to find the pattern. Extend the multiplications based on the observed pattern.

Q35

Observe the number of digits in the two numbers being multiplied and their product in each case. Is there any connection between the numbers being multiplied and the number of digits in their product?

Q36

Roxie says that the product of two 2-digit numbers can only be a 3- or a 4-digit number. Is she correct?

Q37

Should we try all possible multiplications with 2-digit numbers to tell whether Roxie’s claim is true? Or is there a better way to find out?

Q38

Can multiplying a 3-digit number with another 3-digit number give a 4-digit number?

Q39

Can multiplying a 4-digit number with a 2-digit number give a 5-digit number?

Q40

Observe the multiplication statements below. Do you notice any patterns? See if this pattern extends for other numbers as well.

Q41

Calculate the product to uncover the fact:

$1250 \times 380$

Read the resulting number in both Indian and American naming systems.

Also consider the questions:

How many years did he live to compose so many songs?
At what age did he start composing songs?
If he composed 4,75,000 songs, how many songs per year did he have to compose?

Q42

Calculate the product to uncover the fact:

$2100 \times 70,000$

Read the resulting number in both Indian and American naming systems.

Q43

Calculate the quotient to uncover the fact:

$10,50,00,000 \div 700$

Q44

Calculate the quotient to uncover the fact:

$52,00,00,000,000 \div 130$

Q45

The RMS Titanic ship carried about 2500 passengers. Can the population of Mumbai fit into 5000 such ships?

Q46

Inspired by this strange question, Roxie wondered, “If I could travel 100 kilometers every day, could I reach the Moon in 10 years?” (The distance between the Earth and the Moon is 3,84,400 km.)

How far would she have travelled in a year?
How far would she have travelled in 10 years?
Is it not easier to perform these calculations in stages? You can use this method for all large calculations.

Q47

Find out if you can reach the Sun in a lifetime, if you travel 1000 kilometers every day. (You had written down the distance between the Earth and the Sun in a previous exercise.)

Q48

Make necessary reasonable assumptions and answer the questions below:

(a) If a single sheet of paper weighs 5 grams, could you lift one lakh sheets of paper together at the same time?

(b) If 250 babies are born every minute across the world, will a million babies be born in a day?

Q49

Think and create more such fun questions and share them with your class.