Questions & Answers: "Large Numbers Around Us"

First, convert the given number from the Indian System to digits: Nine crore ninety-nine lakh ninety-nine thousand nine hundred ninety-nine is 9,99,99,999. In the International System, numbers are grouped in threes from the right (ones, thousands, millions, billions). So, 9,99,99,999 becomes 99,999,999. This reads as Ninety-nine million nine hundred ninety-nine thousand nine hundred ninety-nine. Option A correctly represents this, as 9 crore is 90 million, and 99 lakh is 9.9 million, summing up to 99.99 million approximately.

To compare large numbers, we start comparing digits from the leftmost place. All numbers have 8 digits.

• City A: 2,34,56,789
• City B: 2,35,46,789
• City C: 2,34,57,689
• City D: 2,35,47,689

All numbers start with '2' in the crores place. The next digit is '3' in the ten lakhs place for all. Now compare the 'lakhs' place (third digit from left): Cities A and C have '4', while Cities B and D have '5'. Therefore, Cities B and D are larger than A and C. Now, compare Cities B (2,35,46,789) and D (2,35,47,689). The 'ten thousands' place: City B has '4' and City D has '4'. The 'thousands' place: City B has '6' and City D has '7'. Since 7 is greater than 6, City D has the largest population.

To find the total number of toys produced in a month, multiply the number of toys produced per day by the number of working days: Total toys = Number of toys per day $\times$ Number of working days Total toys = $15,250 \times 25$ Performing the multiplication: $15250 \times 25 = 381250$. So, 3,81,250 toys are produced in that month. Option A is the correct answer. The other options are incorrect as they result from miscalculation or rounding.

In the Indian System of Numeration, 1 lakh is equal to 1,00,000 (one hundred thousand). In the International System of Numeration, 1 million is equal to 1,000,000 (one thousand thousands). To find how many lakhs make one million, we can divide 1 million by 1 lakh: $\frac{1,000,000}{1,00,000} = 10$ Therefore, 10 lakhs make one million. Option B is correct. Options A, C, and D represent incorrect conversion factors.

Let's analyze the place values in the Indian System for the number 78,92,10,345:

• The digit '8' is in the Crores place (the 8th position from the right). So, its place value is $8 \times 1,00,00,000 = 8,00,00,000$ (which is 80 million in the International System).
• The digit '1' is in the Lakhs place (the 6th position from the right). So, its place value is $1 \times 1,00,000 = 1,00,000$ (which is 100 thousand in the International System).

Comparing these with the options, Option A correctly states the place values: Place value of '8' is 80,000,000 and '1' is 100,000. Option B incorrectly identifies the place value of '8' as 8 million. Option C incorrectly identifies the place value of '1' as ten thousand. Option D misidentifies both place values, placing '8' in the hundred crores and '1' in ten lakhs place.

In the Indian system, numbers are grouped from the right in sets of 3, then 2, then 2, and so on. So, 9,75,00,403 is read as 'Nine crore seventy-five lakh four hundred three'. Option B is incorrect as it uses the International system. Options C and D misinterpret the place values for lakhs and crores.

One lakh is equal to 1,00,000. To find out how many thousands are in 1,00,000, we divide 1,00,000 by 1,000, which gives 100. Therefore, 5 lakhs will have $5 \times 100 = 500$ thousands.

In the International system, the place values from right to left are Ones, Tens, Hundreds, Thousands, Ten Thousands, Hundred Thousands, Millions, Ten Millions, etc. In the number 78,921,065, the digit '5' is in the Ones place, and the digit '6' is in the Tens place.

To form the smallest number, arrange the digits in ascending order. However, a number cannot start with zero if it is to be a 6-digit number. So, place the smallest non-zero digit first (1), followed by 0, and then the remaining digits in ascending order (2, 3, 5, 7). This results in 102357. Option B is incorrect because 012357 is essentially a 5-digit number. Options C and D have incorrect ordering of digits.

To find the increase in population, subtract the population in 2001 from the population in 2011. $2,54,83,900 - 2,35,47,100 = 19,36,800$. The other options are incorrect calculations.

In the Indian System of Numeration, 'Five crore twenty-three lakh forty-five thousand six hundred seventy-eight' is written as 5,23,45,678. To convert this to the International System, we consider the place values. 1 crore is equal to 10 million, and 1 lakh is equal to 100 thousand. So, 5 crore = 50 million, and 23 lakh = 2.3 million. Combining these, the number becomes 50 million + 2.3 million + 45 thousand + 678, which is 52,345,678. In the International System, numbers are grouped in threes from the right (e.g., thousands, millions, billions). Hence, 52,345,678 is the correct representation.

To find the time taken, we use the formula: $\text{Time} = \frac{\text{Distance}}{\text{Speed}}$.\nGiven Distance $= 384,400 \text{ km}$ and Speed $= 4000 \text{ km/hour}$.\n$\text{Time} = \frac{384,400 \text{ km}}{4000 \text{ km/hour}} = \frac{3844}{40} \text{ hours}$.\nDividing $3844$ by $40$ gives $96.1$. Therefore, it would take approximately $96.1 \text{ hours}$.

First, write the given number in numerical form according to the International System of Numeration: 84,532,197. Now, let's identify the place value of each digit from the right:\n7 is in the Ones place.\n9 is in the Tens place.\n1 is in the Hundreds place.\n2 is in the Thousands place.\n3 is in the Ten Thousands place.\n5 is in the Hundred Thousands place.\n4 is in the Millions place.\n8 is in the Ten Millions place.\nTherefore, the digit in the 'tens of thousands' place is 3.

To arrange the numbers in ascending order, we compare their values.\nP = 12,345,678\nR = 31,536,000\nQ = 299,792,458\nComparing the numbers:\n12,345,678 (P) is the smallest.\n31,536,000 (R) is larger than P but smaller than Q.\n299,792,458 (Q) is the largest.\nSo, the ascending order is P, R, Q.

The given digits are 9, 0, 5, 2, 7, 1, 4.\nTo form the largest 7-digit number, arrange the digits in descending order: 9, 7, 5, 4, 2, 1, 0. So, the largest number is 9,754,210.\nTo form the smallest 7-digit number, arrange the digits in ascending order. However, 0 cannot be the first digit. So, we place the smallest non-zero digit first, which is 1, followed by 0, and then the remaining digits in ascending order: 1, 0, 2, 4, 5, 7, 9. So, the smallest number is 1,024,579.\nNow, calculate the difference between the largest and smallest numbers:\nDifference = 9,754,210 - 1,024,579 = 8,729,631.

In the Indian system, numbers are grouped into periods like ones, thousands, lakhs, and crores. The number 7,89,21,092 is read by first placing commas according to the Indian system (after 3 digits from the right, then every 2 digits). So, it's 7 (crore), 89 (lakh), 21 (thousand), 092 (ninety-two). Therefore, it is 'Seven crore eighty-nine lakh twenty-one thousand ninety-two'. Option B describes the International system (78,921,092). Options C and D misrepresent the place values.

To compare large numbers, we first compare the number of digits. All cities have 7-digit populations. Next, we compare the digits from the leftmost place value. All start with '32 lakh'. Moving to the thousands place, City A and City C have '54 thousand', while City B and City D have '45 thousand'. Thus, City A and C are larger than B and D. Now compare City A (32,54,890) and City C (32,54,980). The lakh and thousand places are the same. In the hundreds place, City A has 8 and City C has 9. Since 9 is greater than 8, City C has the largest population.

To find the number of cartons needed, we divide the total number of toys by the capacity of each carton: $45678 \div 100 = 456.78$. The question asks to round this to the nearest whole number of cartons. To round $456.78$ to the nearest whole number, we look at the digit in the tenths place, which is 7. Since 7 is 5 or greater, we round up the ones digit. Therefore, $456.78$ rounded to the nearest whole number is 457 cartons.

First, calculate the production in February: January production + 15,000 = $1,25,000 + 15,000 = 1,40,000$ parts. Next, calculate the total production for both months: January production + February production = $1,25,000 + 1,40,000 = 2,65,000$ parts. Therefore, the total production is 2,65,000 car parts.

To convert kilometers to meters, we multiply the number of kilometers by 1000, since 1 kilometer is equal to 1000 meters. So, $250 \text{ kilometers} \times 1000 \text{ meters/kilometer} = 2,50,000 \text{ meters}$. The correct conversion is 2,50,000 meters.