Exact and Approximate Values
What do you think of this conversation? Have you read or heard such headlines or statements? Very often, exact numbers are not required and just an approximation is sufficient. For example, according to the 2011 census, the population of Chintamani town is 76,068. Instead, saying that the population is about 75,000 is enough to give an idea of how big the quantity is.
There are situations where it makes sense to round up a number (rounding up is when the approximated number is more than the actual number). For example, if a school has 732 people including students, teachers and staff: the principal might order 750 sweets instead of 700 sweets.
There are situations where it is better to round down (rounding down is when the approximated number is less than the actual number). For example, if the cost of an item is ₹470, the shopkeeper may say that the cost is around ₹450 instead of saying it is around ₹500.
Think and share situations where it is appropriate to (a) round up,
(b) round down, (c) either rounding up or rounding down is okay and (d) when exact numbers are needed.
Nearest Neighbours
With large numbers it is useful to know the nearest thousand, lakh or crore. For example, the nearest neighbours of the number 6,72,85,183 are shown in the table below.
| Nearest Neighbours | Nearest Number |
|---|---|
| Nearest thousand | 6,72,85,000 |
| Nearest ten thousand | 6,72,90,000 |
| Nearest lakh | 6,73,00,000 |
| Nearest ten lakh | 6,70,00,000 |
| Nearest crore | 7,00,00,000 |
Similarly, write the five nearest neighbours for these numbers: (a) 3,87,69,957 (b) 29,05,32,481
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?
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? (c) Will the sum be greater than 8,83,128 or less than 8,83,128? Why do you think so? (d) Exact value of 4,63,128 + 4,19,682 = ___________
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? (c) Will the difference be greater than 9,63,128 or less than 9,63,128? Why do you think so? (d) Exact value of 14,63,128 – 4,90,020 = __________
Populations of Cities
Observe the populations of some Indian cities in the table below.
| Rank | City | Population (2011) | Population (2001) |
|---|---|---|---|
| 1 | Mumbai | 1,24,42,373 | 1,19,78,450 |
| 2 | New Delhi | 1,10,07,835 | 98,79,172 |
| 3 | Bengaluru | 84,25,970 | 43,01,326 |
| 4 | Hyderabad | 68,09,970 | 36,37,483 |
| 5 | Ahmedabad | 55,70,585 | 35,20,085 |
| 6 | Chennai | 46,81,087 | 43,43,645 |
| 7 | Kolkata | 44,86,679 | 45,72,876 |
| 8 | Surat | 44,67,797 | 24,33,835 |
| 9 | Vadodara | 35,52,371 | 16,90,000 |
| 10 | Pune | 31,15,431 | 25,38,473 |
| 11 | Jaipur | 30,46,163 | 23,22,575 |
| 12 | Lucknow | 28,15,601 | 21,85,927 |
| 13 | Kanpur | 27,67,031 | 25,51,337 |
| 14 | Nagpur | 24,05,665 | 20,52,066 |
| 15 | Indore | 19,60,631 | 14,74,968 |
| 16 | Thane | 18,18,872 | 12,62,551 |
| 17 | Bhopal | 17,98,218 | 14,37,354 |
| 18 | Visakhapatnam | 17,28,128 | 13,45,938 |
| 19 | Pimpri-Chinchwad | 17,27,692 | 10,12,472 |
| 20 | Patna | 16,84,222 | 13,66,444 |
From the information given in the table, answer the following questions by approximation:
- What is your general observation about this data? Share it with the class.
- What is an appropriate title for the above table?
- How much is the population of Pune in 2011? Approximately, by how much has it increased compared to 2001?
- Which city’s population increased the most between 2001 and 2011?
- Are there cities whose population has almost doubled? Which are they?
- By what number should we multiply Patna’s population to get a number/population close to that of Mumbai?