Large Numbers Around Us · Lesson 4 of 9
Exact Values, Rounding, and Estimation
“Choose useful approximations and check large calculations with bounds and historical data.”
• Decide when an exact value is essential and when an approximation is useful. • Round non-negative whole numbers to a specified place using a stated halfway rule. • Estimate sums and differences and justify comparisons using bounds. • Interpret a dated population table and compare increases and approximate ratios. • Recognise that different rounding choices answer different practical questions.
Does the last digit matter?
Suppose a book fair reports about one lakh visitors. If one visitor had stayed home, the useful message would usually still be “about one lakh”. But changing a digit in a phone number may connect you to a different person. We must decide whether a numeral is measuring a quantity approximately or identifying something exactly.
An exact value specifies the quantity without intentional rounding. An approximation is a nearby value chosen to describe the quantity more conveniently for a particular purpose.
Bills, recorded payments, and phone numbers usually need their exact digits. For a quick description of a crowd, a nearby round number is often enough. The chapter’s exact 2011 Chintamani population figure is 76,068, while its opening story uses the broad estimate 75,000. That convenient estimate is not the standard rounding of 76,068 to the nearest thousand, which is 76,000.
Problem
There are 732 people at an event. Sweets come in convenient quantities of 50. A separate item costs ₹470. Discuss estimates for both situations.
- 1.Ordering 700 sweets leaves 32 people without one. Ordering 750 is a convenient upward choice that covers everyone.
- 2.The upward choice answers “How many should we provide?”, not “Which hundred is closest?”. The nearest hundred to 732 is 700.
- 3.For mental comparison, ₹470 might be described as roughly ₹450 or ₹500, depending on the purpose and the desired accuracy.
- 4.An estimate of ₹450 understates the actual price by ₹20. It cannot replace ₹470 when paying and may be unsuitable for a budget that must cover the cost.
Rounding to the nearest place
To round to the nearest thousand, find the multiples of 1,000 immediately below and above the number. Choose whichever is closer. For a halfway value, this lesson rounds upward; we use non-negative whole numbers throughout.
The same idea works for ten thousand, lakh, ten lakh, and crore. The digit immediately to the right of the chosen place tells which side of the midpoint the number lies on. If it is 0–4, retain the chosen digit and replace the digits to its right with zeros; if it is 5–9, increase the chosen place by one and then write zeros to its right. A carry can affect an earlier place.
Problem
Round 6,72,85,183 to the nearest thousand, ten thousand, lakh, ten lakh, and crore.
- 1.Nearest thousand: the last three digits are 183, below 500, so the result is 6,72,85,000.
- 2.Nearest ten thousand: the remainder after 6,72,80,000 is 5,183, above the 5,000 midpoint, so the result is 6,72,90,000.
- 3.Nearest lakh: 85,183 exceeds 50,000, so the result is 6,73,00,000.
- 4.Nearest ten lakh: 6,72,85,183 lies below the midpoint 6,75,00,000 between 6,70,00,000 and 6,80,00,000, so round to 6,70,00,000.
- 5.Nearest crore: it is nearer 7,00,00,000 than 6,00,00,000, so round to 7,00,00,000.
| Original number | Thousand | Ten thousand | Lakh | Ten lakh | Crore |
|---|---|---|---|---|---|
| 3,87,69,957 | 3,87,70,000 | 3,87,70,000 | 3,88,00,000 | 3,90,00,000 | 4,00,00,000 |
| 29,05,32,481 | 29,05,32,000 | 29,05,30,000 | 29,05,00,000 | 29,10,00,000 | 29,00,00,000 |
Larger rounding units generally give a coarser description. Notice that 29,05,32,481 rounds upward to 29,10,00,000 at the ten-lakh scale but downward to 29,00,00,000 at the crore scale. “Rounding up” is not an unchanging property of the original number; it depends on the selected scale.
Problem
Which whole numbers round to 5,00,00,000 at all five scales: thousand, ten thousand, lakh, ten lakh, and crore?
- 1.At the finest scale, the neighbouring thousands are 4,99,99,000, 5,00,00,000, and 5,00,01,000.
- 2.The lower midpoint is 4,99,99,500; it rounds upward to 5,00,00,000. The upper midpoint is 5,00,00,500; it rounds upward to 5,00,01,000 and must be excluded.
- 3.Thus the allowed whole numbers run from 4,99,99,500 through 5,00,00,499, inclusive.
- 4.All of these numbers are also within the wider midpoint ranges for the four coarser scales. There are 1,000 possible whole numbers, including 4,99,99,999 and 5,00,00,001.
Estimate first, then compare carefully
Estimation gives us a sense of the answer before a full calculation. Bounds make that sense more precise: they state a number the answer must exceed or must stay below. When deciding whether a sum or difference is near a midpoint, use the actual quantities or sufficiently sharp bounds.
Problem
Estimate and then locate 4,63,128 + 4,19,682.
- 1.Using nearest lakhs gives 5,00,000 + 4,00,000 = 9,00,000 as an estimate.
- 2.Adding exactly gives 8,82,810. It lies between 8,00,000 and 9,00,000.
- 3.The midpoint is 8,50,000; the sum is above it and therefore closer to 9,00,000.
- 4.Since 4,19,682 < 4,20,000, the sum is less than 4,63,128 + 4,20,000 = 8,83,128. This sharper upper bound agrees with the exact answer.
Problem
Compare 14,63,128 − 4,90,020 with 9,00,000, 9,50,000, and 10,00,000.
- 1.Since 4,90,020 is slightly below 5,00,000, subtracting it leaves slightly more than 14,63,128 − 5,00,000 = 9,63,128.
- 2.The exact difference is 9,73,108, which confirms that lower bound.
- 3.It exceeds 9,50,000, the midpoint between 9,00,000 and 10,00,000.
- 4.It is therefore closer to 10,00,000. Its distance from 10,00,000 is 26,892, while its distance from 9,00,000 is 73,108.
If you replace a subtracted number by a larger estimate, the estimated difference becomes smaller. This direction is easy to reverse by mistake. Check with a small example, such as 20 − 9 being greater than 20 − 10.
Read a population table as dated evidence
A table lets us compare many quantities using the same units. The following values are the chapter’s city-population data for 2001 and 2011. They are historical figures as printed in the chapter, with its stated city boundaries; they are not current populations or a new census report.
| City | 2001 population | 2011 population |
|---|---|---|
| Mumbai | 1,19,78,450 | 1,24,42,373 |
| New Delhi | 98,79,172 | 1,10,07,835 |
| Bengaluru | 43,01,326 | 84,25,970 |
| Hyderabad | 36,37,483 | 68,09,970 |
| Ahmedabad | 35,20,085 | 55,70,585 |
| Chennai | 43,43,645 | 46,81,087 |
| Kolkata | 45,72,876 | 44,86,679 |
| Surat | 24,33,835 | 44,67,797 |
| Vadodara | 16,90,000 | 35,52,371 |
| Pune | 25,38,473 | 31,15,431 |
| Jaipur | 23,22,575 | 30,46,163 |
| Lucknow | 21,85,927 | 28,15,601 |
| Kanpur | 25,51,337 | 27,67,031 |
| Nagpur | 20,52,066 | 24,05,665 |
| Indore | 14,74,968 | 19,60,631 |
| Thane | 12,62,551 | 18,18,872 |
| Bhopal | 14,37,354 | 17,98,218 |
| Visakhapatnam | 13,45,938 | 17,28,128 |
| Pimpri-Chinchwad | 10,12,472 | 17,27,692 |
| Patna | 13,66,444 | 16,84,222 |
A suitable title is “Population of selected Indian cities in 2001 and 2011”. Read the column labels before drawing conclusions. Almost every listed city increased, but Kolkata decreased from 45,72,876 to 44,86,679 in this table. A statement that every city increased would ignore an important observation.
Problem
Find Pune’s approximate increase and identify the greatest increase among the listed cities.
- 1.Pune’s 2011 value is 31,15,431. Subtract its 2001 value: 31,15,431 − 25,38,473 = 5,76,958.
- 2.To the nearest lakh, the increase is 6,00,000, or about six lakh.
- 3.Bengaluru increased by 84,25,970 − 43,01,326 = 41,24,644. Comparing the differences for all listed cities gives the greatest increase for Bengaluru.
- 4.Mumbai has the largest 2011 total, but Bengaluru has the largest increase. These are different questions.
To investigate “almost doubled”, compare the 2011 value with twice the 2001 value. Bengaluru, Hyderabad, Surat, and Vadodara are roughly twice their earlier figures, although none is exactly double. For example, twice Bengaluru’s earlier population is 86,02,652, close to 84,25,970. Whether a looser comparison such as 1.7 times counts as “almost double” depends on how much approximation you allow; state your criterion.
Problem
Use the 2011 figures to find an approximate multiplier from Patna’s population to Mumbai’s.
- 1.Divide 1,24,42,373 by 16,84,222. The quotient is about 7.3876, or 7.4 to one decimal place.
- 2.Seven times Patna gives 1,17,89,554 and eight times gives 1,34,73,776, so seven is the nearer whole-number multiplier.
- 3.For a closer approximate comparison, use 7.4 rather than 7.
- 4.The quotient compares the listed historical counts, not the physical size or current population of the cities.
Quiz
Which information should normally retain its exact digits?
Round 76,068 to the nearest thousand.
Using halfway-up rounding, 29,05,32,481 rounds to which nearest ten lakh?
Which is the greatest whole number that rounds to 5,00,00,000 to the nearest thousand?
The sum 4,63,128 + 4,19,682 is:
Which city has the greatest increase in the supplied table?
Practice Problems
- Give two situations requiring exact digits and two in which an approximation is useful. Explain your choices.
- Round 3,87,69,957 and 29,05,32,481 to the nearest thousand, ten thousand, lakh, ten lakh, and crore. Explain one upward and one downward choice.
- Write four different whole numbers whose rounding at all five scales gives 5,00,00,000. Test the lower and upper boundaries.
- Estimate 4,63,128 + 4,19,682, then explain why it is above 8,50,000 but below 8,83,128.
- Without beginning with column subtraction, explain why 14,63,128 − 4,90,020 is greater than 9,63,128. Then calculate it.
- Using the dated table, calculate Kolkata’s decrease and Pune’s increase. Give a suitable title for the table.
- For Bengaluru, Hyderabad, Surat, and Vadodara, compare the later figure with twice the earlier one. State what you mean by “almost double”.
- Compare seven and eight times Patna’s 2011 population with Mumbai’s. Explain why 7.4 is a more precise approximate multiplier.
- If 732 guests each need one sweet and boxes contain 50 sweets, decide how many boxes to buy. Distinguish this choice from nearest-hundred rounding.
Key Takeaways
• The purpose determines whether exact digits or an approximation are appropriate. • Nearest-place rounding chooses the closer multiple of the selected unit. • For non-negative values in this lesson, exact halfway cases round upward. • Bounds help justify estimates of sums and differences. • Largest total, largest increase, and greatest proportional increase are different comparisons. • Read dates, units, and column labels before interpreting a table.