1.4 Exact and Approximate Values

“Screen Readable NCERT Class 7 Mathematics Textbook – Ganita Prakash – for Blind and Visually Impaired Students prepared by Professor T K Bansal.”

Boy: "1 Lakh people visited the book fair."

Girl: "Oh, so many people!"

Boy: "If I had not gone, they would have written “99,999 people visited the book fair last week”.

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 population is.

Figure 1.13

Description of the Figure begins:
The first part shows a boy, named Bappi, and
his father. Father says to his son, "Bappi, Call and check if Toofan Express is on time."
In part 2, Bappi is noting down some numbers from a notebook. The numbers shown in the notebook are,
Police: 100 or 112,
Fire: 101,
Ambulance: 102,
Disaster Management Services:108,
National Emergency Number: 112,and
Railway Enquiry: 139.
Part shows that the door bell in Bappi's house rings, "Ding Dong."
Part four shows that the father opens the door, and finds the police at the door.
Father asks Bappi, "What number did you call?"
The last part shows that Bappi replies, "I called the rounded off number, of course! My teacher told us that efficient people deal with rounded off numbers."
End of the description of the figure.

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.

Likewise, 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.

1.4.1 Nearest Neighbors

With large numbers it is useful to know the nearest thousand, lakh or crore. For example, the nearest neighbors of the number 6,72,85,183 are shown in the table below.

Table:

Similarly, write the five nearest neighbors for these numbers:
(a) 3,87,69,957
(b) 29,05,32,481

I have a number for which all five nearest neighbors 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.

1.
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.”

Answer the following questions based on the above discussion:
(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 = □.

2.
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.”

Answer the following questions based on the above discussion:
(a) Are these estimates correct? Whose estimate is closer to the sum?
(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 = □

Note to the Teacher:
Ask the students, questions like — “what could the numbers be if the sum had to be less than 8,50,000."

1.4.2 Populations of Cities

Observe the populations of some Indian cities mentioned in the table below.

Table:

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

1.
What is your general observation about this data? Share it with the class.

2.
What is an appropriate title for the above table?

3.
How much is the population of Pune in 2011? Approximately, by how much has it increased compared to 2001?

4.
Which city’s population increased the most between 2001 and 2011?

5.
Are there cities whose population has almost doubled? Which are they?

6.
By what number should we multiply Patna’s population to get a number/population close to that of Mumbai?