Skill Check XIII IITJEE Foundation Maths: Approximation

Introduction:

(Reference: an old ICSE grade VIII text book, New Guided Mathematics, class 8, Abhijit Mukerjea and Sushil Kumar Sharma, Oxford University Press, India)

Rahul needs to divide a web page into three vertical parts. He wants it to look neat and thus wants the measurements to be perfect. But the web page is 640 pixels wide and 640/3 = 213.3 pixels. Is it necessary for Rahul to be so accurate? If he rounds off each part to 213 pixels , his page can be divided into 3 parts measuring 213, 214, and 213 pixels. Although the parts are not exactly equal, they are approximately equal to each other, a relationship that is denoted by the symbol \approx.

Rounding Off Decimal Fractions

Rounding off a decimal fraction to its decimal places:

  1. If the digit in the (n+1)th decimal place is less than 5, then the digit in the (n+1)th decimal place, and all the digits after it, are simply omitted.
  2. If the digit in the (n+1)th decimal place is 5 or more than 5, the nth digit is increased by 1 and the digit in the (n+1)th decimal place, and all the digits are it, are omitted.

Example 1: Convert \frac{0}{7} into a decimal fraction and round off your answer to:

i) 1 decimal place

ii) 3 decimal places

iii) 5 decimal places

iv) a whole number

Solution: \frac{9}{7}=1.2857142\ldots

(i) In order to round off the answer to the tenths or 1 decimal place, we consider the (1+1)th decimal place digit. As 8>5, the digit in the tenths place is increased by 1 and all the digits after it are omitted. Thus, \frac{9}{7}=1.3

(ii) As the (3+1)th decimal place digit is 7 and 7>5, the digit in the thousandths place or 5 is increased by 1 and all the digits after it are omitted. Thus, \frac{9}{7}=1.286

(iii) As the digit in the (5+1)th decimal place is 4 and 4<5, the digit in the (5+1)th decimal place and all the digits after it are omitted. Thus, \frac{9}{7}=1.28571

(iv) As the digit in the tenths place is 2 and 2<5, it is omitted and all the digits after it are also omitted. Thus, \frac{9}{7}=1

TRY THIS: Convert \frac{10}{20} into a decimal fraction and round off your answer to 4 decimal places.

Rounding Off to a Specified Unit:

We have just learned how to round off our answers to a specified number of decimal places. But in real life we have to use our sense and figure out how much to round off such that our answer makes sense.

  1. Raima buys a pair of sunglasses marked at Rs. 690.75 selling at a discount of 35 %. The shopkeeper charges her Rs. 449. Should Raima complain that she has been charged Rs. 0.0125 more?
  2. A lady wishing to buy a gold ring asks the jeweller the market price of gold. Would it make more sense to the lady if the jeweller quoted the price of gold in kilograms or grams?
  3. A traveller consults a railway timetable to find out how much it would cost to travel from Kolkata to Mumbai. Would it help the traveller if the fare chart quotes fares in paise per meter rather than rupees per kilometer?

Example 2: An ice-cream vendor buys 60 ice-creams for Rs. 210 and wishes to earn a 40% profit. At what price should he sell each ice-cream?

Answer 2: Cost price per ice cream is 210/60=Rs. 3.50

Profit expected per ice-cream is 3.5 x 40/100, that is, Rs. 1.40

Accurate SP per ice-cream is Rs. 3.50 + Rs. 1.40 that is Rs. 4.90

As it would be difficult for all his customers to pay exact change (Rs. 4.90), the vendro could round it off to a reasonable price of Rs. 5 per ice-cream.

Example 3: What is the capacity of a cubical ink pot that is 0.04 m long?

Answer 3: The calculation of capacity with length in meters would give an answer in kilolitres. But the vessel in question is an ink pot and not a tanker !! Thus, it would make more sense to convert the length to centimeters and find the capacity in cubic cm or millilitres.

0.04 m is 4 cm.

Capacity of ink pot is 4 x 4 x 4, that is, 64 cubic cm.

Significant Digits in Decimals

We know that 00002.3=2.3=2.30000

The significant or meaningful digits in the above numbers are only the digits 2 and 3 on either side of the decimal point that give us an idea of its value and location on the number line.

  1. All non-zero digits in a decimal number are significant digits.

(a) 5.695 has four significant digits.

(b) 58.2 has three significant digits.

2. The zeroes between non-zero digits in a decimal number are significant digits.

(a) 3.01 has three significant digits, being 3, 0 and 1.

(b) 5.2001 has five significant digits, being 5, 2, 0, 0 and 1.

3. The zeros to the left of the first non-zero digit are not significant digits.

(a) 0.09 has only one significant digit, being 9.

(b) 0.00012 has only two significant digits, being 1 and 2.

4. The zeros to the right of the last non-zero digit may or may not be significant. The condition depends on the unit of measurement or the need for approximation.

Case I: When zeros to the right of the last non-zero digits are significant.

Example : Convert 1.2000 m into centimeter.

Solution: As 1.2000m is 120 cm, so in 1.2000 m there are three significant digits, being 1, 2 and 0.

Example: Convert 3.6000000 kg into mg.

Solution: As 3.6000000 kg is 3600000 mg, in 3.6000000 kg, there are 7 significant digits being 5, 6, and give zeros.

Case II: When zeros to the right of the last non-zero digit are not significant.

Example: Express Rs. 3.60000 in paise.

Solution: As Rs. 3.60000 is 360 paise, Rs. 3.60000 has only three significant digits, being 3, 6 and 0, the three zeros after it being insignificant.

Example: Given the area of the Arctic Ocean as 13079000 square km, express its area in thousand square km.

Solution: 13079000 sq km x 1/1000 is 13079 thousand sq km. Thus, 13079000 sq km has only 5 significant digits, being 1, 3, 0, 7 and 9, the three zeros after it being insignificant.

Approximation to Signficant Digits

Example 1: Approximate 0.0003801 to 1 significant digit.

Solution 1: The given number has 4 significant digits, being 3, 8, 0 and 1. To approximate it to 1 significant digit, it will have to be rounded off to its ten-thousandths place where its first significant digit is. As 8 in the hundred-thousandths place is greater than 5, 0.0003801 is 0.0004.

Example 2: Approximate 3.1428571 to 3 significant digits.

Solution 2: The given number has 8 significant digits. To approximate it to 3 significant digits, it will have to be rounded off to the hundredths place. As 2 in the thousandths place is less than 5, 3.1428571 \approx 3.14

Representation of Numbers

Let us recall how we expressed a decimal number in expanded form in previous classes:

Example : Write 5873.1264 in expanded form:

Solution: 5873.1264 can be written as

5000+800+70+3+0.1+0.02+0.006+0.0004 or

5 \times 10^{3} + 8 \times 10^{2} + 7 \times 10^{1}+ 3 \times 10^{0}+ 1 \times 10^{-1} + 2 \times 10^{-2} + 6 \times 10^{-3} + 4 \times 10^{-4}

TRY THIS!

Write 5840.183 in expanded form.

Thus, whatever be the place of a digit in the integral or decimal part of a number, its place value can be described as a multiple of a power of 10.

The distance between Earth and Pluto is 575,00,00,000 km. This is too big a number to communicate and remember:

(i) Thus, it is approximated to its significant digits and represented as 5 billion 750 million km. Or as 575 crore km OR

(ii) All the significant digits are considered and it is written in scientific notation as the product of a decimal number less than 10 and a power of 10, that is, 5.8 \times 10^{9} km.

We know that a millilitre is one-thousandth part of a metre. Scientists and industrialists are very excited, about “nanotechnology” now-a-days. Do you know the relationship between a nanometre and a metre?

1 nanometre is 0.000000001 of a metre or a nanometre is a thousand-millionth part of a metre. Such a small decimal number is again written in scientific notation as the product of a decimal number less than 10 and a power of 10 as follows: 1 nm = 1 \times 10^{-9} m.

Example: Write 0.000200100 in scientific notation.

Solution: The significant digits in the number are 2, 0, 0, and 1. The decimal number less than 10 is 2.001 and as a product of power of 10, the scientific notation for 0.000200100 is 2.001 \times 10^{-4}.

Try to figure out the weight of an electron if it is given in scientific notation as equal to 9.10908 \times 10^{-31}kg.

Exercises:

  1. Approximate 0.0004567 to 1 significant digit.
  2. Write 6245.3173 in expanded form.
  3. Round off 22.5% of Rs. 58.50 to the nearest piase.
  4. Express 7 liters 54 ml to the nearest litre.
  5. Write 7.000 40 23587 in scientific notation.

Regards,

Nalin Pithwa

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