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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. This chapter contains many examples of arithmetic techniques that are used directly or indirectly in algebra. Since the chapter is intended as a review, the problem-solving techniques are presented without being developed. Therefore, no work space is provided, nor does the chapter contain all of the pedagogical features of the text. As a review, this chapter can be assigned at the discretion of the instructor and can also be a valuable reference tool for the student.

Overview

  • Decimal Fractions
  • Adding and Subtracting Decimal Fractions
  • Multiplying Decimal Fractions
  • Dividing Decimal Fractions
  • Converting Decimal Fractions to Fractions
  • Converting Fractions to Decimal Fractions

Decimal fractions

Fractions are one way we can represent parts of whole numbers. Decimal fractions are another way of representing parts of whole numbers.

Decimal fractions

A decimal fraction is a fraction in which the denominator is a power of 10.

A decimal fraction uses a decimal point to separate whole parts and fractional parts. Whole parts are written to the left of the decimal point and fractional parts are written to the right of the decimal point. Just as each digit in a whole number has a particular value, so do the digits in decimal positions.

The positions of the digits lying to the left and to the right of the decimal point are labeled. Moving towards left from the decimal point, the positions are labeled: the first as 'ones', the second as 'Tens', the third as 'Hundreds', the fourth as 'Thousands', the fifth as 'Ten Thousands', the sixth as 'Hundred Thousands', and the seventh as 'Millions'. Moving towards right from the decimal point, the positions are labeled: the first position as 'Tenths', the second position as 'Hundredths', the third as 'Thousandths', the fourth as 'Ten Thousandths', the fifth as 'Hundred Thousandths', and the sixth position as 'Millionths'. There is a comment written below the decimal positons as

Sample set a

The following numbers are decimal fractions.

57.9 The 9 is in the t e n t h s position . 57.9 = 57 9 10 .

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6.8014 The 8 is in the  t e n t h s  position .  The 0 is in the  h u n d r e d t h s  position .  The 1 is in the  t h o u s a n d t h s  position .  The 4 is in the ten  t h o u s a n d t h s  position .  6.8014 = 6 8014 10000 .

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Adding and subtracting decimal fractions

Adding/subtracting decimal fractions

To add or subtract decimal fractions,
  1. Align the numbers vertically so that the decimal points line up under each other and corresponding decimal positions are in the same column. Add zeros if necessary.
  2. Add or subtract the numbers as if they were whole numbers.
  3. Place a decimal point in the resulting sum or difference directly under the other decimal points.

Sample set b

Find each sum or difference.

9.183 + 2.140 The decimal points are aligned in the same column .  9 .183 +  2 .140 ¯ 11 .323

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841.0056 + 47.016 + 19.058 The decimal points are aligned in the same column .  841.0056 47.016 Place a 0 into the thousandths position . + 19.058 ¯ Place a 0 into the thousandths position .  The decimal points are aligned in the same column .  841.0056 47.0160 + 19.0580 ¯ 907.0796

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16.01 7.053 The decimal points are aligned in the same column .  16.01 Place a 0 into the thousandths position .  7.053 ¯ The decimal points are aligned in the same column .  16.010 7.053 ¯ 8.957

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Multiplying decimal fractions

Multiplying decimal fractions

To multiply decimals,
  1. Multiply tbe numbers as if they were whole numbers.
  2. Find the sum of the number of decimal places in the factors.
  3. The number of decimal places in the product is the sum found in step 2.

Sample set c

Find the following products.

6.5 × 4.3

The vertical multiplication of two decimals; six point five, and four point three. See the longdesc for a full description.

6.5 × 4.3 = 27.95

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23.4 × 1.96

The vertical multiplication of two decimals; twenty-three point four, and one point nine six. See the longdesc for a full description.

23.4 × 1.96 = 45.864

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Dividing decimal fractions

Dividing decimal fractions

To divide a decimal by a nonzero decimal,
  1. Convert the divisor to a whole number by moving the decimal point to the position immediately to the right of the divisor’s last digit.
  2. Move the decimal point of the dividend to the right the same number of digits it was moved in the divisor.
  3. Set the decimal point in the quotient by placing a decimal point directly above the decimal point in the dividend.
  4. Divide as usual.

Sample set d

Find the following quotients.

32.66 ÷ 7.1

A long division problem showing seven point one dividing thirty-two point six six. See the longdesc for a full description.

32.66 ÷ 7.1 = 4.6 C h e c k : 32.66 ÷ 7.1 = 4.6 if 4.6 × 7.1 = 32.66 4.6 7.1 ¯ 4.6 322 ¯ 32.66 True

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A long division problem showing zero point five one three dividing one point zero seven seven three. See the longdesc for a full description.

Check by multiplying 2.1 and 0.513. This will show that we have obtained the correct result.

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Converting decimal fractions to fractions

We can convert a decimal fraction to a fraction by reading it and then writing the phrase we have just read. As we read the decimal fraction, we note the place value farthest to the right. We may have to reduce the fraction.

Sample set e

Convert each decimal fraction to a fraction.

0.6 0. 6 ¯ tenths position Reading: six tenths 6 10 Reduce: 0.6 = 6 10 = 3 5

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21.903 21.90 3 ¯ thousandths position Reading: twenty-one and nine hundred three thousandths 21 903 1000

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Converting fractions to decimal fractions

Sample set f

Convert the following fractions to decimals. If the division is nonterminating, round to 2 decimal places.

5 6

A long division problem showing six dividing five point zero zero zero. See the longdesc for a full description.

5 6 = 0.833... We are to round to 2 decimal places . 5 6 = 0.83  to 2 decimal places .

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5 1 8 Note that  5 1 8 = 5 + 1 8 .

One point zero zero zero is being divided by eight, using long division method. See the longdesc for a full description.

1 8 = .125 Thus,  5 1 8 = 5 + 1 8 = 5 + .125 = 5.125.

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0.16 1 4

This is a complex decimal. The “6” is in the hundredths position. The number 0.16 1 4 is read as “sixteen and one-fourth hundredths.”

0.16 1 4 = 16 1 4 100 = 16 · 4 + 1 4 100 = 65 4 100 1 = 65 13 4 · 1 100 20 = 13 × 1 4 × 20 = 13 80

Now, convert 13 80 to a decimal.

Thirteen point zero zero zero zero is being divided by eighty, using long division method. See the longdesc for a full description.

0.16 1 4 = 0.1625.

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Exercises

For the following problems, perform each indicated operation.

.0012 + 1.53 + 5.1

6.6312

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5.0004 3.00004 + 1.6837

3.68406

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1.11 + 12.1212 13.131313

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4.26 · 3.2

13.632

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23.05 · 1.1

25.355

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0.1 · 3.24

0.324

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1000 · 12.008

12 , 008

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10 , 000 · 12.008

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75.642 ÷ 18.01

4.2

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0.0000448 ÷ 0.014

0.0032

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0.129516 ÷ 1004

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For the following problems, convert each decimal fraction to a fraction.

For the following problems, convert each fraction to a decimal fraction. If the decimal form is nonterminating,round to 3 decimal places.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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