Unit 3:Decimal numbers

In this lesson you will learn about: Approximating decimal numbers

truncating and rounding. Absolute and relative error Operations with decimal numbers Scientific notation

Vocabulary you need in this lesson:You are going to work with some vocabulary related to thislesson and the problems you will work with. The same wordsappear in the crosswords, in the word scramble and in theword search.

Some theoryDecimal Numbers

Decimal numbers such as 3.762 are used in situations which call for more precisionthan whole numbers provide.

As with whole numbers, a digit in a decimal number has a value which depends onthe place of the digit. The places to the left of the decimal point are ones,tens, hundreds, and so on, just as with whole numbers. This table shows thedecimal place value for various positions:

Note that adding extra zeros to the right of the last decimal digit does notchange the value of the decimal number.

In the number 3.762, the 3 is in the ones place, the 7 is in the tenths place, the 6 is inthe hundredths place, and the 2 is in the thousandths

Truncation:To truncate a decimal number to any decimal place value you eliminate all the digitsafter this place value.5.567 → 5.56 ( to hundretdhs)

Rounding:To round a number to any decimal place value, we want to find the number with zerosin all of the lower places that is closest in value to the original number. As with wholenumbers, we look at the digit to the right of the place we wish to round to. Note:When the digit 5, 6, 7, 8, or 9 appears in the ones place, round up; when the digit 0, 1,2, 3, or 4 appears in the ones place, round down.8.9786 → 8.98 (to hundredths)

Place (underlined) Name of Position

1.234567 Ones (units) position

1.234567 Tenths

1.234567 Hundredths

1.234567 Thousandths

1.234567 Ten thousandths

1.234567 Hundred Thousandths

1.234567 Millionths

Error

When you approximate a decimal number by truncating or rounding youcommit an error.

Absolute error: is the difference between the decimal number and itsapproximation

Relative error: is the quotient between the absolute error and the decimalnumber

Kind of decimal numbers: there are three kind of decimal numbers:

Exact: they have a limited number of digits

Pure recurring decimal numbers: they have infinite digits and they arerepeated just after the comma. The repeated digits form the period

Mixed recurring decimal number: they have infinite digits and they arerepeated but not just after the comma. There are some digits after thecomma and before the period

Now it´s time to practise1. Exercises

1. Approximate the following numbers, by truncation , until tenths, hundredths and thou-sandths:

7.3251

5.556127

17.48216

Find the absolute and relative errors in each case.2. Repeat the exercise nº 1, but approximate by rounding.

3. Write each fraction as a decimal expression and classify the expression:

4. Read the words and write the number down in figures. Don’t forget the decimal point!

H T U ● t h th

4 units and 3 tenths ___________________

7 units and 6 tenths ___________________

9 units and 0 tenths ___________________

8 tens, 2 units and 6 tenths ___________________

6 tens, 3 units and 4 tenths ___________________

5 tens and 3 tenths ___________________

8 units, 3 tenths and 6 hundredths ___________________

9 units, 2 tenths and 7 hundredths ___________________

0 units, 1 tenth and 8 hundredths ___________________

7 hundreds, 5 tens, 1 unit and 6 tenths ___________________

3

2

15

12

9

4

6

5

12

11

15

6

20

6

16

3

14

7

2. Problems

1. Phone Bill

Laura’s got a BIG problem. (She’s got a BIG phone bill). But, she doesn’t knowhow big, because the phone company forgot to fill in the totals. Can you helpher?

1. Put all the information into the spreadsheet.Show your work on a separate sheet:2. Calculate the Amount due for each type of call in part B3. Calculate the subtotals for each area (A, B, C)4. Calculate the Subtotal in part DCalculate the Total Phone Bill

Bonus:Laura went back to work at Bar Louie this summer to pay her bills. Sheworks 5 shifts a week. On a morning shift she earns an average of $9.65/hour. On a night shift she earns and average of $22.14/hour. Morningshifts last 7 hours. Night shifts are 10 hours long. Does Laura have enoughmoney to pay her phone bill if she works 3 day shifts and 2 night shifts?

Super Bonus:What percentage of Laura’s income does she need to pay her phone bill?

AT&T Phone Co.AmountDue Totals

A Monthly Charges 35.25

B Usage Fees# OfCalls

Time(min)

Rate($/min)

Local Calls 87 289 0.3Long Distance Calls 1 1 0.17International Calls 16 423 0.76

C DiscountsInternational Plus Package -10.55

Bonus Bucks -6.46D

Account Number: 8476340695Sub Total:Tax (11%):

Address: 123 America street Chicago,USA 98765 Total:

2. A tailor needs a piece of cloth to make a t-shirt for Falete.

After measuring Falete’s body, Angelo Marujonetti (this is the tailor's name) , realized hewas going to need a lot of cloth , so , he bought a piece of cloth whose measurements

were 8.163m 4.367m.

How much is it in m2?

Approximate the area of the cloth, by rounding, until hundredths. Calculate the absoluteand relative errors that have been committed.

Suppose the price of the cloth is 7.12 €/m2. How much does it cost? Round until hun-dredths.

3. The director of Sealife Aquariums wants to make a big aquarium to show the typical eco-system of coral reefs.

The Technical Staff need to make a container like a biguncovered shoe box, made of glass. Its measurementsare the following:

LENGTH → 17.21 m

WIDTH → 12.543 m

DEPTH → 4.632 m

Find the exact value of the volume in m3.

Approximate the value of the volume until hundredths. Calculate absolute and relative er-rors.

Suppose the price of crystal is $27,95 per squared meter.How much did it cost? Approximate until hundredths.

4. What is the distance Jesse has covered after 6 hours, if he was travelling at a rate of49.4 miles per hour?

5.The cost for a field trip (viaje escolar) is £99.9 and 31 students have to pay for it. Howmuch does each student need to pay? (round to the nearest cent)

6. In 6 hours Tom travelled 633.9 miles. Calculate the speed at which Tom was travelling.

7. A pack of 4 marbles weighs 0.75 lbs. What is the weight of one mar-ble?

8. If you travel at a speed of 70.2 miles per hour, what distance will you cover in 5.2 hours?

9. Tom scored 28 out of 70 in the spelling test. What is his percentage? (Round to nearest tenths)

10. In Matt’s school 52% of the students are boys. The total number of students in the schoolis 8000. Find the number of boys. How many girls are there?

11. In a town, two years ago, there were 2500 people. The population increased by 5%. What isthe population of the town now?

12. A basketball team won in 8 games, lost in 9 games. What percent did the team win? Whatpercent did they lose?

13. Kate saved £ 5394. She spent 10% of it on her dress and 21% on games for her sister andbrother. How much did she spend on games and how much money does she have now? (Round tonearest pence)

14. 83 kids went on a field trip, which is 62% of the total number of kids in the first course.Find the total number of kids in the first course?

15. Find the increased or decreased percentage:

64 bags sold on Thursday, 86 bags sold on Friday.

70 computers sold on Tuesday, 14 computers sold on Wednesday

30 dollars spent last week, 16 dollars spent this week.

16 cents spent on Friday, 89 cents spent on Saturday.

Some interesting web sites!

If you want to read more about decimal numbers and what you have le-arnt in this unit and you want to practice with more exercises go to:

http://www.mathsisfun.com/decimals-menu.html

To practice rounding decimal numbers after some examples clickhere:

http://www.augusta.k12.va.us/668712562122824/lib/668712562122824/Rounding_Decimal_Numbers.ppt

To play a game where you have to operate with decimals to get theresult cliclk here:

http://www.math-play.com/decimal-math-games.html

To practise with decimal numbers solving some problems go to:

http://www.angelfire.com/ca5/hwforkids/10decimal.html

To take a test where you can check what you have learnt go to:

http://www.thatquiz.org/tq/practice.html?fraction

If you want to read about π, which is one of the most important and fa-mous decimal numbers click here:

http://en.wikipedia.org/wiki/Pi