Unit 1: Working with Decimals
Topic B: Reading & Writing Decimals
Remember the place value chart of whole numbers? Complete the following exercise for a refresher.
Exercise 1
352 is the number on the first line of the chart below. The 3 is in the hundreds spot, the 5 is in the tens spot, and the 2 is in the ones spot.
Place the following numbers on the place value chart:
- 4984
- 836466
- 70834
| Thousands | Ones | |||||
|---|---|---|---|---|---|---|
| Hundred thousands | Ten thousands | One thousand | Hundreds | Tens | Ones | . (decimal point) |
| 3 | 5 | 2 | . | |||
Answers to Exercise
Check with your instructor to see if you have placed the numbers in the chart correctly.
Turning a Decimal into Words
Have you ever wondered what goes to the right of the decimal in a place value chart? That is where the decimal numbers go! (The parts of the whole.)
Here is a place value chart for decimals:
| Hundreds | Tens | Ones | Decimal | Tenths | Hundredths | Thousandths | Ten thousandths | Hundred thousandths |
|---|---|---|---|---|---|---|---|---|
| 3 | . | 4 | 5 | 3 | ||||
| 0 | . | 9 | 6 |
See the words in to the right of the decimal point? They look different than the usual whole number words you are used to. These are all the names for the decimal places. You will see them in the next lesson.
The first number on the chart above is 3.453. We say “Three point four five three” or “Three and four hundred fifty-three thousandths.”
- 3 is in the ones spot
- 4 is in the tenths spot
- 5 is in the hundredths spot
- 3 is in the thousandths spot
The second number is 0.96. We say “Zero point nine six” or “Zero and ninety six hundredths.”
- 0 is in the ones spot
- 9 is in the tenths spot
- 6 is in the hundredths spot
Tenths Place
Common fractions with a denominator of 10 are written as a decimal with one place to the right of the decimal point. This is the tenths place.
We often shorten “places to the right of the decimal point” to “decimal places.” We can say that tenths have one decimal place.
- [latex]\frac{6}{10}[/latex] = 0.6 = six tenths
- [latex]\frac{3}{10}[/latex] = 0.3 three tenths
Exercise 2
- [latex]\frac{4}{10}[/latex] = 0.4 = four tenths
- [latex]\frac{1}{10}[/latex] = 0.1 = one tenth
- [latex]\frac{2}{10}[/latex] = =
- [latex]\frac{9}{10}[/latex] = =
- [latex]\frac{7}{10}[/latex] = =
- Now enter each common fraction in the place value chart below. The first one is done for you. Please show this to your instructor once you are finished to make sure you are on the right track.
| Hundreds | Tens | Ones | Decimal | Tenths | Hundredths | Thousandths | Ten thousandths | Hundred thousandths |
|---|---|---|---|---|---|---|---|---|
| 0 | . | 4 | ||||||
Answers to Exercise 2
- 0.4, four tenths
- 0.1, one tenth
- 0.2, two tenths
- 0.9, nine tenths
- 0.7, seven tenths
Decimals with one digit to the right of the decimal point have an unwritten denominator of ten. This means that the whole thing is broken into 10 equal parts. Each part is called a tenth.
When we write decimals, we put a zero to the left of the decimal point to show there is no whole number. This zero keeps the decimal point from being “lost” or not noticed.
For example, .2 should be written as 0.2.
Exercise 3
Write each decimal as a common fraction and in words.
- 0.3 = [latex]\frac{3}{10}[/latex] = three tenths
- 0.4 = [latex]\frac{4}{10}[/latex] = four tenths
- 0.8 = =
- 0.7 = =
- 0.1 = =
Answers to Exercise 3
- [latex]\frac{3}{10}[/latex] three tenths
- [latex]\frac{4}{10}[/latex] four tenths
- [latex]\frac{8}{10}[/latex] eight tenths
- [latex]\frac{7}{10}[/latex] seven tenths
- [latex]\frac{1}{10}[/latex] one tenth
Hudredths Place
Decimals with two digits to the right of the decimal point have an unwritten denominator of one hundred. This means that the whole thing is broken into 100 equal parts. Each part is called a hundredth.
Exercise 4
Write each decimal as a common fraction and in words.
- 0.34 = [latex]\frac{34}{100}[/latex] = thirty-four hundredths
- 0.71 = [latex]\frac{71}{100}[/latex] = seventy-one hundredths
- 0.06 = =
- 0.56 = =
- Now place the above decimal numbers in the place value chart. The first two are done for you. Then ask your instructor to mark it.
| Hundreds | Tens | Ones | Decimal | Tenths | Hundredths | Thousandths | Ten thousandths | Hundred thousandths |
|---|---|---|---|---|---|---|---|---|
| 0 | . | 3 | 4 | |||||
| 0 | . | 7 | 1 | |||||
Answers to Exercise 4
- [latex]\frac{34}{100}[/latex] thirty-four hundredths
- [latex]\frac{71}{100}[/latex] seventy-one hundredths
- [latex]\frac{6}{100}[/latex] six hundredths
- [latex]\frac{56}{100}[/latex] fifty-six hundredths
- [latex]\frac{33}{100}[/latex] thirty-three hundredths
- [latex]\frac{40}{100}[/latex] forty hundredths
Common fractions with a denominator of one hundred are written as decimals with two decimal places.
- [latex]\frac{23}{100}[/latex] = 0.23
- [latex]\frac{99}{100}[/latex] = 0.99
- [latex]\frac{4}{100}[/latex] = 0.04
Look at that last example. The 0 must be used after the decimal point in 0.04 to hold the tenths place. This makes it clear that the denominator is one hundred. There are two zeros in the denominator, so there must be two decimal places taken up.
This is called prefixing zeros.
Exercise 5
Write these common fractions as decimals.
- [latex]\frac{34}{100}[/latex] = 0.34
- [latex]\frac{70}{100}[/latex] = 0.70
- [latex]\frac{85}{100}[/latex]
- [latex]\frac{11}{100}[/latex]
- [latex]\frac{21}{100}[/latex]
- [latex]\frac{5}{100}[/latex]
- [latex]\frac{6}{100}[/latex]
- [latex]\frac{45}{100}[/latex]
- [latex]\frac{50}{100}[/latex]
- [latex]\frac{1}{100}[/latex]
Answers to Exercise 5
- 0.34
- 0.70
- 0.85
- 0.11
- 0.21
- 0.05
- 0.06
- 0.45
- 0.50
- 0.01
Thousandths Place
Decimals with three digits to the right of the decimal point (three decimal places) have an unwritten denominator of one thousand. This means that the whole thing is broken into 1000 equal parts. Each part is one thousandth.
Look carefully at how thousandths are written. Watch for the prefixing zeros that may be needed to hold the tenth decimal place or the hundredth decimal place.
- 0.472 = four hundred seventy-two thousandths = [latex]\tfrac{472}{1000}[/latex]
- 0.085 = eighty-five thousandths = [latex]\tfrac{85}{1000}[/latex]
- 0.003 = three thousandths = [latex]\tfrac{3}{1000}[/latex]
There are three zeros in the denominator, so there must be three decimal places taken up.
Exercise 6
Write each decimal as a common fraction and in words. Practice saying them out loud.
- 0.006 = [latex]\frac{6}{1000}[/latex] = six thousandths
- 0.142 = [latex]\frac{142}{1000}[/latex] = one hundred forty-two thousandths
- 0.238 = =
- 0.562 = =
- 0.600 = =
Answers to Exercise 6
- [latex]\frac{6}{1000}[/latex], six thousandths
- [latex]\frac{142}{1000}[/latex], one hundred forty-two thousandths
- [latex]\frac{238}{1000}[/latex], two hundred thirty-eight thousandths
- [latex]\frac{562}{1000}[/latex], five hundred sixty-two thousandths
- [latex]\frac{600}{1000}[/latex], six hundred thousandths
Exercise 7
Write each common fraction as a decimal.
- [latex]\frac{736}{1000}[/latex] = 0.736
- [latex]\frac{84}{1000}[/latex] = 0.084
- [latex]\frac{210}{1000}[/latex]
- [latex]\frac{6}{1000}[/latex]
- [latex]\frac{106}{1000}[/latex]
- [latex]\frac{116}{1000}[/latex]
- [latex]\frac{592}{1000}[/latex]
- [latex]\frac{962}{1000}[/latex]
- Now place the above decimal numbers in the place value chart on the next page. The first two are done for you. Then ask your instructor to mark it.
| Hundreds | Tens | Ones | . | Tenths | Hundredths | Thousandths | Ten thousandths | Hundred thousandths |
|---|---|---|---|---|---|---|---|---|
| 0 | . | 7 | 3 | 6 | ||||
| 0 | . | 1 | 4 | 2 | ||||
Answers to Exercise 7
- 0.736
- 0.084
- 0.210
- 0.006
- 0.106
- 0.116
- 0.592
- 0.962
Ten-thousandths Place
Decimals with four decimal places have an unwritten denominator of ten-thousand. The whole thing is being thought of as having 10000 equal parts. Each part is one ten-thousandth.
- 0.1458 = [latex]\frac{1458}{10000}[/latex] = one thousand four hundred fifty-eight ten-thousandths
- 0.0581 = [latex]\frac{581}{10000}[/latex] = five hundred eighty-one ten-thousandths
Exercise 8
Write each decimal as a common fraction and in words. Practice saying these aloud to someone else; they can be real tongue-twisters!
- 0.2489 = [latex]\frac{2489}{10000}[/latex] = two thousand four hundred eighty-nine ten-thousandths
- 0.1111 = [latex]\frac{1111}{10000}[/latex] = one thousand one hundred eleven ten-thousandths
- 0.0236
- 0.4015
- 0.2306
- 0.0003
Answers to Exercise 8
- [latex]\frac{236}{10000}[/latex], two hundred thirty-six ten-thousandths
- [latex]\frac{4015}{10000}[/latex], four thousand fifteen ten-thousandths
- [latex]\frac{2306}{10000}[/latex], two thousand three hundred six ten-thousandths
- [latex]\frac{3}{10000}[/latex], three ten-thousandths
Exercise 9
Write these common fractions as decimals.
- [latex]\frac{1489}{10000}[/latex] = 0.1489
- [latex]\frac{2}{10000}[/latex] = 0.0002
- [latex]\frac{386}{10000}[/latex]
- [latex]\frac{9137}{10000}[/latex]
- [latex]\frac{4}{10000}[/latex]
- [latex]\frac{916}{10000}[/latex]
Now place the above decimal numbers in the place value chart below. The first two are done for you. Then ask your instructor to mark it.
| Hundreds | Tens | Ones | . | Tenths | Hundredths | Thousandths | Ten thousandths | Hundred thousandths |
|---|---|---|---|---|---|---|---|---|
| 0 | . | 1 | 4 | 8 | 9 | |||
| 0 | . | 0 | 0 | 0 | 2 | |||
Answers to Exercise 9
- 0.1489
- 0.0002
- 0.0386
- 0.9137
- 0.0004
- 0.0916
Mixed Decimals
Mixed decimals are a whole number and a decimal written together.
4.3 = [latex]4\frac{3}{10}[/latex] = four and three tenths
27.27 = [latex]27\frac{27}{100}[/latex] = twenty-seven and twenty-seven hundredths
8.104 = [latex]8\frac{104}{1000}[/latex] = eight and one hundred four thousandths
Digits to the left of the decimal point are whole numbers.
Digits to the right of the decimal point are fractions.
We say “and” for the decimal point.
Look at the mixed decimals from the examples above in the place value chart below:
| Hundreds | Tens | Ones | Decimal | Tenths | Hundredths | Thousandths | Ten thousandths | Hundred thousandths |
|---|---|---|---|---|---|---|---|---|
| 4 | . | 3 | ||||||
| 2 | 7 | . | 2 | 7 | ||||
| 8 | . | 1 | 0 | 4 |
Turning Words into a Decimal
- Read the number. Does the word “and” show that this is a mixed decimal?
- If it does, the whole number is before the word “and.” Write the whole number with the decimal point after it.
- If there is no whole number, write a 0 with the decimal point after it.
- Decide how many decimal places you need. Look and listen for the “ths” ending. The word with “ths” is the understood denominator. It may help if you draw a little line for each decimal place that you need.
- Tenths need one decimal place.
- Hundredths need two decimal places.
- Thousandths need three decimal places.
- Ten-thousandths need four decimal places.
- Write the decimal so the last digit is on the last little line and fill any remaining lines with zeros.
Example A
- Seven hundredths
- It it not a mixed decimal, so write: 0.
- “Hundredths” means two decimal places: 0. _ _
- Fill in the numbers: 0.07
- Eight thousandths
- It it not a mixed decimal, so write: 0.
- “Thousands” means three decimal places: 0. _ _ _
- Fill in the numbers: 0.008
- Twenty-six thousandths
- It it not a mixed decimal, so write: 0.
- “Thousandths” means three decimal places: 0. _ _ _
- Fill in the numbers: 0.026
- Four hundred six thousandths
- It it not a mixed decimal, so write: 0.
- “Thousandths” means three decimal places: 0. _ _ _
- Fill in the numbers: 0.406
Reading and Writing Money
Dollars
We write money with a dollar sign, a whole number, and a decimal with two decimal places.
$1.00 = 1 dollar
What do we call [latex]\frac{1}{100}[/latex] of a dollar? Right! One cent.
- $2.33 = two dollars and thirty-three cents
- $427.05 = four hundred twenty-seven dollars and five cents
- $0.62 = sixty-two cents
- $0.03 = three cents
Exercise 10
Write the amount of money in words.
- $212.63 – two hundred twelve dollars and sixty-three cents
- $47.01
- $9.28
- $82.50
- $100.05
Write with numerals, using $.
- twenty-seven dollars and six cents – $27.06
- one hundred sixty-two dollars
- thirteen dollars and sixty cents
- one thousand dollars and seventy-seven cents
- sixty-nine cents
- seven cents
- five hundred dollars and ninety cents
Answers to Exercise 10
- two hundred twelve dollars and sixty-three cents
- forty-seven dollars and one cent
- nine dollars and twenty-eight cents
- eighty-two dollars and fifty cents
- one hundred dollars and five cents
- $27.06
- $162.00
- $13.60
- $1000.77
- $0.69
- $0.07
- $500.90
“Centum” is a Latin word that means hundred! Here are English words that start with “cent”:
- centurion – commander of a hundred soldiers
- century – a hundred years
- centennial – a hundredth anniversary
- centigrade – having a hundred degrees
- cent – one hundredth of a dollar
- centimetre – one hundredth of a metre
- centipede – wormlike creatures with a hundred legs
When we read $12.25 as “twelve dollars and twenty-five cents” we are using the Latin word for “one hundredths.”
We could also write our money like this, as we do on cheques (although it looks funny!):
$14.75 = [latex]\$14\frac{75}{100}[/latex]
$12.25 = [latex]\$12\frac{25}{100}[/latex]
Cents
We have another way of writing money. We often write money that is less than one dollar using a cent sign which is a c for cent with a line through it ¢.
- $0.05 = 5¢
- $0.33 = 33¢
- $0.10 = 10¢
- $0.25 = 25¢
- $0.99 = 99¢
- $1.08 = 108¢
It is incorrect to use both a dollar sign and a cent sign. Instead of $4.53¢, do $4.53 or 453¢.
It is incorrect to use a cent sign with a decimal point. Instead of 4.53¢, do $4.53 or 453¢.
Important Information
We do not need to use a decimal point with the cent sign. A decimal point would indicate a fraction or part of one cent.
For example, If a sign said “Ice cream cones 0.50¢” those ice cream cones would only cost half a cent each!
Pay careful attention to the way amounts of money are written.
Exercise 11
Rewrite these using the other common way of writing money. Remember to use the ¢ or $ as needed.
- $0.75 = 75¢
- 83¢ = $0.83
- $0.01 =
- 47¢ =
- $0.04 =
- 3¢ =
- $0.40 =
- 101¢ =
- $0.29 =
- 50¢ =
- $0.80 =
- 99¢ =
- $1.00 =
- 175¢ =
- $1.10 =
Answers to Exercise 11
- 75¢
- $0.83
- 1¢
- $0.47
- 4¢
- $0.03
- 40¢
- $1.01
- 29¢
- $0.50
- 80¢
- $0.99
- 100¢
- $1.75
- 110¢
Exercise 12
Correct the following ways of writing money.
- .50¢ = 50¢
- .99¢ =
- .20¢ =
- ¢0.40 =
Answers to Exercise 12
- 50¢
- 99¢
- 20¢
- 40¢
Exercise 13: Review
Complete the chart so that each question has the amount written as a decimal, a common fraction, and in words. The first two are done.
| # | Decimal | Fraction | In words |
|---|---|---|---|
| a | .048 | [latex]\frac{48}{1000}[/latex] | forty eight thousandths |
| b | 0.7 | [latex]\frac{7}{10}[/latex] | seven tenths |
| c | four hundredths | ||
| d | 0.006 | ||
| e | [latex]16\frac{2}{1000}[/latex] | ||
| f | twelve and fifteen hundredths | ||
| g | 463.03 | ||
| h | [latex]213\frac{25}{1000}[/latex] | ||
| i | seventy-five and twenty-eight thousandths | ||
| j | 1833.018 | ||
| k | [latex]12\frac{418}{10000}[/latex] | ||
| l | nine tenths |
Answer to Exercise 13
| # | Decimal | Fraction | In words |
|---|---|---|---|
| c | 0.04 | [latex]\frac{4}{100}[/latex] | four hundredths |
| d | 0.006 | [latex]\frac{6}{1000}[/latex] | six thousandths |
| e | 16.002 | [latex]16\frac{2}{1000}[/latex] | sixteen and two thousandths |
| f | 12.15 | [latex]12\frac{15}{100}[/latex] | twelve and fifteen hundredths |
| g | 463.03 | [latex]463\frac{3}{100}[/latex] | four hundred sixty-three and three hundredths |
| h | 213.025 | [latex]213\frac{25}{1000}[/latex] | two hundred thirteen and twenty-five thousandths |
| i | 75.028 | [latex]75\frac{28}{1000}[/latex] | seventy-five and twenty-eight thousandths |
| j | 1833.018 | [latex]1833\frac{18}{1000}[/latex] | one thousand eight hundred thirty-three and eighteen thousandths |
| k | 12.0418 | [latex]12\frac{418}{10000}[/latex] | twelve and four hundred eighteen ten-thousandths |
| l | 0.9 | [latex]\frac{9}{10}[/latex] | nine tenths |
Topic B: Self-Test
Mark /41 Aim 36/41
- Write as decimals. (10 marks)
- [latex]\dfrac{3}{10}[/latex]
- [latex]\dfrac{24}{1000}[/latex]
- [latex]\dfrac{36}{1000}[/latex]
- [latex]\dfrac{206}{10000}[/latex]
- [latex]3\dfrac{123}{1000}[/latex]
- [latex]\dfrac{2}{100}[/latex]
- [latex]6\dfrac{3}{10}[/latex]
- [latex]4\dfrac{11}{1000}[/latex]
- [latex]6\dfrac{250}{1000}[/latex]
- [latex]9\dfrac{47}{10000}[/latex]
- Change these decimals to common fractions. (8 marks)
- 0.5
- 0.04
- 0.37
- 0.010
- 3.0918
- 3.025
- 0.164
- 2.1498
- Write as common fractions and as decimals. (20 marks)
- one hundredth
- forty-seven hundredths
- two hundred seventy-one thousandths
- forty-one thousandths
- one hundred twenty ten-thousandths
- four and four tenths
- two hundred sixty and fourteen ten-thousandths
- seven and two hundred eleven thousandths
- forty and six hundredths
- five dollars and sixty-three cents
- Write the amount of money with numerals, using a $ sign. (3 marks)
- Five dollars and sixty cents
- Seventy-two cents
- Fifty-six cents
- Six cents
- One hundred twenty-four cents
Answers to Topic B Self-Test
- Write as decimals.
- 0.3
- 0.24
- 0.036
- 0.0206
- 3.123
- 0.02
- 6.3
- 4.011
- 6.250
- 9.0047
- Change these decimals to common fractions.
- [latex]\dfrac{5}{10}[/latex]
- [latex]\dfrac{4}{100}[/latex]
- [latex]\dfrac{37}{100}[/latex]
- [latex]\dfrac{10}{1000}[/latex]
- [latex]3\dfrac{918}{10000}[/latex]
- [latex]3\dfrac{25}{1000}[/latex]
- [latex]\dfrac{164}{1000}[/latex]
- [latex]2\dfrac{1498}{10000}[/latex]
- Write as common fractions and as decimals.
- [latex]\dfrac{1}{100}\text{ and }0.01[/latex]
- [latex]\dfrac{47}{100}\text{ and }0.47[/latex]
- [latex]\dfrac{271}{1000}\text{ and }0.271[/latex]
- [latex]\dfrac{41}{1000}\text{ and }0.041[/latex]
- [latex]\dfrac{120}{10000}\text{ and }0.0120[/latex]
- [latex]4\dfrac{4}{10}\text{ and }4.4[/latex]
- [latex]260\dfrac{14}{10000}\text{ and }260.0014[/latex]
- [latex]7\dfrac{211}{1000}\text{ and }7.211[/latex]
- [latex]40\dfrac{6}{100}\text{ and }40.06[/latex]
- [latex]\$5\dfrac{63}{100}\text{ and }\$5.63[/latex]
- Write the amount of money with numerals, using a $ sign.
- $5.60
- $0.72
- $0.56
- $0.06
- $1.24
We understand numbers by the way the digits (numerals) are arranged in relationship to each other and to the decimal point. Each position has a certain value. Our number system is a decimal system. The place value is based on ten.
e.g., ⅔, ³⁄₇ , ⁴⁹⁄₅₀
The bottom number in a common fraction; tells into how many equal parts the whole thing has been divided.
Any of the ten numerals (0 to 9) are digits. This term comes from our ten fingers which are called digits. The numerals came to be called "digits" from the practice of counting on the fingers!
A whole number and a decimal fraction. 1.75