Chapter Notes: Decimal Fractions
Money is a big part of our daily lives! When we buy something that costs three dollars and fifty cents, we write $3.50. When we measure height in meters and get 2 meters and 3 tenths, we can write 2.3 meters. Numbers like 3.50 and 2.3 are called decimal fractions, or just decimals. They are a special way to write fractions that have denominators of 10, 100, 1000, and so on. Learning about decimals helps us work with money, measure things more precisely, and understand numbers between whole numbers. In this chapter, you will discover how decimals work, how to read them, write them, and use them in everyday situations.
Understanding Place Value with Decimals
Before we work with decimals, we need to understand place value. Place value tells us what each digit in a number means based on where it sits. You already know that in the number 345, the 5 is in the ones place, the 4 is in the tens place, and the 3 is in the hundreds place.
When we add a decimal point (the dot), we can show parts of a whole number. The decimal point separates the whole number part from the fractional part. To the right of the decimal point, we have places for tenths, hundredths, thousandths, and so on.
Here is how the place value system works with decimals:
Let's look at the number 34.56. The 3 is in the tens place, the 4 is in the ones place, the 5 is in the tenths place, and the 6 is in the hundredths place. This means:
- 3 tens = 30
- 4 ones = 4
- 5 tenths = 5/10
- 6 hundredths = 6/100
When we add them all together: 30 + 4 + 5/10 + 6/100 = 34.56
Think of a decimal point as a fence. Everything to the left of the fence shows whole things. Everything to the right shows parts of one whole thing.
Reading and Writing Decimals
To read a decimal out loud, we follow these steps:
- Read the whole number part (the digits to the left of the decimal point)
- Say "and" for the decimal point
- Read the digits to the right as a whole number
- Say the place value of the last digit
Example: Read the decimal 2.3
How do we say this number out loud?
Solution:
The whole number part is 2.
The decimal point is read as "and."
The digit after the decimal point is 3, and it is in the tenths place.
We read this as two and three tenths.
Example: Read the decimal 45.67
How do we say this number out loud?
Solution:
The whole number part is 45, which we read as "forty-five."
The decimal point is read as "and."
The digits after the decimal point are 67, and the last digit is in the hundredths place.
We read this as forty-five and sixty-seven hundredths.
Example: Read the decimal 0.8
How do we say this number out loud?
Solution:
The whole number part is 0, which we read as "zero."
The decimal point is read as "and."
The digit after the decimal point is 8, and it is in the tenths place.
We read this as zero and eight tenths or simply eight tenths.
Writing Decimals from Words
Sometimes you need to write a decimal when someone tells you the number in words. Let's practice writing decimals from words.
Example: Write "five and two tenths" as a decimal.
What decimal matches these words?
Solution:
The whole number part is 5.
We write the decimal point after the 5.
Two tenths means 2 in the tenths place.
The decimal is 5.2.
Example: Write "twelve and thirty-four hundredths" as a decimal.
What decimal matches these words?
Solution:
The whole number part is 12.
We write the decimal point after the 12.
Thirty-four hundredths means 34 in the tenths and hundredths places.
The decimal is 12.34.
Connecting Decimals to Fractions
Decimals and fractions are two different ways to show the same thing! When we write a decimal, we are actually writing a fraction with a denominator of 10, 100, 1000, or another power of 10.
The place value tells us what denominator to use:
- One place after the decimal point = tenths = denominator of 10
- Two places after the decimal point = hundredths = denominator of 100
- Three places after the decimal point = thousandths = denominator of 1000
Example: Write 0.7 as a fraction.
What fraction is equal to this decimal?
Solution:
The 7 is in the tenths place.
This means 7 tenths.
As a fraction, we write this as 7/10.
Example: Write 0.25 as a fraction.
What fraction is equal to this decimal?
Solution:
The 5 is in the hundredths place.
This means 25 hundredths.
As a fraction, we write this as 25/100.
Example: Write 3.6 as a mixed number.
What mixed number is equal to this decimal?
Solution:
The whole number part is 3.
The 6 is in the tenths place, which means 6/10.
As a mixed number, we write this as 3 6/10 or in simplest form 3 3/5.
Writing Fractions as Decimals
We can also change fractions into decimals. If the fraction has a denominator of 10, 100, or 1000, this is easy!
Example: Write 3/10 as a decimal.
What decimal is equal to this fraction?
Solution:
The denominator is 10, so we need one place after the decimal point.
The numerator is 3, so we put 3 in the tenths place.
The decimal is 0.3.
Example: Write 47/100 as a decimal.
What decimal is equal to this fraction?
Solution:
The denominator is 100, so we need two places after the decimal point.
The numerator is 47, so we put 47 in the tenths and hundredths places.
The decimal is 0.47.
Example: Write 5 8/100 as a decimal.
What decimal is equal to this mixed number?
Solution:
The whole number part is 5.
The fraction part is 8/100, which has two places after the decimal point.
We write 08 in the tenths and hundredths places (we need the 0 to show there are no tenths).
The decimal is 5.08.
Comparing and Ordering Decimals
Sometimes we need to figure out which decimal is larger or smaller. To compare decimals, we look at the digits from left to right, starting with the whole number part.
Here are the steps to compare two decimals:
- Line up the decimal points
- Compare the digits in each place value, starting from the left
- The first place where the digits are different tells you which number is larger
Example: Which is greater: 3.45 or 3.5?
Which decimal is larger?
Solution:
Line up the decimal points: 3.45 and 3.5 (we can write 3.5 as 3.50 to make it easier).
Compare the ones place: both have 3, so they are equal so far.
Compare the tenths place: both have 5, so they are still equal.
Compare the hundredths place: 3.45 has 5, and 3.50 has 0.
Since 5 > 0, we know 3.5 is greater than 3.45.
A helpful trick: you can add zeros to the end of a decimal without changing its value. So 3.5 is the same as 3.50 or 3.500. This makes comparing easier!
Example: Order these decimals from least to greatest: 2.8, 2.08, 2.80, 2.18
What is the correct order?
Solution:
Write all decimals with two places: 2.80, 2.08, 2.80, 2.18.
Compare the ones place: all have 2.
Compare the tenths place: 2.08 has 0, 2.18 has 1, 2.80 and 2.80 have 8.
The order is 2.08, 2.18, 2.8, 2.80 (note that 2.8 and 2.80 are equal).
Decimals on a Number Line
A number line is a great tool for understanding decimals. It helps us see where decimals are located between whole numbers. Each space between two whole numbers can be divided into 10 equal parts (tenths) or 100 equal parts (hundredths).
Imagine a ruler. The marks between each inch can represent tenths or hundredths, depending on how closely spaced they are.
When we place 2.3 on a number line, we:
- Find the whole number 2
- Move 3 tenths (or 3 out of 10 spaces) toward 3
- Mark the spot
Example: Where is 1.6 on a number line between 1 and 2?
How do we locate this decimal?
Solution:
Start at 1 on the number line.
Divide the space between 1 and 2 into 10 equal parts (each part is one tenth).
Count 6 parts to the right of 1.
The point at 1.6 is 6 tenths of the way from 1 to 2.
Rounding Decimals
Rounding means finding a nearby number that is easier to work with. We round decimals to a certain place value, just like we round whole numbers. Rounding is useful when we don't need an exact answer or when we estimate.
Here are the steps to round a decimal:
- Find the place value you are rounding to
- Look at the digit to the right of that place
- If that digit is 5 or more, round up (add 1 to the rounding place)
- If that digit is less than 5, round down (keep the rounding place the same)
- Drop all digits to the right of the rounding place
Example: Round 4.67 to the nearest tenth.
What is 4.67 rounded to the nearest tenth?
Solution:
The tenths place has the digit 6.
Look at the digit to the right: 7.
Since 7 ≥ 5, we round up.
6 + 1 = 7, so the tenths place becomes 7.
Rounded to the nearest tenth, 4.67 is 4.7.
Example: Round 3.24 to the nearest tenth.
What is 3.24 rounded to the nearest tenth?
Solution:
The tenths place has the digit 2.
Look at the digit to the right: 4.
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Rounded to the nearest tenth, 3.24 is 3.2.
Example: Round 7.89 to the nearest whole number.
What is 7.89 rounded to the nearest whole number?
Solution:
The ones place has the digit 7.
Look at the digit to the right (the tenths place): 8.
Since 8 ≥ 5, we round up.
7 + 1 = 8.
Rounded to the nearest whole number, 7.89 is 8.
Using Decimals with Money
One of the most common places we see decimals is with money. In the United States, we use dollars and cents. One dollar equals 100 cents. When we write money amounts, we use a decimal point to separate dollars from cents.
For example:
- $5.25 means 5 dollars and 25 cents
- $0.75 means 0 dollars and 75 cents (or just 75 cents)
- $12.08 means 12 dollars and 8 cents
Notice that money always has exactly two decimal places (for the cents). If there are fewer than 10 cents, we write a 0 in the tenths place. For example, 8 cents is written as $0.08, not $0.8.
Example: Sarah has three quarters, one dime, and four pennies.
How much money does she have?Write the total amount as a decimal.
Solution:
Three quarters = 3 × 25 cents = 75 cents.
One dime = 10 cents.
Four pennies = 4 cents.
Total = 75 + 10 + 4 = 89 cents.
As a decimal, this is $0.89.
Example: A notebook costs $2.35 and a pen costs $1.50.
How much do they cost together?What is the total cost?
Solution:
Add the dollars: 2 + 1 = 3 dollars.
Add the cents: 35 + 50 = 85 cents.
Total cost = 3 dollars and 85 cents.
The total cost is $3.85.
Equivalent Decimals
Equivalent decimals are decimals that have the same value but look different. Adding zeros to the end of a decimal does not change its value.
For example:
- 0.5 = 0.50 = 0.500
- 3.2 = 3.20
- 7 = 7.0 = 7.00
This is helpful when comparing decimals or when working with money.
Example: Are 0.7 and 0.70 equivalent?
Do these decimals have the same value?
Solution:
Write 0.7 as a fraction: 7/10.
Write 0.70 as a fraction: 70/100.
Simplify 70/100: 70 ÷ 10 = 7, and 100 ÷ 10 = 10, so 70/100 = 7/10.
Both fractions equal 7/10, so 0.7 and 0.70 are equivalent.
Think of it like this: if you cut a pizza into 10 slices and take 5, that's the same as cutting it into 100 tiny pieces and taking 50. Either way, you get half the pizza! So 0.5 = 0.50.
Real-World Applications of Decimals
Decimals are everywhere in real life! Here are some common places where we use decimals:
- Money: Prices, bank balances, and making change
- Measurement: Height, weight, length (such as 1.5 meters or 3.75 liters)
- Sports: Scores, times, and records (such as running 100 meters in 12.4 seconds)
- Temperature: Weather reports (such as 72.5 degrees Fahrenheit)
- Cooking: Recipes with measurements like 2.5 cups of flour
Example: A runner finished a race in 15.8 seconds.
Another runner finished in 15.75 seconds.Who was faster?
Solution:
Write both times with two decimal places: 15.80 and 15.75.
Compare the tenths place: both have 7 in the tenths place.
Compare the hundredths place: 15.80 has 8, and 15.75 has 5.
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The runner who finished in 15.75 seconds was faster.
Understanding decimals opens up a whole new world of numbers! You can now work with amounts between whole numbers, measure things more precisely, handle money confidently, and solve problems that involve parts of wholes. Decimals are just another way to write fractions, and they make many everyday tasks much easier. As you continue practicing, reading, writing, comparing, and using decimals will become natural and easy!
