Connecting Fractions to Decimal Notation - Fourth Grade Mathematics
Throughout history, mathematicians have created different writing systems to represent identical ideas. When you write the Roman numeral V or the Arabic digit 5, both symbols represent five units. In fourth-grade mathematics, fractions and decimals share this exact relationship: decimals are simply a specialized, elegant way to write fractions whose denominators are powers of ten without needing a fraction bar. When you write 0.7, you are writing 7/10 in decimal shorthand. When you write 0.45, you are writing 45/100. In this chapter, you will discover how the decimal point acts as a gateway separating whole numbers from fractional parts, master reading and writing decimals in tenths and hundredths, and connect decimal notation to base-ten place value.
The Decimal Point as a Place-Value Gateway
Our base-ten place-value system does not stop at the ones place; it continues to the right into fractional parts, with the decimal point serving as the landmark marker.
+-----------------------------------------------------------------------------------+
| THE PLACE VALUE CHART WITH DECIMALS |
+-----------------------------------+-----------------------------------------------+
| WHOLE NUMBERS | FRACTIONAL PARTS |
+-----------------+-----------------+---------------+---------------+---------------+
| Hundreds (100) | Tens (10) | Ones (1) [.]| Tenths (1/10)|Hundredths(1/100
+-----------------+-----------------+---------------+---------------+---------------+
| 2 | 4 | 7 . | 6 | 3 |
+-----------------+-----------------+---------------+---------------+---------------+
| |
| Standard Form: 247.63 |
| Word Form : Two hundred forty-seven AND sixty-three hundredths |
| Fraction Form: 247 and 63/100 |
| |
+-----------------------------------------------------------------------------------+
The Solemn Rule of the Word "AND"
Notice where the word "and" appears in word form: right at the decimal point! In mathematics: "Two hundred forty-seven" represents the whole number. "AND" represents the decimal point [.]. "Sixty-three hundredths" represents the fractional decimal portion. This is why you must never use the word "and" when reading whole numbers like 247; "and" is strictly reserved to signal the transition across the decimal point into tenths and hundredths!
Tenths in Decimal Notation
The first place value column to the right of the decimal point is the tenths place.
+-----------------------------------------------------------------------------------+
| TENTHS: FRACTION TO DECIMAL CHART |
+-----------------------+-----------------------+-----------------------------------+
| Fraction Form | Decimal Form | Word Form |
+-----------------------+-----------------------+-----------------------------------+
| 1/10 | 0.1 | One tenth |
| 4/10 | 0.4 | Four tenths |
| 7/10 | 0.7 | Seven tenths |
| 9/10 | 0.9 | Nine tenths |
| 10/10 = 1 whole | 1.0 | One whole |
+-----------------------+-----------------------+-----------------------------------+
The Leading Zero: Why We Write 0.4 Instead of .4
Why do mathematicians write a zero in front of the decimal point (0.4) rather than writing just .4? A small decimal point standing alone can easily be missed or mistaken for a speck of pencil dust on a page. Placing a zero in the ones place (0.4) announces clearly that there are zero whole units and alerts the reader that a decimal follows immediately.
Hundredths in Decimal Notation
The second place value column to the right of the decimal point is the hundredths place.
+-----------------------------------------------------------------------------------+
| HUNDREDTHS: FRACTION TO DECIMAL CHART |
+-----------------------+-----------------------+-----------------------------------+
| Fraction Form | Decimal Form | Word Form |
+-----------------------+-----------------------+-----------------------------------+
| 1/100 | 0.01 | One hundredth |
| 8/100 | 0.08 | Eight hundredths |
| 25/100 | 0.25 | Twenty-five hundredths |
| 70/100 = 7/10 | 0.70 = 0.7 | Seventy hundredths (Seven tenths)|
| 99/100 | 0.99 | Ninety-nine hundredths |
+-----------------------+-----------------------+-----------------------------------+
The Single-Digit Hundredths Trap: 0.08 Versus 0.8
Pay close attention to single-digit hundredths like 8/100! To write 8/100 as a decimal, the digit 8 must sit in the second place value to the right of the decimal point: 0.08. The 0 in the tenths place acts as an essential placeholder! If you accidentally omit the zero and write 0.8, you have written eight tenths (8/10 = 80/100), which is ten times larger than 0.08! Always verify: Tenths have ONE digit after the decimal point (0.8). Hundredths have TWO digits after the decimal point (0.08).
Chapter Practice Exercises
Exercise 1: Convert each fraction to its equivalent decimal form: 6/10, 35/100, 4/100, and 90/100.
Exercise 2: Convert each decimal to its equivalent fraction form (with denominator 10 or 100): 0.3, 0.07, 0.62, and 0.50.
Exercise 3: Write the number 48.09 in word form and as a mixed number.
Exercise 4: Explain why 0.6 and 0.60 represent the exact same mathematical value, using both place value and fraction reasoning.
Exercise 5: A student is asked to write 5/100 as a decimal and writes 0.5. Explain why this answer is incorrect and state the correct decimal representation.
Solutions and Step-by-Step Explanations
Solution 1: 6/10 = 0.6 (6 in tenths place). 35/100 = 0.35 (3 tenths and 5 hundredths). 4/100 = 0.04 (0 tenths placeholder, 4 in hundredths place). 90/100 = 0.90 (or 0.9).
Solution 2:
0.3 = 3/10.
0.07 = 7/100.
0.62 = 62/100 (which simplifies to 31/50).
0.50 = 50/100 (or 5/10 = 1/2).
Solution 3: In word form, 48.09 is written as "forty-eight and nine hundredths." As a mixed number, it is written as 48 and 9/100.
Solution 4: In 0.6, the digit 6 sits in the tenths place, representing 6/10. In 0.60, there are 6 tenths and 0 hundredths, representing 60/100. As fractions, 6/10 is equivalent to 60/100 (multiplying numerator and denominator by 10/10). On a place-value grid, 6 tenths covers 6 full columns of 10 squares, which is exactly 60 squares out of 100. Adding a trailing zero to a decimal does not change its value.
Solution 5: The decimal 0.5 represents 5 tenths, which equals 5/10 (or 50/100). The fraction 5/100 represents 5 hundredths, which is ten times smaller than 5 tenths. To represent 5 hundredths, the digit 5 must sit in the second column after the decimal point, requiring a placeholder zero in the tenths column: 0.05.