Understanding Decimals for 5th Grade: Place Value and Meaning

Decimals represent parts of a whole using our base-ten number system extended to the right of the decimal point. In Grade 5, students deepen their understanding of decimals to tenths, hundredths, and thousandths. Decimals are used everywhere: prices ($3.99), distances (4.5 miles), measurements (2.75 liters), sports statistics (3.14 batting average), and scientific data. Understanding decimals helps students interpret real-world quantities that fall between whole numbers.

A decimal has two main parts: the whole number part (to the left of the decimal point) and the fractional part (to the right). The decimal point separates these parts. Each place to the right of the decimal point is one-tenth of the place to its left. So tenths, hundredths, and thousandths follow the same pattern as tens, hundreds, and thousands—just in the opposite direction. This symmetry makes decimals a natural extension of place value.

DETAILED EXPLANATION

Place values to the right of the decimal point:

• Tenths: \(\frac{1}{10}\) or 0.1 (one place to the right)

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• Hundredths: \(\frac{1}{100}\) or 0.01 (two places)

• Thousandths: \(\frac{1}{100}\) or 0.001 (three places)

Each place is 10 times smaller than the place to its left. So \(0.1 = 10 \times 0.01\) and \(0.01 = 10 \times 0.001\).

In the number 3.456: The 3 is in the ones place (value 3), the 4 is in the tenths place (value 0.4), the 5 is in the hundredths place (value 0.05), and the 6 is in the thousandths place (value 0.06). The total value is \(3 + 0.4 + 0.05 + 0.006 = 3.456\).

Expanded form for decimals: \(2.34 = 2 + \frac{3}{10} + \frac{4}{100}\) or \(2 + 0.3 + 0.04\). Writing decimals in expanded form reinforces place value understanding.

WORKED EXAMPLES WITH STEP BY STEP SOLUTIONS

Example 1

Maria ran 3.5 miles. What does the 5 represent?

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Solutions:

Step 1: In 3.5, identify the place of each digit. The 3 is to the left of the decimal point (ones place). The 5 is to the right of the decimal point.

Step 2: The 5 is in the first place to the right of the decimal point, which is the tenths place.

Step 3: The value of the digit 5 in the tenths place is \(5 \times 0.1 = 0.5\), or 5 tenths, or \(\frac{5}{10}\).

Step 4: So the 5 represents 0.5 mile—half a mile—or 5 tenths of a mile.

Answer: The 5 represents 5 tenths (0.5 mile).

Example 2

Write 2.34 in expanded form.

Solutions:

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Step 1: Identify each digit and its place value. In 2.34: 2 is in the ones place, 3 is in the tenths place, 4 is in the hundredths place.

Step 2: Write each digit’s value. Ones: \(2 \times 1 = 2\). Tenths: \(3 \times 0.1 = 0.3\). Hundredths: \(4 \times 0.01 = 0.04\).

Step 3: Combine as a sum: \(2 + 0.3 + 0.04\).

Step 4: Optional fraction form: \(2 + \frac{3}{10} + \frac{4}{100}\).

Step 5: Verify: \(2 + 0.3 + 0.04 = 2.34\). ✓

Answer: \(2.34 = 2 + 0.3 + 0.04\) or \(2 + \frac{3}{10} + \frac{4}{100}\)

Example 3

A rope is 4.567 meters long. What is the value of the 6?

Solutions:

Step 1: In 4.567, list the places from left to right: ones (4), tenths (5), hundredths (6), thousandths (7).

Step 2: The digit 6 is in the hundredths place.

Step 3: The value of 6 in the hundredths place is \(6 \times 0.01 = 0.06\).

Step 4: 0.06 is 6 hundredths, or \(\frac{6}{100}\).

Answer: The value of the 6 is 0.06 (6 hundredths).

Example 4

What decimal is represented by \(5 + \frac{2}{10} + \frac{7}{100}\)?

Solutions:

Step 1: The whole number part is 5. Write 5, then a decimal point.

Step 2: \(\frac{2}{10} = 0.2\), so we have 2 in the tenths place. \(\frac{7}{100} = 0.07\), so we have 7 in the hundredths place.

Step 3: Combine: 5.27.

Step 4: Verify: \(5 + 0.2 + 0.07 = 5.27\). ✓

Answer: 5.27