3.1 Decimal Place Value, Rounding & Comparing Decimals

3.1 Decimal Place Value, Rounding & Comparing Decimals

The decimal number system is an extension of the base-10 positional place value system used for whole numbers. While whole number place values increase by factors of 10 as you move to the left of the ones place (tens, hundreds, thousands), decimal place values decrease by factors of 10 as you move to the right of the decimal point. Understanding decimal place values, knowing how to round decimals accurately, and mastering methods to compare decimal magnitudes are fundamental skills tested extensively on the ACCUPLACER Arithmetic exam.

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The Base-10 Decimal Place Value System

The decimal point serves as the separator between the whole number part (to its left) and the fractional part (to its right). Each position to the right of the decimal point represents a fraction whose denominator is a power of 10:

• Tenths place ($10^{-1} = \frac{1}{10} = 0.1$): The first digit to the right of the decimal point.
• Hundredths place ($10^{-2} = \frac{1}{100} = 0.01$): The second digit to the right of the decimal point.
• Thousandths place ($10^{-3} = \frac{1}{1000} = 0.001$): The third digit to the right of the decimal point.
• Ten-thousandths place ($10^{-4} = \frac{1}{10000} = 0.0001$): The fourth digit to the right of the decimal point.

Decimal Place Value Chart

To visualize how place values align across the decimal point, examine the number 4,523.0769 in the table below:

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Reading and Writing Decimals

Converting decimal numbers between standard form, expanded form, and word form requires strict adherence to mathematical naming conventions.

Rules for Writing Decimals in Word Form

1. Read the whole number portion to the left of the decimal point as you normally would.
2. Express the decimal point using the word 'and'.
3. Read the number to the right of the decimal point as if it were a whole number.
4. State the place value of the last digit on the far right.

Mathematical Convention Note: In formal mathematics, the word 'and' is reserved exclusively for the decimal point. For example, $105$ is read as "one hundred five," whereas $100.05$ is read as "one hundred and five hundredths."

Examples of Decimal Forms

• Standard Form: $0.047$

  • Word Form: Forty-seven thousandths
  • Expanded Form: $(4 \times 0.01) + (7 \times 0.001) = \frac{4}{100} + \frac{7}{1000}$

• Standard Form: $12.308$

  • Word Form: Twelve and three hundred eight thousandths
  • Expanded Form: $(1 \times 10) + (2 \times 1) + (3 \times 0.1) + (0 \times 0.01) + (8 \times 0.001)$

• Standard Form: $5.0064$

  • Word Form: Five and sixty-four ten-thousandths
  • Expanded Form: $5 + (6 \times 0.001) + (4 \times 0.0001)$

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Rounding Decimals to Specified Place Values

Rounding decimals simplifies numbers while maintaining an approximation close to the original value. Rounding is frequently required after division or percentage calculations.

The Standard 4-Step Rounding Algorithm

1. Identify the Target Place: Locate the digit in the place value to which you are rounding (e.g., tenths, hundredths).
2. Examine the Test Digit: Look at the single digit immediately to the right of the target place.
3. Apply the Rounding Decision Rule:
   • If the test digit is 5, 6, 7, 8, or 9, round UP (add 1 to the target digit).
   • If the test digit is 0, 1, 2, 3, or 4, round DOWN (keep the target digit unchanged).
4. Drop Remaining Digits: Eliminate all digits to the right of the target place. Do not add zeros after the decimal point unless maintaining a specific decimal precision (such as cents in money).

Step-by-Step Worked Rounding Examples

Example 1: Standard Rounding

Round 38.4729 to (a) the nearest tenth, (b) the nearest hundredth, and (c) the nearest thousandth.

• (a) Nearest Tenth:

  • Target digit: 4 (tenths place).
  • Test digit: 7 (hundredths place).
  • Decision: Since $7 \ge 5$, add 1 to the target digit ($4 + 1 = 5$).
  • Result: 38.5

• (b) Nearest Hundredth:

  • Target digit: 7 (hundredths place).
  • Test digit: 2 (thousandths place).
  • Decision: Since $2 < 5$, keep the target digit unchanged (7).
  • Result: 38.47

• (c) Nearest Thousandth:

  • Target digit: 2 (thousandths place).
  • Test digit: 9 (ten-thousandths place).
  • Decision: Since $9 \ge 5$, add 1 to the target digit ($2 + 1 = 3$).
  • Result: 38.473

Example 2: Cascading Rounding Across Nines

Round 4.9962 to the nearest hundredth.

• Target digit: 9 (hundredths place).
• Test digit: 6 (thousandths place).
• Decision: Since $6 \ge 5$, add 1 to the hundredths digit ($9 + 1 = 10$).
• Carrying: Write 0 in the hundredths place and carry 1 to the tenths place ($9 + 1 = 10$). Write 0 in the tenths place and carry 1 to the ones place ($4 + 1 = 5$).
• Result: 5.00 (Keep the trailing zeros to explicitly show rounding to the hundredths place).

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Comparing and Ordering Decimals

A common misconception is assuming that decimals with more digits are larger in value (e.g., mistakenly thinking $0.089$ is greater than $0.4$). To compare decimals accurately, follow a systematic place-value alignment strategy.

The Decimal Comparison Procedure

1. Align Decimal Points: Write the numbers vertically, ensuring the decimal points are aligned in a straight column.
2. Annex Trailing Zeros: Add zeros to the right of the numbers so that all numbers have the same number of decimal places.
3. Compare Left-to-Right: Starting from the left (highest place value), compare digits in each column.
4. Identify First Difference: The number with the larger digit at the first point of difference is the larger decimal.

Step-by-Step Worked Comparison Example

Order the following decimals from least to greatest: $0.4$, $0.089$, $0.405$, $0.45$, $0.09$.

• Step 1: Align vertically and annex zeros to 3 decimal places:

  $$\begin{array}{rl} 0.400 & (\text{annexed two zeros}) \\ 0.089 & (\text{already 3 places}) \\ 0.405 & (\text{already 3 places}) \\ 0.450 & (\text{annexed one zero}) \\ 0.090 & (\text{annexed one zero}) \end{array}$$

• Step 2: Compare the tenths column:

  • $0.089$ and $0.090$ have 0 in the tenths place (smallest group).
  • $0.400$, $0.405$, and $0.450$ have 4 in the tenths place (larger group).

• Step 3: Compare within the tenths = 0 group:

  • Look at the hundredths place: $0.089$ has 8, whereas $0.090$ has 9.
  • Since $8 < 9$, $0.089 < 0.090$.

• Step 4: Compare within the tenths = 4 group:

  • Look at hundredths and thousandths: $0.400 < 0.405 < 0.450$.

• Final Ordered List (Least to Greatest): $0.089 < 0.09 < 0.4 < 0.405 < 0.45$

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