2. Decimal Place Value and Equivalent Forms

Session 2: Decimal Place Value and Equivalent Forms

Learning targets

• Read decimals through thousandths and beyond.
• Write decimals in expanded and fraction form.
• Use trailing zeros without changing value.
• Relate decimal shifts to multiplication and division by powers of ten.

Why this matters

Decimals extend the same place-value system to quantities smaller than one. This session builds the precision needed for money, measurement, scientific data, and calculator work.

Core lesson

To the right of the decimal point are tenths, hundredths, thousandths, and smaller places. In \(7.049\), the 0 is in the tenths place, the 4 is in the hundredths place, and the 9 is in the thousandths place:
\[ 7.049=7+\frac{4}{100}+\frac{9}{1000}. \]

Trailing zeros do not change a decimal's value, so \(3.5=3.50=3.500\). They can make place alignment clearer. Leading zeros after the decimal do matter: \(0.06\) is six hundredths, not six tenths. Multiplying by 10 shifts every digit one place toward a greater value; dividing by 10 shifts every digit one place toward a smaller value. It is more accurate to think about digits changing place than to memorize decimal-point movement.

Foundations in depth

The decimal point marks the boundary between ones and tenths; it does not create a new number system. Every place still follows a factor of ten. This explains why 0.4 is larger than 0.39: four tenths is forty hundredths, and forty hundredths is greater than thirty-nine hundredths. Writing equivalent trailing zeros is a powerful comparison tool because 0.4 can be displayed as 0.40 without changing its value.

Precision describes how finely a measurement is recorded. A price of $6.50 and a measurement of 6.500 meters may use zeros for different reasons: the money is stated to cents, while the measurement is stated to thousandths. Do not remove zeros blindly when the stated precision matters. When multiplying or dividing by powers of ten, explain the change through place value: digits shift to positions worth ten times as much or one tenth as much.

A reliable method

1. Place the decimal in a chart with ones on its left.
2. Name the last occupied place to read the decimal correctly.
3. Add trailing zeros when matching precision or comparing values.
4. Use a power-of-ten estimate to verify the direction of a shift.

Ten GED®-style worked examples

Worked example 1

GED®-style problem. Write \(0.407\) in expanded form.

Step 1 — Understand. Identify the given quantities, the exact target, and any unit, sign, or precision requirement.

Step 2 — Plan and set up. Mark the decimal place and rewrite with aligned or expanded places.

Step 3 — Solve carefully. \(0.407=\frac{4}{10}+\frac{7}{1000}=0.4+0.007\). The zero holds the hundredths place.

Step 4 — Verify and state the answer. Use trailing zeros or a power-of-ten estimate to verify scale.

Why this method works. The setup preserves the numerical relationship while making place value, operation structure, or unit cancellation visible.

Worked example 2

GED®-style problem. Write \(6.25\) as a fraction.

Step 1 — Understand. Identify the given quantities, the exact target, and any unit, sign, or precision requirement.

Step 2 — Plan and set up. Mark the decimal place and rewrite with aligned or expanded places.

Step 3 — Solve carefully. \(6.25=\frac{625}{100}=\frac{25}{4}=6\frac14\).

Step 4 — Verify and state the answer. Use trailing zeros or a power-of-ten estimate to verify scale.

Why this method works. The setup preserves the numerical relationship while making place value, operation structure, or unit cancellation visible.

Worked example 3

GED®-style problem. Find \(0.084\times100\).

Step 1 — Understand. Identify the given quantities, the exact target, and any unit, sign, or precision requirement.

Step 2 — Plan and set up. Mark the decimal place and rewrite with aligned or expanded places.

Step 3 — Solve carefully. Each digit becomes one hundred times as valuable, so \(0.084\times100=8.4\).

Step 4 — Verify and state the answer. Use trailing zeros or a power-of-ten estimate to verify scale.

Why this method works. The setup preserves the numerical relationship while making place value, operation structure, or unit cancellation visible.

Worked example 4

GED®-style problem. Write 12.305 in expanded form.

Step 1 — Understand. Identify the given quantities, the exact target, and any unit, sign, or precision requirement.

Step 2 — Plan and set up. Name every occupied place to the right and left of the decimal point.

Step 3 — Solve carefully. 12.305 = 10 + 2 + 3/10 + 0/100 + 5/1000 = 10 + 2 + 0.3 + 0.005.

Step 4 — Verify and state the answer. Adding 12 + 0.3 + 0.005 gives 12.305.

Why this method works. The setup preserves the numerical relationship while making place value, operation structure, or unit cancellation visible.

Worked example 5

GED®-style problem. What value does the digit 6 have in 0.0064?

Step 1 — Understand. Identify the given quantities, the exact target, and any unit, sign, or precision requirement.

Step 2 — Plan and set up. Count tenths, hundredths, then thousandths from the decimal point.

Step 3 — Solve carefully. The first two decimal positions are zeros; 6 is in the thousandths place. Its value is 6/1000 = 0.006.

Step 4 — Verify and state the answer. 0.0064 = 0.006 + 0.0004, confirming the contribution.

Why this method works. The setup preserves the numerical relationship while making place value, operation structure, or unit cancellation visible.

Worked example 6

GED®-style problem. Compare 4.7 and 4.069.

Step 1 — Understand. Identify the given quantities, the exact target, and any unit, sign, or precision requirement.

Step 2 — Plan and set up. Append trailing zeros so equal places line up.

Step 3 — Solve carefully. Write 4.7 as 4.700. Ones match; in the tenths place, 7 > 0. Therefore 4.700 > 4.069.

Step 4 — Verify and state the answer. Both are between 4 and 5, and 4.7 lies much closer to 5.

Why this method works. The setup preserves the numerical relationship while making place value, operation structure, or unit cancellation visible.

Worked example 7

GED®-style problem. Order 0.405, 0.45, 0.045, and 0.4501 from least to greatest.

Step 1 — Understand. Identify the given quantities, the exact target, and any unit, sign, or precision requirement.

Step 2 — Plan and set up. Write each value to four decimal places.

Step 3 — Solve carefully. The aligned values are 0.4050, 0.4500, 0.0450, and 0.4501. Comparing tenths then later places gives 0.045 < 0.405 < 0.45 < 0.4501.

Step 4 — Verify and state the answer. The first is below one tenth; the others are near four or five tenths, supporting the order.

Why this method works. The setup preserves the numerical relationship while making place value, operation structure, or unit cancellation visible.

Worked example 8

GED®-style problem. A medication label shows 0.08 gram. Express this amount in thousandths of a gram.

Step 1 — Understand. Identify the given quantities, the exact target, and any unit, sign, or precision requirement.

Step 2 — Plan and set up. Rewrite hundredths as equivalent thousandths.

Step 3 — Solve carefully. 0.08 = 8/100. Multiply numerator and denominator by 10: 80/1000 = 0.080. The amount is 80 thousandths of a gram.

Step 4 — Verify and state the answer. 0.080 and 0.08 occupy the same point; only the displayed precision changes.

Why this method works. The setup preserves the numerical relationship while making place value, operation structure, or unit cancellation visible.

Worked example 9

GED®-style problem. Compute 3.407 × 1,000 using place value.

Step 1 — Understand. Identify the given quantities, the exact target, and any unit, sign, or precision requirement.

Step 2 — Plan and set up. Multiplication by 1,000 gives every digit a place worth 1,000 times as much.

Step 3 — Solve carefully. Three place shifts change 3.407 to 3,407. Thus 3.407 × 1,000 = 3,407.

Step 4 — Verify and state the answer. Dividing 3,407 by 1,000 returns 3.407.

Why this method works. The setup preserves the numerical relationship while making place value, operation structure, or unit cancellation visible.

Worked example 10

GED®-style problem. A measurement was entered as 62.5 but should have been divided by 100. What is the corrected value?

Step 1 — Understand. Identify the given quantities, the exact target, and any unit, sign, or precision requirement.

Step 2 — Plan and set up. Dividing by 100 moves every digit two places toward smaller value.

Step 3 — Solve carefully. 62.5 ÷ 100 = 0.625. Add placeholder zeros as needed while shifting the digits.

Step 4 — Verify and state the answer. Multiplying 0.625 by 100 restores 62.5.

Why this method works. The setup preserves the numerical relationship while making place value, operation structure, or unit cancellation visible.

Adult-life and GED® applications

This session's reasoning applies to:

• prices measured to cents
• laboratory and construction readings
• metric unit changes

When the setting changes, keep the mathematical relationship and unit logic visible. Do not choose an operation from a keyword alone. First decide what the quantities mean and how they relate.

Common traps and repairs

• Reading 0.04 as four tenths: The 4 is two places right of the decimal, so it is four hundredths.
• Dropping an internal zero: Only trailing zeros may be removed without changing place value.
• Shifting in the wrong direction: Multiplication by a number greater than one should increase a positive value.

Retrieval and mixed practice

Retrieve the whole-number place ladder from Session 1, then extend it three places to the right of ones.

After recalling an answer, compare it with the worked reasoning and correct the explanation, not only the final number. Then mix this session's skill with at least one earlier topic so method selection becomes part of the practice.

Session check

Exit reflection

Complete these two sentences: The main decision in this session is... and I can check my answer by... Keep both statements specific enough to use on a new problem.