Decimals and Place Value | CSE Reviewer

What Is a Decimal?

A decimal is a way to write a number that is not a whole number. It uses a dot (called a decimal point) to separate the whole number part from the part smaller than 1.

For example, in \(3.75\):

• The whole number part is 3 (left of the dot)
• The decimal part is 75 (right of the dot)

Decimals are based on powers of 10: tenths, hundredths, thousandths, and so on.

Tip: Read the decimal point as "and." \(3.75\) is read as "3 and 75 hundredths."

Example: If you buy something for PHP 25.50, you read it as "25 and 50 hundredths" or "25 pesos and 50 centavos."

Meaning: A decimal splits a number into a whole part and a part smaller than 1, using a dot to separate them.

Place Value in Decimals

Place value tells us what each digit means. In decimals, the place value goes to the right of the decimal point as tenths, hundredths, thousandths, etc.

Let me show you the place values step by step for \(4.562\):

• 4 → ones place (whole number)
• 5 → tenths place (1st digit after the dot)
• 6 → hundredths place (2nd digit after the dot)
• 2 → thousandths place (3rd digit after the dot)

Place Value Chart:

Position	Name	Value
Left of dot	Ones	1
1st after dot	Tenths	\(\frac{1}{10}\) or 0.1
2nd after dot	Hundredths	\(\frac{1}{100}\) or 0.01
3rd after dot	Thousandths	\(\frac{1}{1000}\) or 0.001

Tip: Count the digits after the decimal point. 1 digit = tenths, 2 digits = hundredths, 3 digits = thousandths.

Example 1: In \(0.7\), what is the place value of 7?

Step 1: Count digits after dot → 1 digit
Step 2: 1 digit = tenths place

Answer: 7 is in the tenths place

Example 2: In \(0.34\), what is the place value of 4?

Step 1: Count digits after dot → 2 digits
Step 2: 2 digits = hundredths place

Answer: 4 is in the hundredths place

Meaning: Place value in decimals tells you the name of each position: tenths (1st), hundredths (2nd), thousandths (3rd), and so on.

Writing Decimals in Words

To write a decimal in words, write the whole number first, then "and," then write the decimal part using the place value name.

Step-by-step for \(5.62\):

• Step 1: Write whole number → "5"
• Step 2: Write "and" for the decimal point
• Step 3: Write decimal part with place value → 62 is in the hundredths place
• Step 4: Combine → "5 and 62 hundredths"

Tip: The last digit's place value tells you the name for the whole decimal part.

Example 1: Write \(0.8\) in words.

Step 1: Whole number = 0 (skip writing 0)
Step 2: Decimal part = 8 in tenths place

Answer: "8 tenths"

Example 2: Write \(3.25\) in words.

Step 1: Whole number = 3 → "3"
Step 2: Write "and"
Step 3: Decimal part = 25 in hundredths place → "25 hundredths"

Answer: "3 and 25 hundredths"

Meaning: Write decimals in words by saying the whole number, "and," then the decimal part with its place value name.

Comparing Decimals

To compare decimals, look at the digits from left to right. The first place where they differ tells you which is bigger.

Step-by-step to compare \(0.45\) and \(0.437\):

• Step 1: Make the same number of digits (add zeros if needed)
• Step 2: Compare digit by digit from left
• Step 3: The bigger digit means the bigger number

Shortcut: Add zeros to make the same length. \(0.45 = 0.450\), now compare \(0.450\) vs \(0.437\).

Example: Which is bigger: \(0.45\) or \(0.437\)?

Step 1: Make same length → \(0.45 = 0.450\)
Step 2: Compare \(0.450\) vs \(0.437\)
Step 3: Look at digits:
- 1st digit: 4 = 4 (same)
- 2nd digit: 5 vs 3 → 5 is bigger

Answer: \(0.45\) is bigger

Meaning: Compare decimals by adding zeros to make the same length, then compare digit by digit from the left.

Rounding Decimals

Rounding means making a decimal simpler by keeping fewer digits. You look at the digit after the place you want to keep.

Step-by-step to round \(3.674\) to the nearest hundredth:

• Step 1: Identify the place to keep → hundredths (2nd digit after dot)
• Step 2: Look at the next digit (the one to round) → 4
• Step 3: If it is 5 or more, round up. If it is 4 or less, keep the same
• Step 4: Write the rounded number

Rule: 5 or more → round up. 4 or less → keep the same.

Example 1: Round \(3.674\) to the nearest hundredth.

Step 1: Keep hundredths → 7 (2nd digit)
Step 2: Look at next digit → 4
Step 3: 4 is less than 5 → keep 7 the same
Step 4: Write rounded number → \(3.67\)

Answer: \(3.67\)

Example 2: Round \(2.586\) to the nearest hundredth.

Step 1: Keep hundredths → 8 (2nd digit)
Step 2: Look at next digit → 6
Step 3: 6 is 5 or more → round up 8 to 9
Step 4: Write rounded number → \(2.59\)

Answer: \(2.59\)

Meaning: To round, look at the digit after the place you want. If it is 5 or more, round up. If 4 or less, keep the same.

Adding Decimals

Adding decimals is easy if you line up the decimal points. Then add like normal numbers.

Step-by-step to add \(2.45 + 1.3\):

• Step 1: Line up the decimal points
• Step 2: Add zeros to make the same length (optional)
• Step 3: Add the numbers
• Step 4: Put the decimal point in the same place

Shortcut: Add zeros to make the same length. \(1.3 = 1.30\), now add \(2.45 + 1.30\).

Example: \(2.45 + 1.3\)

Step 1: Line up decimals:
\(\begin{array}{r} 2.45 \\ + 1.30 \\ \hline \end{array}\)
Step 2: Add zeros → \(1.3 = 1.30\)
Step 3: Add:
\(\begin{array}{r} 2.45 \\ + 1.30 \\ \hline 3.75 \\ \end{array}\)

Answer: \(3.75\)

Meaning: Line up decimal points, add zeros if needed, then add normally and keep the decimal in the same place.

Subtracting Decimals

Subtracting decimals works the same as adding. Line up the decimal points, add zeros if needed, then subtract.

Step-by-step to subtract \(5.6 - 2.34\):

• Step 1: Line up decimal points
• Step 2: Add zeros to make same length
• Step 3: Subtract normally
• Step 4: Keep decimal in the same place

Example: \(5.6 - 2.34\)

Step 1: Line up:
\(\begin{array}{r} 5.60 \\ - 2.34 \\ \hline \end{array}\)
Step 2: Add zero → \(5.6 = 5.60\)
Step 3: Subtract:
\(\begin{array}{r} 5.60 \\ - 2.34 \\ \hline 3.26 \\ \end{array}\)

Answer: \(3.26\)

Meaning: Line up decimal points, add zeros if needed, then subtract normally.

Multiplying Decimals

To multiply decimals, multiply like whole numbers first. Then count how many digits are after the decimal points in both numbers. Put the decimal in the answer so it has the same total number of digits after it.

Step-by-step to multiply \(2.4 × 0.3\):

• Step 1: Ignore decimals and multiply → \(24 × 3 = 72\)
• Step 2: Count decimal places → 2.4 has 1, 0.3 has 1 → total 2
• Step 3: Put decimal in answer so it has 2 digits after → \(0.72\)

Shortcut: Count total decimal places first, then place the decimal in the answer at the end.

Example: \(2.4 × 0.3\)

Step 1: Multiply without decimals → \(24 × 3 = 72\)
Step 2: Count decimal places → 1 + 1 = 2
Step 3: Put decimal → \(0.72\)

Answer: \(0.72\)

Example 2: \(1.25 × 0.4\)

Step 1: Multiply without decimals → \(125 × 4 = 500\)
Step 2: Count decimal places → 2 + 1 = 3
Step 3: Put decimal → \(0.500 = 0.5\)

Answer: \(0.5\)

Meaning: Multiply as whole numbers, count total decimal places, then put the decimal in the answer.

Dividing Decimals

To divide decimals, make the divisor a whole number first by moving the decimal point. Move the decimal in the dividend the same number of places. Then divide normally.

Step-by-step to divide \(4.8 ÷ 0.2\):

• Step 1: Move decimal in divisor to make it whole → 0.2 becomes 2 (move 1 place)
• Step 2: Move decimal in dividend the same → 4.8 becomes 48 (move 1 place)
• Step 3: Divide as whole numbers → 48 ÷ 2 = 24

Shortcut: Move both decimals the same number of places so the divisor becomes whole.

Example: \(4.8 ÷ 0.2\)

Step 1: Divide 0.2 by moving 1 place → 2
Step 2: Move 4.8 the same → 48
Step 3: Divide → \(48 ÷ 2 = 24\)

Answer: 24

Example 2: \(3.6 ÷ 0.15\)

Step 1: Move 0.15 by 2 places → 15
Step 2: Move 3.6 by 2 places → 360
Step 3: Divide → \(360 ÷ 15 = 24\)

Answer: 24

Meaning: Move both decimals the same number of places to make the divisor whole, then divide normally.

Tips and Shortcuts for Decimals

Decimals can be fast if you use shortcuts. Here are the best tips to save time.

Tips:

• Compare: Add zeros to make same length, then compare left to right.
• Round: Look at the digit after the place. 5+ round up, 4- keep same.
• Add/Subtract: Line up decimal points, add zeros if needed.
• Multiply: Multiply as whole numbers, count total decimal places, then place decimal.
• Divide: Move both decimals same places to make divisor whole, then divide.
• Money: .50 = 50 centavos, .25 = 25 centavos, etc.

Example Shortcut: When dividing by 0.5, it is the same as multiplying by 2. When dividing by 0.25, it is the same as multiplying by 4.

Example: \(10 ÷ 0.5 = 10 × 2 = 20\)
Example: \(8 ÷ 0.25 = 8 × 4 = 32\)

More Practice Examples

Example 1: Place value of 6 in \(0.462\)?
Step 1: 6 is 2nd digit → hundredths
Answer: hundredths

Example 2: Which is bigger: \(0.7\) or \(0.69\)?
Step 1: Add zero → \(0.7 = 0.70\)
Step 2: \(0.70 > 0.69\)
Answer: \(0.7\)

Example 3: Round \(4.567\) to nearest tenth?
Step 1: Keep 5 (1st digit)
Step 2: Look at 6 → 6 is 5+ → round up
Answer: \(4.6\)

Example 4: \(3.25 + 1.4\)
Step 1: Line up → \(3.25 + 1.40\)
Step 2: Add → \(4.65\)
Answer: \(4.65\)

Example 5: \(0.6 × 0.5\)
Step 1: Multiply → \(6 × 5 = 30\)
Step 2: Decimal places → 1 + 1 = 2
Step 3: Place decimal → \(0.30 = 0.3\)
Answer: \(0.3\)

What To Remember

• A decimal splits a number into whole and part smaller than 1 using a dot.
• Place values: tenths (1st), hundredths (2nd), thousandths (3rd).
• Write decimals: whole number, "and," decimal part with place value name.
• Compare: Add zeros to make same length, compare left to right.
• Round: 5+ round up, 4- keep same.
• Add/Subtract: Line up decimal points.
• Multiply: Multiply as whole, count decimal places, then place decimal.
• Divide: Move decimals to make divisor whole, then divide.

Multiple Choice Questions

Answer these in the comment section. Choose the best answer for each item.

1. What is the place value of 5 in \(3.452\)?

A. Tenths
B. Hundredths
C. Thousandths
D. Ones

2. Which is bigger: \(0.67\) or \(0.669\)?

A. \(0.67\)
B. \(0.669\)
C. They are equal
D. Cannot tell

3. Round \(5.836\) to the nearest hundredth.

A. \(5.83\)
B. \(5.84\)
C. \(5.80\)
D. \(5.90\)

4. What is \(2.5 × 0.4\)?

A. \(1.0\)
B. \(10\)
C. \(0.1\)
D. \(1.5\)

5. What is \(6.4 ÷ 0.8\)?

A. 6
B. 7
C. 8
D. 9