Solved Exercises on Decimal Notation in Grade 4

Master decimal notation: place value, converting fractions to decimals, comparing decimals through these 3 detailed exercises.

Solution: Exercises 1 to 3
1 Place value understanding
Exercise 1
What does each digit represent in the number 347.82?
Write the place value of each digit.
Definition:

Decimal notation: A way to write numbers using digits 0-9, where each position has a specific value (place value). The decimal point separates whole numbers from fractional parts.

Place value method:
  1. Identify the decimal point position
  2. Count positions from the decimal point to the left (ones, tens, hundreds...)
  3. Count positions from the decimal point to the right (tenths, hundredths...)
  4. Assign each digit its corresponding place value
Number
347.82
Place Values
H T O . t h
Step 1: Identify the decimal point

In 347.82, the decimal point is between 7 and 8

Step 2: Count positions to the left

7 is in the ones place, 4 is in the tens place, 3 is in the hundreds place

Step 3: Count positions to the right

8 is in the tenths place, 2 is in the hundredths place

Step 4: Assign place values

3 = 300 (hundreds), 4 = 40 (tens), 7 = 7 (ones), 8 = 0.8 (tenths), 2 = 0.02 (hundredths)

3 = 300 (hundreds)
4 = 40 (tens)
7 = 7 (ones)
8 = 0.8 (tenths)
2 = 0.02 (hundredths)
Final answer:

347.82 = 300 + 40 + 7 + 0.8 + 0.02

Each digit represents: 3=hundreds, 4=tens, 7=ones, 8=tenths, 2=hundredths

Applied rules:

• Place Value: Each position represents 10 times more than the position to its right

• Decimal Point: Separates whole numbers (left) from fractional parts (right)

• Position Names: Left of decimal: ones, tens, hundreds; Right: tenths, hundredths, thousandths

2 Converting fractions to decimals
Exercise 2
Convert these fractions to decimal notation:
a) 3/10
b) 7/100
c) 25/100
Definition:

Converting fractions to decimals: Fractions with denominators of 10, 100, 1000, etc. convert directly to decimal notation by placing the numerator in the appropriate decimal place.

Fractions
3/10, 7/100, 25/100
Decimals
0.3, 0.07, 0.25
Step 1: Analyze denominator for 3/10

Denominator is 10, so the numerator goes in the tenths place

Step 2: Convert 3/10

3/10 = 0.3 (3 in the tenths place)

Step 3: Analyze denominator for 7/100

Denominator is 100, so the numerator goes in the hundredths place

Step 4: Convert 7/100

7/100 = 0.07 (7 in the hundredths place, with 0 in tenths)

Step 5: Convert 25/100

25/100 = 0.25 (2 in tenths, 5 in hundredths)

a) 3/10 = 0.3
b) 7/100 = 0.07
c) 25/100 = 0.25
Final answer:

a) 3/10 = 0.3

b) 7/100 = 0.07

c) 25/100 = 0.25

Applied rules:

• Denominator 10: Numerator goes in tenths place (one decimal place)

• Denominator 100: Numerator goes in hundredths place (two decimal places)

• Denominator 1000: Numerator goes in thousandths place (three decimal places)

3 Comparing decimals
Exercise 3
Compare these decimal numbers using >, <, or =:
a) 0.45 and 0.405
b) 0.7 and 0.70
c) 0.32 and 0.23
Definition:

Comparing decimals: Start from the leftmost digit and compare each place value until a difference is found. Equal numbers may have different representations (e.g., 0.7 = 0.70).

Comparisons
0.45 vs 0.405, 0.7 vs 0.70, 0.32 vs 0.23
Results
>, =, >
Step 1: Compare 0.45 and 0.405

Align by adding zeros: 0.450 vs 0.405. Compare tenths: 4=4. Compare hundredths: 5>0. So 0.45 > 0.405

Step 2: Compare 0.7 and 0.70

0.7 = 0.70 because adding zeros after the last non-zero digit doesn't change the value

Step 3: Compare 0.32 and 0.23

Compare tenths: 3>2. So 0.32 > 0.23

Step 4: Final results

a) 0.45 > 0.405, b) 0.7 = 0.70, c) 0.32 > 0.23

a) 0.45 > 0.405
b) 0.7 = 0.70
c) 0.32 > 0.23
Final answer:

a) 0.45 > 0.405

b) 0.7 = 0.70

c) 0.32 > 0.23

Applied rules:

• Comparison Method: Align decimals by adding zeros, then compare from left to right

• Equality: Adding zeros after the last significant digit doesn't change the value

• Place Value Priority: Higher place values (leftmost) determine the relationship

Decimal Notation Rules and Methods
Decimal = Whole + Fractional Part
Decimal Structure
Place Values Left
..., Millions, Hundred-Thousands, Ten-Thousands, Thousands, Hundreds, Tens, Ones
Whole Numbers
Decimal Point
.
Separates Whole and Fractional
Place Values Right
Tenths, Hundredths, Thousandths, ...
Fractional Parts
Key definitions:

Decimal Notation: A way to write numbers using base-10 system with a decimal point

Place Value: The value of a digit based on its position in the number

Decimal Point: Symbol that separates whole numbers from fractional parts

Complete methodology:
  1. Identify the decimal point: Locate where whole numbers end and fractions begin
  2. Recognize place values: Understand the value of each position
  3. Convert when needed: Change fractions to decimals or vice versa
  4. Compare systematically: Align decimals and compare from left to right
Tip 1: Remember that 0.7 = 0.70 = 0.700 (adding zeros after the last significant digit doesn't change the value).
Tip 2: When comparing decimals, align them by adding zeros so all have the same number of decimal places.
Tip 3: The decimal point is always between the ones place and the tenths place.
Tip 4: Fractions with denominators of 10, 100, 1000 convert directly to decimals.
Common errors: Misreading place values, forgetting that adding trailing zeros doesn't change the value, misaligning decimals when comparing.
Exam preparation: Practice identifying place values, converting fractions to decimals, and comparing decimal numbers.
Decimal Place Value Chart & Visual Guide
..., H-Th, T-Th, Th, H, T, O . t, h, th, ...
Decimal Place Value System
Example Number
543.21
H T O . t h
Expanded Form
500 + 40 + 3 + 0.2 + 0.01
Value Breakdown
Fraction Form
543 + 21/100
Decimal to Fraction
Place Value Positions:

Left of decimal: Hundreds (H), Tens (T), Ones (O)

Right of decimal: Tenths (t), Hundredths (h), Thousandths (th)

Further left: Ten Thousands (T-Th), Hundred Thousands (H-Th), Millions (M)

Place Value Visualization:
H-Th T-Th Th H T O . t h th 100,000 10,000 1,000 100 10 1 . 0.1 0.01 0.001
Conversion Rule: To convert fraction to decimal, divide numerator by denominator.
Comparison Rule: Add zeros to make decimals have equal length before comparing.
Equivalent Forms: 0.5 = 0.50 = 0.500 = 1/2 = 5/10 = 50/100
Ordering: Compare from left to right: 0.3 < 0.31 < 0.4 < 0.40
Essential Decimal Facts:

• Each place value is 10 times smaller than the one to its left

• The decimal point is always between ones and tenths

• Adding zeros after the decimal point doesn't change the number's value

• Decimal notation is another way to write fractions with denominators of 10, 100, 1000, etc.

Decimal Notation Laws, Methods, Rules, and Expressions
a.bcd = a + b/10 + c/100 + d/1000
Decimal to Fraction Conversion
Key definitions:

Decimal Notation: A representation of numbers using the base-10 system with a decimal point to separate whole and fractional parts

Place Value: The value of each digit depends on its position relative to the decimal point

Decimal Fraction: A fraction whose denominator is a power of 10 (10, 100, 1000, etc.)

Complete methodology:
  1. Reading Decimals: Read the whole number part, say "and" for the decimal point, read the fractional part as individual digits followed by the place value of the last digit
  2. Writing Decimals: Place each digit in its correct position relative to the decimal point
  3. Converting Fractions: If denominator is 10, 100, 1000, etc., write the numerator in the appropriate decimal places
  4. Comparing Decimals: Align decimal points and compare from left to right
Tip 1: When converting fractions to decimals, count decimal places: 1 place for tenths, 2 for hundredths, 3 for thousandths.
Tip 2: When comparing decimals, it helps to add zeros so all numbers have the same number of decimal places.
Tip 3: Remember that 0.5 is the same as 0.50 and 0.500 - adding zeros after the last significant digit doesn't change the value.
Tip 4: The decimal point always separates the ones place from the tenths place.
Common errors: Confusing place values, forgetting that trailing zeros don't change the value, misaligning decimals when comparing.
Exam preparation: Practice reading and writing decimals, converting between fractions and decimals, and ordering decimals.
Decimal Notation Laws and Rules:

• Base-10 System: Each place value is 10 times the value of the place to its right

• Decimal Point: Acts as a separator between whole and fractional parts

• Place Value Progression: Moving right past the decimal: tenths (1/10), hundredths (1/100), thousandths (1/1000)

• Equivalence: Adding zeros after the last non-zero digit creates equivalent decimals

• Comparison: Compare digits from left to right until a difference is found

Questions & Answers

Question: I don't understand why 0.7 and 0.70 are the same value. Doesn't adding the zero make it bigger?

Answer: Great question! This is a common misconception. Adding zeros AFTER the last significant digit doesn't change the value:

  • 0.7 means 7 tenths
  • 0.70 means 7 tenths AND 0 hundredths
  • Since there are 0 hundredths, it's still just 7 tenths

Think of money: $0.70 is the same as $0.7. Both represent 70 cents. The extra zero just shows the precision of measurement but doesn't add value.

Examples: 0.3 = 0.30 = 0.300, and 1.5 = 1.50 = 1.500

Question: How do I know where to put the decimal point when converting fractions like 45/100 to decimal?

Answer: The denominator tells you where to place the decimal point:

  • Denominator of 10 → 1 decimal place (tenths)
  • Denominator of 100 → 2 decimal places (hundredths)
  • Denominator of 1000 → 3 decimal places (thousandths)

For 45/100: Since the denominator is 100 (2 zeros), you need 2 decimal places. Write the numerator (45) in the hundredths and tenths places: 0.45

More examples: 7/10 = 0.7, 234/1000 = 0.234, 8/100 = 0.08

Question: Why is 0.32 greater than 0.23? It seems like 23 should be bigger than 32.

Answer: You need to compare place by place from left to right:

  • Compare tenths place: 0.32 has 3 tenths, 0.23 has 2 tenths
  • Since 3 > 2, we already know 0.32 > 0.23
  • We don't even need to look at the hundredths place

Think of it like this: 32 cents ($0.32) is more than 23 cents ($0.23). The tenths place (3 vs 2) determines the relationship because it represents a larger value than the hundredths place.

Always compare decimals digit by digit from left to right!