Solved Exercises on Relating Fractions to Decimals (Tenths and Hundredths) in Grade 4

Master fraction-to-decimal relationships: rules for converting fractions with denominators 10 and 100 to decimals through these 5 detailed exercises with visual representations.

Solution: Exercises 1 to 3
1 Convert tenths to decimals
Exercise 1
Convert \(\frac{3}{10}\) to a decimal
Definition:

Tenths: Fractions with denominator 10, representing one part out of ten equal parts

Method for converting tenths to decimals:
  1. Identify the numerator of the fraction
  2. Place the numerator in the tenths place of the decimal
  3. Write the decimal as 0.numerator
Fraction
\(\frac{3}{10}\)
Numerator
3
Decimal
0.3
Step 1: Identify the fraction

We have \(\frac{3}{10}\), where the numerator is 3 and the denominator is 10

Step 2: Recognize the denominator

The denominator is 10, which means we're dealing with tenths

Step 3: Place the numerator in tenths position

The numerator (3) goes in the tenths place: 0.3

3/10
→
0.3
Ones
0
Tenths
3
\(\frac{3}{10} = 0.3\)
\(\frac{3}{10}\)
=
0.3
\(\frac{3}{10} = 0.3\)
Final answer:

\(\frac{3}{10} = 0.3\)

Applied rules:

• Tenths conversion: \(\frac{a}{10} = 0.a\)

• Decimal placement: The numerator goes in the tenths position after the decimal point

• Ones position: For fractions less than 1, the ones position is 0

2 Convert hundredths to decimals
Exercise 2
Convert \(\frac{47}{100}\) to a decimal
Definition:

Hundredths: Fractions with denominator 100, representing one part out of one hundred equal parts

Fraction
\(\frac{47}{100}\)
Numerator
47
Decimal
0.47
Step 1: Identify the fraction

We have \(\frac{47}{100}\), where the numerator is 47 and the denominator is 100

Step 2: Recognize the denominator

The denominator is 100, which means we're dealing with hundredths

Step 3: Place the numerator in hundredths position

The numerator (47) goes in the tenths and hundredths places: 0.47

47/100
→
0.47
Ones
0
Tenths
4
Hundredths
7
\(\frac{47}{100} = 0.47\)
\(\frac{47}{100}\)
=
0.47
\(\frac{47}{100} = 0.47\)
Final answer:

\(\frac{47}{100} = 0.47\)

Applied rules:

• Hundredths conversion: \(\frac{a}{100} = 0.0a\) (when a < 10) or \(0.ab\) (when a ≥ 10)

• Two-digit placement: For two-digit numerators, the first digit goes in tenths, second in hundredths

• Ones position: For fractions less than 1, the ones position is 0

3 Convert decimals to fractions
Exercise 3
Convert 0.6 to a fraction
Definition:

Decimal to fraction conversion: Expressing a decimal number as a fraction with denominator 10 or 100 based on the number of decimal places

Decimal
0.6
Digits After Point
6
Denominator
10
Result
\(\frac{6}{10}\)
Step 1: Identify the decimal

We have 0.6, which has one digit after the decimal point

Step 2: Determine the denominator

One decimal place means the denominator is 10

Step 3: Write the fraction

The numerator is the digit(s) after the decimal: 6, so we have \(\frac{6}{10}\)

0.6
→
6/10
Ones
0
Tenths
6
\(0.6 = \frac{6}{10}\)
0.6
=
\(\frac{6}{10}\)
\(0.6 = \frac{6}{10}\)
Final answer:

\(0.6 = \frac{6}{10}\)

Applied rules:

• Decimal to tenths: \(0.a = \frac{a}{10}\)

• Decimal to hundredths: \(0.ab = \frac{ab}{100}\)

• Denominator determination: Number of decimal places determines the denominator (1 decimal = 10, 2 decimals = 100)

Rules and methods, laws,...
\(\frac{a}{10} = 0.a\)
Tenths Conversion Rule
\(\frac{a}{100} = 0.0a\) or \(0.ab\)
Hundredths Conversion Rule
\(0.a = \frac{a}{10}\)
Decimal to Tenths Rule
Rule 1
\(\frac{a}{10} = 0.a\)
Convert tenths fraction to decimal
Rule 2
\(\frac{ab}{100} = 0.ab\)
Convert hundredths fraction to decimal
Rule 3
\(0.a = \frac{a}{10}\)
Convert tenths decimal to fraction
Rule 4
\(0.ab = \frac{ab}{100}\)
Convert hundredths decimal to fraction
Key definitions:

Decimal: A number that contains a decimal point, representing fractions with denominators that are powers of 10 (10, 100, 1000, etc.)

Decimal point: The dot used to separate the whole number part from the fractional part in a decimal number

Tenths place: The first position to the right of the decimal point, representing \(\frac{1}{10}\)

Hundredths place: The second position to the right of the decimal point, representing \(\frac{1}{100}\)

Tenths fraction: A fraction with denominator 10, such as \(\frac{3}{10}\), \(\frac{7}{10}\), etc.

Hundredths fraction: A fraction with denominator 100, such as \(\frac{23}{100}\), \(\frac{87}{100}\), etc.

Equivalent forms: Different ways to express the same value (e.g., \(\frac{5}{10} = 0.5\))

Place value: The value of a digit based on its position in a number, determined by powers of 10

Conversion: The process of changing from one form (fraction) to another (decimal) while maintaining the same value

Complete methodology:
  1. Identify the form: Determine if starting with a fraction or a decimal
  2. Recognize the denominator: For fractions, identify if denominator is 10 or 100
  3. Count decimal places: For decimals, count digits after the decimal point
  4. Apply the conversion rule: Use the appropriate rule based on the identified form
  5. Verify the result: Check that the conversion maintains the same value
  6. Simplify if possible: Reduce fractions to lowest terms when appropriate
Tip 1: Fractions with denominator 10 always convert to decimals with 1 digit after the decimal point.
Tip 2: Fractions with denominator 100 always convert to decimals with 2 digits after the decimal point.
Tip 3: The numerator of a tenths fraction becomes the tenths digit in the decimal.
Tip 4: The numerator of a hundredths fraction becomes the tenths and hundredths digits in the decimal.
Tip 5: The number of decimal places tells you the denominator (1 decimal = 10, 2 decimals = 100).
Tip 6: Use visual models like grids or circles to see the equivalence between forms.
Common errors: Misplacing decimal points, confusing tenths with hundredths, forgetting that fractions less than 1 start with 0 in decimal form, not recognizing that denominators of 10 or 100 are required for simple conversion, confusing numerator with denominator in the conversion process.
Memory aid: "Tenths have tenths, hundredths have hundredths" - the denominator name matches the decimal place name.
Rules to know by heart:

• Tenths conversion: \(\frac{a}{10} = 0.a\)

• Hundredths conversion: \(\frac{ab}{100} = 0.ab\)

• Decimal to tenths: \(0.a = \frac{a}{10}\)

• Decimal to hundredths: \(0.ab = \frac{ab}{100}\)

• Denominator-place relationship: Number of decimal places matches the power of 10 in the denominator

• Ones position: For values less than 1, the ones digit is 0

Solution: Exercises 4 to 5
4 Word problem with fraction-decimal conversion
Exercise 4
Sarah ate \(\frac{7}{10}\) of an apple. Express this as a decimal.
Definition:

Word problems with conversion: Real-world scenarios where fractions need to be expressed as decimals for measurement, comparison, or calculation purposes

Fraction
\(\frac{7}{10}\)
Decimal
0.7
Step 1: Identify the fraction

Sarah ate \(\frac{7}{10}\) of an apple, where the numerator is 7 and the denominator is 10

Step 2: Recognize the denominator

The denominator is 10, which means we're dealing with tenths

Step 3: Convert to decimal

The numerator (7) goes in the tenths place: 0.7

7/10
→
0.7
Ones
0
Tenths
7
\(\frac{7}{10} = 0.7\)
\(\frac{7}{10}\)
=
0.7
Sarah ate 0.7 of an apple
Final answer:

Sarah ate 0.7 of an apple

Applied rules:

• Tenths conversion: \(\frac{a}{10} = 0.a\)

• Real-world application: Fraction-decimal conversions occur in measurement and sharing scenarios

• Decimal placement: Numerator goes in tenths place when denominator is 10

5 Advanced conversion problem
Exercise 5
Convert 0.83 to a fraction and explain the process.
Definition:

Advanced conversion: Converting decimals with two digits after the decimal point to fractions with denominator 100

Decimal
0.83
Digits After Point
83
Denominator
100
Result
\(\frac{83}{100}\)
Step 1: Identify the decimal

We have 0.83, which has two digits after the decimal point

Step 2: Determine the denominator

Two decimal places mean the denominator is 100

Step 3: Write the fraction

The numerator is the digits after the decimal: 83, so we have \(\frac{83}{100}\)

0.83
→
83/100
Ones
0
Tenths
8
Hundredths
3
\(0.83 = \frac{83}{100}\)
0.83
=
\(\frac{83}{100}\)
\(0.83 = \frac{83}{100}\)
Final answer:

\(0.83 = \frac{83}{100}\)

Applied rules:

• Decimal to hundredths: \(0.ab = \frac{ab}{100}\)

• Digit counting: Number of decimal places determines the denominator

• Two-digit numerator: Both digits after decimal point become the numerator

Complete Summary: Relating Fractions to Decimals (Tenths and Hundredths)
\(\frac{a}{10} = 0.a\)
Tenths Conversion
\(\frac{ab}{100} = 0.ab\)
Hundredths Conversion
\(0.a = \frac{a}{10}\)
Decimal to Tenths
\(0.ab = \frac{ab}{100}\)
Decimal to Hundredths
Key definitions:

Decimal: A number that contains a decimal point, representing fractions with denominators that are powers of 10 (10, 100, 1000, etc.)

Decimal point: The dot used to separate the whole number part from the fractional part in a decimal number

Tenths place: The first position to the right of the decimal point, representing \(\frac{1}{10}\)

Hundredths place: The second position to the right of the decimal point, representing \(\frac{1}{100}\)

Tenths fraction: A fraction with denominator 10, such as \(\frac{3}{10}\), \(\frac{7}{10}\), etc.

Hundredths fraction: A fraction with denominator 100, such as \(\frac{23}{100}\), \(\frac{87}{100}\), etc.

Equivalent forms: Different ways to express the same value (e.g., \(\frac{5}{10} = 0.5\))

Place value: The value of a digit based on its position in a number, determined by powers of 10

Conversion: The process of changing from one form (fraction) to another (decimal) while maintaining the same value

Base-10 system: Our number system based on powers of 10, where each position is 10 times the previous

Decimal expansion: Writing a fraction as a decimal number

Fraction notation: Writing a decimal as a fraction with denominator 10 or 100

Complete methodology:
  1. Identify the form: Determine if starting with a fraction or a decimal
  2. Recognize the denominator: For fractions, identify if denominator is 10 or 100
  3. Count decimal places: For decimals, count digits after the decimal point
  4. Apply the conversion rule: Use the appropriate rule based on the identified form
  5. Verify the result: Check that the conversion maintains the same value
  6. Simplify if possible: Reduce fractions to lowest terms when appropriate
Tip 1: Fractions with denominator 10 always convert to decimals with 1 digit after the decimal point.
Tip 2: Fractions with denominator 100 always convert to decimals with 2 digits after the decimal point.
Tip 3: The numerator of a tenths fraction becomes the tenths digit in the decimal.
Tip 4: The numerator of a hundredths fraction becomes the tenths and hundredths digits in the decimal.
Tip 5: The number of decimal places tells you the denominator (1 decimal = 10, 2 decimals = 100).
Tip 6: Use visual models like grids or circles to see the equivalence between forms.
Tip 7: When converting decimals to fractions, the numerator is the digits after the decimal point.
Tip 8: For values less than 1, the ones digit in the decimal is always 0.
Common errors: Misplacing decimal points, confusing tenths with hundredths, forgetting that fractions less than 1 start with 0 in decimal form, not recognizing that denominators of 10 or 100 are required for simple conversion, confusing numerator with denominator in the conversion process, miscounting decimal places.
Memory aid: "Tenths have tenths, hundredths have hundredths" - the denominator name matches the decimal place name; "What you see is what you get" for decimal to fraction conversion.
Visualization: Draw 10x10 grids to see how hundredths fractions correspond to decimal positions.
Real-world connection: Money (cents) and measurements (centimeters) provide practical examples of decimal-fraction relationships.
Rules to know by heart:

• Tenths conversion: \(\frac{a}{10} = 0.a\)

• Hundredths conversion: \(\frac{ab}{100} = 0.ab\)

• Decimal to tenths: \(0.a = \frac{a}{10}\)

• Decimal to hundredths: \(0.ab = \frac{ab}{100}\)

• Denominator-place relationship: Number of decimal places matches the power of 10 in the denominator

• Ones position: For values less than 1, the ones digit is 0

• Value preservation: The conversion maintains the same numerical value

• Place value system: Each position represents a power of 10

Fraction-Decimal Conversion Framework

📊
Tenths Conversion

\(\frac{3}{10} = 0.3\)

Numerator becomes tenths digit

3
7
Hundredths Conversion

\(\frac{47}{100} = 0.47\)

Numerator fills tenths and hundredths places

47
53
Decimal to Fraction

\(0.6 = \frac{6}{10}\)

Digits after decimal become numerator

6
4
Conversion Process
1
Identify
2
Recognize
3
Apply
4
Verify
Base-10 Relationships

Decimal places correspond to fraction denominators that are powers of 10

Master Fraction-Decimal Conversions!

Questions & Answers

Question: Why do \(\frac{3}{10}\) and 0.3 represent the same value? They look completely different!

Answer: Great question! Even though \(\frac{3}{10}\) and 0.3 look different, they represent exactly the same amount:

  • \(\frac{3}{10}\) means 3 parts out of 10 equal parts of a whole
  • 0.3 means 3 tenths, which is the same as 3 parts out of 10 equal parts

Think of it like this: If you have a pizza cut into 10 equal slices and you eat 3 of them, you've eaten \(\frac{3}{10}\) of the pizza, which is the same as 0.3 of the pizza.

The decimal system is based on fractions with denominators that are powers of 10. So 0.3 literally means "zero point three tenths," which is the same as \(\frac{3}{10}\).

They're just two different ways of writing the same number!

Question: How do I convert \(\frac{7}{100}\) to a decimal? It seems harder than tenths.

Answer: Converting \(\frac{7}{100}\) to a decimal is actually quite straightforward once you understand the pattern:

  • The denominator is 100, which means we're dealing with hundredths
  • The numerator is 7
  • Since we have 7 hundredths, we put the 7 in the hundredths place: 0.07

Notice that we need two digits after the decimal point for hundredths. Since 7 is only one digit, we write it as 07 in the hundredths place, which becomes 0.07.

The general rule is: \(\frac{a}{100} = 0.0a\) when a is a single digit, or \(\frac{ab}{100} = 0.ab\) when ab is two digits.

So \(\frac{7}{100} = 0.07\), \(\frac{23}{100} = 0.23\), \(\frac{5}{100} = 0.05\), etc.

Question: What happens if I have a fraction like \(\frac{15}{10}\)? Can I still convert it to a decimal?

Answer: Yes! \(\frac{15}{10}\) can definitely be converted to a decimal. Let's work through it:

  • The denominator is 10, so we're dealing with tenths
  • The numerator is 15, which is greater than the denominator
  • This means we have 15 tenths, which is more than 1 whole

We can think of this as: \(\frac{15}{10} = \frac{10}{10} + \frac{5}{10} = 1 + 0.5 = 1.5\)

So \(\frac{15}{10} = 1.5\). This makes sense because 15 tenths equals 1 whole plus 5 more tenths.

When the numerator is larger than the denominator, the decimal will be greater than 1. For example:

  • \(\frac{12}{10} = 1.2\)
  • \(\frac{25}{10} = 2.5\)
  • \(\frac{107}{100} = 1.07\)