Relating Hundredths and Decimals: Complete Educational Worksheets

Master the relationship between hundredths and decimals through these 5 detailed exercises with step-by-step solutions.

Solution: Exercises 1 to 3
1 Converting hundredths to decimals
Exercise 1
Convert the fraction \(\frac{47}{100}\) to a decimal.
Definition:

Hundredths: A fraction with denominator 100, representing parts of a whole divided into 100 equal parts

Decimal: A number with a decimal point representing parts of a whole, where the second digit after the decimal point represents hundredths

Conversion method:
  1. Recognize that hundredths have denominator 100
  2. Write the numerator as the digits after the decimal point
  3. Place the decimal point two positions from the right
Fraction
\(\frac{47}{100}\)
Hundredths
47 hundredths
Decimal
\(0.47\)
Step 1: Identify the fraction

\(\frac{47}{100}\) means 47 parts out of 100 equal parts

Step 2: Understand place value

When the denominator is 100, the numerator becomes the number of hundredths

This means 47 hundredths

Step 3: Write as decimal

Since hundredths is the second place after the decimal point, 47 hundredths = 0.47

Step 4: Verify the result

0.47 means 47 hundredths, which equals \(\frac{47}{100}\)

\(\frac{47}{100} = 0.47\)
Final answer:

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

Applied rules:

• Hundredths rule: When denominator is 100, numerator becomes hundredths place

• Place value: Second digit after decimal point represents hundredths

• Equivalence: \(\frac{a}{100} = 0.a0\) if a is single digit, or \(0.ab\) if a has two digits

Tip: Think of a dollar bill as representing 1 whole. Then 47 cents = 47 hundredths of a dollar = $0.47
2 Converting decimals to hundredths
Exercise 2
Convert the decimal \(0.83\) to a fraction in hundredths form.
Definition:

Hundredths form: A fraction with denominator 100

Decimal
\(0.83\)
Hundredths
83 hundredths
Fraction
\(\frac{83}{100}\)
Step 1: Identify the decimal places

\(0.83\) has two digits after the decimal point, so it represents hundredths

Step 2: Read the decimal aloud

"Zero point eight three" is the same as "eighty-three hundredths"

Step 3: Write as fraction

Eighty-three hundredths = \(\frac{83}{100}\)

Step 4: Check if simplification is needed

GCD of 83 and 100 is 1, so \(\frac{83}{100}\) is already in simplest form

\(0.83 = \frac{83}{100}\)
Final answer:

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

Applied rules:

• Place value rule: 2 decimal places = hundredths, so denominator is 100

• Reading rule: Read decimal as "X hundredths" to get the fraction

• Equivalence: \(0.ab = \frac{ab}{100}\) where ab are the digits after the decimal

Tip: Count decimal places: 1 place = tenths, 2 places = hundredths, 3 places = thousandths
3 Comparing hundredths and decimals
Exercise 3
Which is greater: \(\frac{5}{100}\) or \(0.06\)?
Definition:

Comparing rational numbers: Numbers must be in the same form (both fractions or both decimals) to compare them accurately

Fraction
\(\frac{5}{100}\)
Convert to decimal
\(0.05\)
Compare
\(0.05 < 0.06\)
Step 1: Convert to same form

Convert \(\frac{5}{100}\) to decimal: Since the denominator is 100, \(\frac{5}{100} = 0.05\)

Step 2: Compare decimals

Now compare \(0.05\) and \(0.06\)

Step 3: Analyze place values

Both have 0 in the ones place

Both have 0 in the tenths place

But \(0.05\) has 5 in the hundredths place, while \(0.06\) has 6 in the hundredths place

Step 4: Make comparison

Since \(5 < 6\) in the hundredths place, \(0.05 < 0.06\)

\(\frac{5}{100} < 0.06\)
Final answer:

\(0.06\) is greater than \(\frac{5}{100}\)

Applied rules:

• Conversion rule: Convert to the same form before comparing

• Place value comparison: Compare digits starting from leftmost place value

• Hundredths rule: \(\frac{a}{100} = 0.0a\) when a is a single digit

Alternative method: Convert both to hundredths: \(\frac{5}{100}\) vs \(\frac{6}{100}\), then compare numerators.
Key Rules, Laws, and Methods
\(\frac{a}{100} = 0.0a\) (single digit) or \(0.ab\) (double digit)
Hundredths to Decimal Conversion
Rule 1
\(\frac{a}{100} = 0.ab\) when \(a\) has two digits
Hundredths to decimal conversion
Rule 2
\(0.ab = \frac{ab}{100}\)
Decimal to hundredths conversion
Rule 3
\(\frac{a}{100} = \frac{a×n}{100×n}\)
Equivalent fractions
Key definitions:

Hundredths: One part when a whole is divided into 100 equal parts (\(\frac{1}{100}\))

Decimal place value: The position of a digit after the decimal point determines its value (tenths, hundredths, thousandths)

Equivalent: Two numbers that represent the same value, even though they may look different

Place value: The value of a digit based on its position in a number (ones, tens, hundreds, tenths, hundredths)

Complete methodology:
  1. Identify the operation needed: Are you converting hundredths to decimal, decimal to hundredths, or comparing?
  2. Choose the method: Use place value recognition or conversion to same form
  3. Apply the method: Follow the steps systematically
  4. Verify the result: Check if the conversion makes sense
Tip 1: Remember: 1 hundredth = \(\frac{1}{100} = 0.01\). This is your base unit for understanding hundredths.
Tip 2: When converting decimals to hundredths, the digits after the decimal point become the numerator of the fraction with denominator 100.
Tip 3: For fractions with denominator 100, the numerator directly becomes the number of hundredths.
Tip 4: When comparing, always convert to the same form first (either both fractions or both decimals).
Common errors: Forgetting that hundredths require two digits after decimal, miscounting decimal places, comparing different forms directly.
Exam preparation: Practice converting between forms, work on comparison problems, understand place value relationships.
Formulas to know by heart:

• Conversion: \(\frac{a}{100} = 0.ab\) (where ab are the digits of a)

• Place value: 2 decimal places → hundredths (\(\frac{1}{100}\)), 1 decimal place → tenths (\(\frac{1}{10})

• Equivalence: \(\frac{a}{100} = \frac{a×n}{100×n}\) for any non-zero n

• Decimal reading: Read as "X hundredths" when there are 2 decimal places

Visual Understanding: Hundredths Place Value
Exercise 6: Hundredths Visualization
Visual representation of common hundredths-decimal equivalents:
\(\frac{25}{100} = 0.25\), \(\frac{50}{100} = 0.50\), \(\frac{75}{100} = 0.75\), \(\frac{1}{100} = 0.01\), \(\frac{99}{100} = 0.99\)

Analysis: The chart shows how hundredths fractions and decimals represent the same quantities visually.

  • \(\frac{25}{100} = 0.25\) (twenty-five hundredths)
  • \(\frac{50}{100} = 0.50\) (fifty hundredths)
  • \(\frac{75}{100} = 0.75\) (seventy-five hundredths)
  • \(\frac{1}{100} = 0.01\) (one hundredth)
  • \(\frac{99}{100} = 0.99\) (ninety-nine hundredths)

Questions & Answers

Question: Why does \(\frac{5}{100} = 0.05\) and not 0.5? Isn't 5 bigger than 0.05?

Answer: Great question! Let's break this down:

  • \(\frac{5}{100}\) means 5 parts out of 100 equal parts
  • Think of it as a pizza cut into 100 pieces - you get 5 of those pieces
  • That's much less than half the pizza!
  • 0.5 would mean 50 hundredths, or \(\frac{50}{100}\), which is half

Let's look at the place value:

  • In 0.05, the 5 is in the hundredths place (second digit after decimal)
  • In 0.5, the 5 is in the tenths place (first digit after decimal)
  • So 0.5 = \(\frac{5}{10} = \frac{50}{100}\), which is ten times larger than \(\frac{5}{100}\)

Think of money: $0.05 is 5 cents, while $0.50 is 50 cents. Five cents is much less than fifty cents!

Question: How do I know if a fraction is in hundredths form?

Answer: A fraction is in hundredths form when:

  • The denominator is exactly 100 - This is the key identifier!
  • Examples: \(\frac{3}{100}\), \(\frac{45}{100}\), \(\frac{99}{100}\), \(\frac{1}{100}\)

However, some fractions are equivalent to hundredths but don't initially look like it:

  • \(\frac{1}{2} = \frac{50}{100}\) (multiply top and bottom by 50)
  • \(\frac{1}{4} = \frac{25}{100}\) (multiply top and bottom by 25)
  • \(\frac{3}{5} = \frac{60}{100}\) (multiply top and bottom by 20)

To convert any fraction to hundredths, try to change the denominator to 100 by finding an equivalent fraction.

If you can't make the denominator 100, then the fraction is not in hundredths form, though it can still be converted to a decimal.

Question: My child is struggling with the concept of hundredths. How can I help them visualize it better?

Answer: Here are several ways to help visualize hundredths:

  1. Grid paper: Use a 10x10 grid (100 squares total). Shade in the number of squares equal to the numerator of your hundredths fraction.
  2. Money: Use cents as hundredths of a dollar. 25 cents = 25 hundredths of a dollar = $0.25
  3. Number line: Draw a line from 0 to 1 and divide it into 100 equal parts. Mark specific hundredths.
  4. Base-10 blocks: If available, use flats (100 units), rods (10 units), and units (1 unit) to show the relationship.

Practical activities:

  • Take a dollar bill and show that 1 penny is \(\frac{1}{100}\) of a dollar or 0.01 of a dollar
  • Measure lengths using centimeters and meters (1 cm = \(\frac{1}{100}\) m)
  • Use a 100-bead abacus or draw 100 dots and group them

The key is connecting the abstract concept to concrete, real-world examples that children can touch, see, or experience.