Understanding how to Compare Decimals is a fundamental skill in mathematics. Just like comparing whole numbers, comparing decimals helps us determine which of two or more decimal numbers is larger, smaller, or if they are equal. This skill is crucial not only for math class but also for real-life situations involving measurements, money, and more.
Decimal comparison is introduced to students as early as 4th grade as part of understanding fractions and numbers, and it’s further developed in 5th grade. This guide will walk you through the process of comparing decimals, making it easy to understand and apply.
What Does Comparing Decimals Mean?
Comparing decimals is the process of figuring out the relative size of two or more decimal numbers. When we compare decimals, we’re essentially determining their value and deciding which one represents a greater quantity, a smaller quantity, or if they represent the same quantity.
One of the most effective tools for comparing decimals is the place value chart. This chart helps us organize the digits in each decimal number according to their place value – ones, tenths, hundredths, thousandths, and so on.
Let’s take an example to illustrate this. Suppose we want to compare 0.78 and 0.783.
We start by placing these numbers in a place value chart, ensuring that the decimal points are aligned:
To compare, we begin from the leftmost digit, which has the largest place value – the tenths place in this case. Both numbers have ‘7’ in the tenths place. Since these digits are the same, we move to the next place value to the right, the hundredths place. Again, both numbers have ‘8’.
Now, we proceed to the thousandths place. 0.783 has ‘3’ in the thousandths place, while 0.78 effectively has a ‘0’ because we can add zeros as placeholders to the right of the last digit in a decimal without changing its value.
Since 3 is greater than 0, we can conclude that 0.783 is greater than 0.78. Conversely, 0.78 is less than 0.783.
We can express this comparison using mathematical symbols:
- 0.78 < 0.783 (0.78 is less than 0.783)
- 0.783 > 0.78 (0.783 is greater than 0.78)
This same method applies when comparing decimals with whole numbers or when arranging multiple decimals in ascending (smallest to largest) or descending (largest to smallest) order. For ordering, you would simply place all the decimals you are comparing into the place value chart and compare them pair by pair.
Understanding Decimal Comparison
Connecting to Math Standards
Comparing decimals is a key concept within elementary math curricula. Here’s how it aligns with common grade-level standards:
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4th Grade: In the Number and Operations – Fractions domain (4.NF.C.7), students are expected to compare two decimals to hundredths by thinking about their size. They need to understand that comparisons are only accurate when the decimals refer to the same whole. Students also learn to use symbols like >, =, and < to record comparisons and justify their conclusions using visual models.
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5th Grade: Expanding on this in Number and Operations in Base Ten (5.NBT.A.3b), 5th graders compare two decimals to thousandths. They focus on the meaning of digits in each place value position and use >, =, and < symbols to record comparison results.
How to Compare Decimals: Step-by-Step
To effectively compare decimals, follow these straightforward steps:
- Align by Decimal Point: The most crucial step is to align the numbers vertically by their decimal points. This ensures that you are comparing digits in the same place value positions.
- Compare the Largest Place Value: Begin comparing the digits from the largest place value (farthest to the left). This could be the ones place, tens place, or even hundreds place if you are dealing with larger numbers, or the tenths place if comparing numbers less than one.
- Continue to the Right: If the digits in the largest place value are the same, move to the next place value to the right and compare those digits. Repeat this process until you find a place value where the digits are different.
- Determine Greater or Lesser: Once you find a place value with different digits, the decimal with the larger digit in that place value is the larger decimal.
- Write the Comparison Statement: Use the symbols > (greater than), < (less than), or = (equal to) to write a comparison statement that accurately reflects the relationship between the decimals.
Examples of Comparing Decimals
Let’s walk through some examples to solidify your understanding of comparing decimals.
Example 1: Comparing Decimals to the Hundredths Place
Compare 0.65 and 0.46.
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Align the decimals:
0.65 0.46 -
Start with the largest place value: In this case, it’s the tenths place. Compare the digits in the tenths place: 6 and 4.
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Compare the digits: Since 6 is greater than 4, 0.65 is greater than 0.46.
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Comparison statement: 0.65 > 0.46
Example 2: Comparing Decimals to the Thousandths Place
Compare 0.135 and 0.167.
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Align the decimals:
0.135 0.167 -
Start with the tenths place: Both have 1 in the tenths place.
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Move to the hundredths place: Compare the digits in the hundredths place: 3 and 6.
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Compare the digits: Since 3 is less than 6, 0.135 is less than 0.167.
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Comparison statement: 0.135 < 0.167
Example 3: Comparing Decimals with Whole Numbers
Compare 3.456 and 3.018.
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Align the decimals:
3.456 3.018 -
Start with the ones place: Both have 3 in the ones place.
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Move to the tenths place: Compare the digits in the tenths place: 4 and 0.
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Compare the digits: Since 4 is greater than 0, 3.456 is greater than 3.018.
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Comparison statement: 3.456 > 3.018
Example 4: Comparing Larger Decimals
Compare 104.76 and 104.22.
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Align the decimals:
104.76 104.22 -
Start from the hundreds place: Both have 1.
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Tens place: Both have 0.
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Ones place: Both have 4.
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Move to the tenths place: Compare the digits in the tenths place: 7 and 2.
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Compare the digits: Since 7 is greater than 2, 104.76 is greater than 104.22.
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Comparison statement: 104.76 > 104.22
Example 5: Real-World Problem – Comparing Money
Frederick saved $27.98 for vacation, and Samantha saved $27.89. Who saved more?
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Align the amounts:
27.98 27.89 -
Start from the tens place: Both have 2.
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Ones place: Both have 7.
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Tenths place: Compare the digits: 9 and 8.
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Compare the digits: Since 9 is greater than 8, $27.98 is greater than $27.89.
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Comparison statement: $27.89 < $27.98
Frederick has saved the greatest amount of money.
Example 6: Real-World Problem – Comparing Time
Peter swam 800m in 9.324 minutes in week 1 and 9.243 minutes in week 2. In which week was he faster? (Faster means less time).
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Align the times:
9.324 9.243 -
Start from the ones place: Both have 9.
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Tenths place: Compare the digits: 3 and 2.
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Compare the digits: Since 2 is less than 3, 9.243 is less than 9.324.
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Comparison statement: 9.243 < 9.324
Peter swam faster in Week 2.
Teaching Tips for Comparing Decimals
- Visual Aids: Using place value charts, number lines, base-ten blocks, or decimal tiles can significantly aid students in visualizing the value of each place and making comparisons more concrete.
- Hands-on Activities: While worksheets are useful for practice, incorporate hands-on activities. For instance, using play money (like pennies and dimes) allows students to physically manipulate and compare decimal values.
- Real-Life Context: Connect decimal comparison to real-life scenarios like comparing prices, measurements in cooking, or sports statistics to enhance relevance and engagement.
Common Mistakes to Watch Out For
- Incorrect Decimal Point Alignment: A frequent error is not aligning decimal points correctly, leading to comparing digits of different place values. Always emphasize aligning the decimal points first.
- Misunderstanding Place Value: Some students may incorrectly assume decimal place value mirrors whole number place value, thinking that digits further from the decimal are larger in value. Clarify that to the right of the decimal, place values decrease.
- Confusion with Comparison Symbols: Students sometimes mix up the > (greater than) and < (less than) symbols. Use memory aids or visual cues to help them remember which symbol means “greater than” and which means “less than.”
Practice Questions
Test your understanding with these practice questions:
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Which comparison statement is correct for 0.76 and 1.23?
- 0.76 > 1.23
- 1.23 < 0.76
- 0.76 < 1.23
- 0.76 = 1.23
Explanation: Comparing the ones place, 0 is less than 1, so 0.76 < 1.23.
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Which comparison statement is correct for 0.882 and 0.9?
- 0.882 > 0.9
- 0.882 < 0.9
- 0.882 = 0.9
- 0.9 < 0.882
Explanation: Comparing the tenths place, 8 is less than 9, so 0.882 < 0.9.
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Which comparison statement is correct for 23.87 and 23.871?
- 23.87 > 23.871
- 23.87 = 23.871
- 23.871 < 23.87
- 23.87 < 23.871
Explanation: Adding a zero as a placeholder to 23.87 makes it 23.870. Comparing the thousandths place, 0 is less than 1, so 23.87 < 23.871.
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Which comparison statement is correct for 11.98 and 1.198?
- 11.98 = 1.198
- 11.98 > 1.198
- 11.98 < 1.198
- 1.198 > 11.98
Explanation: Comparing the tens place, 1 is greater than 0 (in 1.198, the tens place is 0), so 11.98 > 1.198.
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Ruby spent $23.17, and Cassie spent $32.71. Which statement correctly compares their spending?
- $32.71 < $23.17
- $32.71 = $23.17
- $32.71 > $23.17
- $23.17 > $32.71
Explanation: Comparing the tens place, 3 is greater than 2, so $32.71 > $23.17.
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Times for 4th graders running laps:
Student Time (seconds) Nydia 49.14 Carlos 45.63 Maria 44.23 Sam 40.16 
Which comparison statement correctly compares Nydia’s and Carlos’s times?
- 49.14 < 45.63
- 44.23 < 45.63
- 40.16 > 45.63
- 49.14 > 45.63
Explanation: Comparing the tens place, 4 is the same for both. Moving to the ones place, 9 is greater than 5, so 49.14 > 45.63.
Comparing Decimals: FAQs
What is the difference between comparing numbers and ordering numbers?
Comparing numbers typically involves assessing two numbers to determine which is greater or if they are equal. Ordering numbers, on the other hand, involves arranging three or more numbers from greatest to least (descending order) or least to greatest (ascending order). Comparing is a binary operation, while ordering is for a sequence of numbers.
How can I compare decimals and fractions?
To compare decimals and fractions, you have two main options:
- Convert fractions to decimals: Divide the numerator of the fraction by the denominator to convert it into a decimal. Then, compare the decimal with the other decimal using the steps outlined above.
- Convert decimals to fractions: Convert the decimal to its fractional form. For example, 0.75 becomes 75/100. Once both numbers are fractions, you can compare them by finding a common denominator or by cross-multiplication if needed.
Conclusion
Mastering the comparison of decimals is a vital step in building a strong foundation in mathematics. By following the step-by-step methods and practicing regularly, both students and adults can confidently compare decimal numbers in any context. Remember to align decimal points, compare digits place by place from left to right, and use visual aids when needed to reinforce understanding.
