Comparing and ordering decimals are important skills since they are applicable in almost all areas of real life, such as monetary issues, measurements, and the interpretation of data.
- When we compare decimals, we determine which number is larger or smaller by examining the digits from left to right, starting with the highest place value.
- Ordering decimals involves arranging them in either ascending (smallest to largest) or descending (largest to smallest) order.
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What are Decimals?
Decimals are a way of representing numbers that are not whole. They are an essential component of the number system used to express fractions, and values between whole numbers, and for precision in measurements. A decimal number consists of a whole number part and a fractional part, separated by a decimal point.
Decimals are numbers that contain a decimal point which acts as a separator between whole and fractional parts.
The digits after the decimal point signify proportions that are smaller parts of a whole number expressed in terms of powers of ten.
For instance, in the decimal number 3.456, the whole number part is 3 whereas ‘456’ is the fractional part which is equivalent to 456 thousandth.
Comparing Decimals
Comparing decimals means identifying which of two or more decimal numbers is bigger or is rather smaller. This process is very important for many cases in mathematics and real-life situations like financial calculations and measurements. You would be able to evaluate the size of each of these numbers by comparing digits starting with the tenth place from the left and going to the right. This technique helps in accurately evaluating and ranking decimal values, ensuring precise and meaningful comparisons.
Steps for Comparing Decimals
Step 1: Write the numbers in a vertical list, ensuring the decimal points are directly aligned.
Step 2: Add zeros to the ends of the shorter decimals to ensure all numbers have the same number of digits after the decimal point.
Step 3: Start from the leftmost digit and compare digits in each place value column. Move to the next column to the right if the digits are equal until you find a difference.
Example of Comparing Decimals
1: Compare 1.245 and 1.24.
Align: 1.245, 1.240
Compare digits: 1 = 1, 2 = 2, 4 = 4, 5 > 0
Result: 1.245 > 1.24
2: Compare 0.789 and 0.7891.
Align: 0.7890, 0.7891
Compare digits: 0 = 0, 7 = 7, 8 = 8, 9 = 9, 0 < 1
Result: 0.789 < 0.7891
3: Compare 5.062 and 5.06.
Align: 5.062, 5.060
Compare digits: 5 = 5, 0 = 0, 6 = 6, 2 > 0
Result: 5.062 > 5.06
Ordering Decimals
Ordering decimals means arranging decimal numbers in ascending (from smallest to largest) or descending (from largest to smallest) order. This process involves comparing the values of decimals to determine their relative positions.
When ordering decimals they have to be arranged according to the decimal point then each digit can be compared starting with the leftmost digit. This process enables you to systematically sort decimals, facilitating clearer data analysis and interpretation.
Steps for Comparing Decimals
Step 1: Write down the numbers, aligning the decimal points vertically.
Step 2: If the numbers have different lengths, add zeros to the end of the shorter numbers to make them the same length.
Step 3: Starting from the leftmost digit, compare the numbers:
- If the digits to the left of the decimal point differ, order them as you would whole numbers.
- If the digits to the left of the decimal point are the same, compare the digits to the right of the decimal point one by one.
Examples on Ordering Decimals
1: Order 2.5, 2.45, 2.456, and 2.4 in Ascending Order.
Align the numbers
2.500
2.450
2.456
2.400
- Compare the digits to the left of the decimal point: All the numbers have the same integer part, which is 2.
- Compare the digits to the right of the decimal point, starting from the tenths place:
- 2.5 (5 in the tenths place)
- 2.45 (4 in the tenths place)
- 2.456 (4 in the tenths place)
- 2.4 (4 in the tenths place)
Since 2.5 has the highest tenths digit (5), it is the largest among them. We now focus on the other three numbers (2.45, 2.456, and 2.4).
- Compare the hundredths place for the remaining numbers:
- 2.45 (5 in the hundredths place)
- 2.456 (5 in the hundredths place)
- 2.4 (0 in the hundredths place)
Since 2.4 has the lowest hundredths digit (0), it is the smallest of the remaining numbers. Now we compare 2.45 and 2.456.
- Compare the thousandths place:
- 2.45 (assume 0 in the thousandths place to make it 2.450)
- 2.456 (6 in the thousandths place)
Since 2.450 is less than 2.456, we can determine the order.
The given numbers 2.5, 2.45, 2.456, and 2.4 ordered in ascending order are:
2.4,2.45,2.456,2.5
Decimal Place Value
The place values for decimals are completely analogous to the place values of whole numbers and continue to the right of the decimal point. All of them are positioned in terms of power of tens: starting from tenths, hundredths, thousandths, etc.
In the number 4. 572, ‘4’ appears in the ones place, ‘5’ in the tenth place, ‘7’ in the hundredth place, and, finally, ‘2 in the thousandth place.
Rules for Comparing and Ordering Decimals
Comparing and ordering decimals can be straightforward if you follow specific rules. This section provides an overview of these rules and details on comparing decimals with the same and different numbers of decimal places.
Comparing Decimals with the Same Number of Decimal Places (Like Decimals)
- Step 1: Align the decimal points.
- Step 2: Compare the digits from left to right.
- Step 3: The first differing digit determines the order.
Example: Compare 2.45 and 2.47
Solution:
Step 1: Align the decimal points.
2.45
2.47
Step 2: Compare the digits from left to right.
The digits to the left of the decimal point are both 2, so they are equal.
Step 3: The first differing digit determines the order.
- Compare the tenths place: Both have 4, so they are equal.
- Compare the hundredths place: 5 (in 2.45) vs. 7 (in 2.47).
Since 5 is less than 7, 2.45 is less than 2.47.
Thus, 2.45 is less than 2.47.
Comparing Decimals with Different Numbers of Decimal Places (Unlike Decimals)
- Step 1: Align the decimal points and add zeros to equalize the number of decimal places(Convert them into like decimals).
- Step 2: Compare the digits from left to right.
- Step 3: The first differing digit determines the order.
Example: Compare 3.2 and 3.19.
Solution:
Step 1: Align the decimal points and add zeros:
3.20
3.19
Step 2: Compare from left to right:
- The digits to the left of the decimal point are both 3, so they are equal.
- Compare the tenths place: Both have 2, so they are equal.
- Compare the hundredths place: 0 (in 3.20) vs. 9 (in 3.19).
Step 3: The first differing digit determines the order.
Since 0 is less than 9, 3.20 is greater than 3.19.
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Solved Examples of Comparing and Ordering Decimals
Question 1: A carpenter needs to measure the length of a wooden plank. The plank measures 2.4 meters, 2.7 meters, and 2.9 meters. Which length is the longest?
To compare and order these lengths, we need to line up the decimal points and compare the digits from left to right:
- 2.4 meters
- 2.7 meters
- 2.9 meters
The first digit is the same in all three numbers (2), so we move to the next digit. The second digit is 4, 7, and 9, respectively. Since 9 is the largest, 2.9 meters is the longest.
Question 2: A customer buys a product that costs Rs. 12.50. They pay with a Rs. 20 note and receive Rs. 7.50 in change. If they also buy another product that costs Rs. 9.25, how much change will they receive?
To find the total change, we need to calculate the total amount paid and the total cost of the products:
Total amount paid = Rs. 20 (initial payment) + Rs. 7.50 (change) = Rs. 27.50
Total cost of products = Rs. 12.50 (first product) + Rs. 9.25 (second product) = Rs. 21.75The change due to the customer is the difference between the total amount paid and the total cost of the products:
Change = Rs. 27.50 - Rs. 21.75 = Rs. 5.75So, the customer will receive Rs. 5.75 in change.
Practice Problems on Comparing and Ordering Decimals
1: Compare the following pairs of decimals and determine which is greater:
- 0.75 and 0.57
- 1.23 and 1.3
- 0.409 and 0.49
2: Arrange the following decimals in ascending order:
- 0.56, 0.65, 0.506, 0.605
3: Which is smaller, 3.45 or 3.405?
4: Compare and order the following decimals from greatest to least:
- 2.34, 2.3, 2.403, 2.04
5:Determine if the following statements are true or false:
- 0.9 > 0.89
- 0.34 < 0.304
- 1.23 = 1.230
6: Which is greater, 7.07 or 7.7?