1. What Does It Mean to Compare Numbers?
When comparing numbers, you determine which number is larger, smaller, or if both numbers are equal. This process involves examining the digits in corresponding place values, starting from the leftmost digit.
2. Steps to Compare Numbers
Step 1: Line Up the Digits
Write the numbers vertically, aligning their place values (ones, tens, hundreds, thousands, etc.).
Step 2: Compare from the Highest Place Value
Begin comparing digits starting from the leftmost place value (highest place).
- If the digits are different, the larger digit indicates the larger number.
- If the digits are the same, move to the next place value to the right.
Step 3: Make a Conclusion
Once a difference is found, determine whether the first number is greater, smaller, or equal to the second number.
3. Comparing Numbers Using Place Value
Let’s compare two numbers:
Example 1:
Compare: 54,231 and 53,876
- Look at the ten-thousands place:
- 5 = 5 (move to the next place).
- Look at the thousands place:
- 4 > 3, so 54,231 is greater than 53,876.
Example 2:
Compare: 123,579 and 123,451
- Look at the hundred-thousands place:
- 1 = 1 (move to the next place).
- Look at the ten-thousands place:
- 2 = 2 (move to the next place).
- Look at the thousands place:
- 3 = 3 (move to the next place).
- Look at the hundreds place:
- 5 > 4, so 123,579 is greater than 123,451.
Activity:
Compare the following numbers:
- 76,483 and 75,934
- 9,835 and 9,853
- 482,391 and 482,251
4. Using Symbols to Compare Numbers
There are three key symbols used when comparing numbers:
- Greater than (>): This means the first number is larger.
- Less than (<): This means the first number is smaller.
- Equal to (=): This means both numbers are the same.
Example:
- 67,892 > 67,489 (67,892 is greater than 67,489).
- 12,345 < 13,456 (12,345 is less than 13,456).
- 89,123 = 89,123 (both numbers are equal).
Activity:
Use the correct symbol (<, >, =) to compare:
- 35,672 and 35,762
- 48,129 and 48,091
- 91,425 and 91,425
5. More Examples
5.1: Comparing Populations
Question:
Two cities have populations of 537,928 and 529,431. Which city has the larger population?
Solution:
- Compare the hundred-thousands place:
- 5 = 5 (move to the next place).
- Compare the ten-thousands place:
- 3 > 2, so 537,928 has a larger population.
5.2: Comparing Distances Traveled
Question:
A truck traveled 4,587 kilometers, while a car traveled 4,659 kilometers. Which vehicle traveled a greater distance?
Solution:
- Compare the thousands place:
- 4 = 4 (move to the next place).
- Compare the hundreds place:
- 6 > 5, so the car traveled a greater distance.
5.3: Comparing Annual Sales
Question:
A company sold 789,543 units of product A and 798,234 units of product B in a year. Which product sold more?
Solution:
- Compare the hundred-thousands place:
- 7 = 7 (move to the next place).
- Compare the ten-thousands place:
- 9 > 8, so product B sold more.
5.4: Largest of Four Numbers
Question:
Compare the numbers 45,732, 47,821, 45,723, and 46,981. Which is the largest?
Solution:
- 47,821: Largest ten-thousands place is 4, and in the thousands place, 7 is the largest.
Answer: 47,821 is the largest number.
5.5: Comparing Bank Balances
Question:
Alice has $8,453 in her savings account, and Bob has $8,532 in his. Who has more savings?
Solution:
- Compare the thousands place:
- 8 = 8 (move to the next place).
- Compare the hundreds place:
- 5 > 4, so Bob has more savings.
5.6: Comparing Mountain Heights
Question:
Mount Everest is 8,848 meters high, and K2 is 8,611 meters high. Which is taller?
Solution:
- Compare the thousands place:
- 8 = 8 (move to the next place).
- Compare the hundreds place:
- 8 > 6, so Mount Everest is taller.
6. Dealing with Numbers in Different Formats
Sometimes, numbers are given in word form or different units (thousands, millions). Convert these numbers into standard form before comparing.
Example:
Compare: 1.5 million and 1,200,000
- Convert 1.5 million into numbers: 1,500,000.
- Now compare: 1,500,000 > 1,200,000.
Activity:
Compare these numbers:
- 2.3 million and 2,300,000
- 75,000 and 74.8 thousand
- 9.2 million and 920,000
7. Comparing Decimal Numbers
When comparing decimal numbers, line up the decimal points and compare digit by digit, just like whole numbers.
Example 1:
Compare: 56.43 and 56.38
- Compare the ones place:
- 5 = 5 (move to the next place).
- Compare the tenths place:
- 6 = 6 (move to the next place).
- Compare the hundredths place:
- 4 > 3, so 56.43 is greater than 56.38.
Example 2:
Compare: 4.09 and 4.1
- Compare the ones place:
- 4 = 4 (move to the next place).
- Compare the tenths place:
- 1 > 0, so 4.1 is greater than 4.09.
Activity:
Compare the following decimal numbers:
- 8.54 and 8.45
- 12.37 and 12.307
- 4.902 and 4.92
8. Practice Problems
- Compare 48,713 and 48,731. Which is larger?
- Use the correct symbol (<, >, =) to compare:
- 9,675 and 9,765
- 1,450,320 and 1.45 million
- 76.48 and 76.84
- Arrange the following numbers in ascending order: 23,451, 23,154, 23,541.
- The distance between two cities is 3,948 km. Another route is 3,984 km. Which route is shorter?
- Compare 5.673 and 5.67. Which is greater?
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