Consider the following two numbers:
[tex]$x=0.649 \text { and } y=0 . \overline{649}$[/tex]
Select all of the true statements below:
[tex]$x$[/tex] is a larger value than [tex]$y$[/tex].
[tex]$y$[/tex] is a rational number.
[tex]$x$[/tex] is a rational number.
[tex]$y$[/tex] is an irrational number.
[tex]$x$[/tex] is an irrational number.
[tex]$y$[/tex] is a larger value than [tex]$x$[/tex].
Answer (2)
Compare x = 0.649 and y = 0. 649 = 0.649649649... and determine that x"> y > x .
Recognize that x = 0.649 = 1000 649 is a rational number.
Convert the repeating decimal y = 0. 649 to a fraction y = 999 649 , confirming that y is a rational number.
Conclude that the true statements are ' y is a larger value than x ', ' y is a rational number', and ' x is a rational number'.
Explanation
Problem Analysis We are given two numbers, x = 0.649 and y = 0. 649 = 0.649649649... . We need to determine which of the given statements are true. Let's analyze each statement.
Comparing x and y First, let's compare the values of x and y . We have x = 0.649 and y = 0.649649649... . Since y has a '6' in the thousandths place while x has a '0' (implicitly) in the thousandths place, y is larger than x . So, the statement ' y is a larger value than x ' is true.
Is x rational? Next, let's determine if x is a rational number. A rational number can be expressed as a fraction q p , where p and q are integers and q = 0 . We can write x = 0.649 as 1000 649 . Since 649 and 1000 are integers, x is a rational number. So, the statement ' x is a rational number' is true.
Is y rational? Now, let's determine if y is a rational number. The number y = 0. 649 is a repeating decimal. Repeating decimals are rational numbers because they can be expressed as a fraction. To convert y to a fraction, let y = 0. 649 . Then 1000 y = 649. 649 . Subtracting y from 1000 y , we get 999 y = 649 , so y = 999 649 . Since 649 and 999 are integers, y is a rational number. So, the statement ' y is a rational number' is true.
Irrationality check Since x and y are rational numbers, the statements ' x is an irrational number' and ' y is an irrational number' are false.
Final Answer Finally, we have determined that the true statements are: ' y is a larger value than x ', ' y is a rational number', and ' x is a rational number'.
Examples
Understanding rational numbers and their decimal representations is crucial in many real-world applications. For instance, when calculating financial transactions involving fractional amounts of currency, it's important to know whether the decimal representation terminates or repeats. If you're dividing a bill of $649 among 999 people, each person owes 999 649 dollars, which is approximately $0.649649...$. Knowing this is a rational number allows for precise calculations and fair distribution of costs.
The true statements about the numbers x and y are that y is larger than x , and both x and y are rational numbers. Therefore, the correct answers are the statements regarding their size and rationality. The statements claiming x or y is irrational are false.
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