Kareem wrote the expressions shown below.
`3.27 Ц2.7` `2.97 $\lfloor 4.8\rfloor$`
Which pair is equivalent?
$\lfloor 2.7\rfloor$ and $\lceil 3.2\rceil$
$\lceil 2.9\rceil$ and $\lfloor 2.7\rfloor$
$\lceil 3.2\rceil$ and $\lceil 2.9\rceil$
$\lfloor 4.8\rfloor$ and $\lceil 3.2\rceil$
Answer (2)
Evaluate the floor function ⌊ 2.7 ⌋ = 2 , ⌊ 4.8 ⌋ = 4 .
Evaluate the ceiling function ⌈ 3.2 ⌉ = 4 , ⌈ 2.9 ⌉ = 3 .
Compare the pairs: ⌊ 2.7 ⌋ = 2 and ⌈ 3.2 ⌉ = 4 (not equal), ⌈ 2.9 ⌉ = 3 and ⌊ 2.7 ⌋ = 2 (not equal), ⌈ 3.2 ⌉ = 4 and ⌈ 2.9 ⌉ = 3 (not equal), ⌊ 4.8 ⌋ = 4 and ⌈ 3.2 ⌉ = 4 (equal).
The equivalent pair is ⌊ 4.8 ⌋ and ⌈ 3.2 ⌉ .
Explanation
Understanding Floor and Ceiling Functions We need to evaluate the floor and ceiling functions for the given numbers and then compare the pairs to find the equivalent one. The floor function, denoted by ⌊ x ⌋ , gives the greatest integer less than or equal to x . The ceiling function, denoted by ⌈ x ⌉ , gives the smallest integer greater than or equal to x .
Evaluating the Expressions Let's evaluate each expression:
⌊ 2.7 ⌋ = 2 (The greatest integer less than or equal to 2.7 is 2). ⌈ 3.2 ⌉ = 4 (The smallest integer greater than or equal to 3.2 is 4). ⌈ 2.9 ⌉ = 3 (The smallest integer greater than or equal to 2.9 is 3). ⌊ 4.8 ⌋ = 4 (The greatest integer less than or equal to 4.8 is 4).
Comparing the Pairs Now, let's check each pair for equivalence:
Pair 1: ⌊ 2.7 ⌋ and ⌈ 3.2 ⌉ . We have 2 and 4 , which are not equal. Pair 2: ⌈ 2.9 ⌉ and ⌊ 2.7 ⌋ . We have 3 and 2 , which are not equal. Pair 3: ⌈ 3.2 ⌉ and ⌈ 2.9 ⌉ . We have 4 and 3 , which are not equal. Pair 4: ⌊ 4.8 ⌋ and ⌈ 3.2 ⌉ . We have 4 and 4 , which are equal.
Finding the Equivalent Pair Therefore, the equivalent pair is ⌊ 4.8 ⌋ and ⌈ 3.2 ⌉ .
Final Answer The equivalent pair is ⌊ 4.8 ⌋ and ⌈ 3.2 ⌉ , both of which evaluate to 4.
Examples
Imagine you're organizing a movie night with friends and need to decide on a start time. You want to start the movie after everyone arrives, but also not too late. If some friends say they'll arrive around 7:20 PM, you can use the floor function ⌊ 7.33 ⌋ to round down to 7 PM, ensuring you don't start too late for those who arrive earlier. On the other hand, if you need to book a venue that charges by the hour and your event is expected to last 2.2 hours, you can use the ceiling function ⌈ 2.2 ⌉ to round up to 3 hours, ensuring you book enough time. These functions help in making practical decisions by providing the nearest whole number that fits your needs, whether it's rounding down to avoid starting too late or rounding up to ensure sufficient booking time.
The equivalent pair from the provided expressions is ⌊ 4.8 ⌋ and ⌈ 3.2 ⌉ , both equating to 4. This shows that both functions give the same result for these numbers. Therefore, the correct option is ⌊ 4.8 ⌋ and ⌈ 3.2 ⌉ .
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