Question

kareem wrote the expressions shown below.
3.2 2.7 2.9 4.8
which pair is equivalent?
○ 2.7 and 3.2
○ 2.9 and 2.7
○ 3.2 and 2.9
○ 4.8 and 3.2

Explanation:

Step1: Define floor/ceiling functions

• Floor function $\lfloor x

floor$: greatest integer ≤ $x$

• Ceiling function $\lceil x

ceil$: least integer ≥ $x$

Step2: Calculate each expression

$\lceil 3.2
ceil = 4$, $\lfloor 2.7
floor = 2$, $\lceil 2.9
ceil = 3$, $\lfloor 4.8
floor = 4$
Sum: $4 + 2 + 3 + 4 = 13$

Step3: Evaluate Option 1

$\lfloor 2.7
floor + \lceil 3.2
ceil = 2 + 4 = 6$ (≠13)

Step4: Evaluate Option 2

$\lceil 2.9
ceil + \lfloor 2.7
floor = 3 + 2 = 5$ (≠13)

Step5: Evaluate Option 3

$\lceil 3.2
ceil + \lceil 2.9
ceil = 4 + 3 = 7$ (≠13)

Step6: Evaluate Option 4

$\lfloor 4.8
floor + \lceil 3.2
ceil = 4 + 4 = 8$ (≠13)
Correction: Re-interpret as matching single pairs

Step7: Match equivalent single pairs

$\lceil 3.2
ceil = 4$, $\lfloor 4.8
floor = 4$; $\lfloor 2.7
floor = 2$, $\lceil 2.9
ceil$ is not 2. The only pair with equal values: $\lfloor 4.8
floor$ and $\lceil 3.2
ceil$

Answer:

○ $\lfloor 4.8
floor$ and $\lceil 3.2
ceil$