Let's go through the problem step by step to determine the correct answers.
1. Favian received a [tex]$100 gift card.
2. He bought 1 belt costing $[/tex]20.
3. He used the remaining amount to buy [tex]\( n \)[/tex] sweaters, each costing [tex]$32.
First, let's find the remaining amount after buying the belt:
\[ 100 - 20 = 80 \]
Now, we need to determine how many sweaters (n) he can buy with the remaining $[/tex]80:
[tex]\[ n \times 32 = 80 \][/tex]
[tex]\[ n = \frac{80}{32} \][/tex]
[tex]\[ n = 2.5 \][/tex]
Since [tex]\( n \)[/tex] must be an integer (you can only buy whole sweaters), we reconsider and understand that he most likely bought 2 sweaters. We correct the facts and resolve:
[tex]\[ n = 2 \][/tex]
[tex]\[ 2 \times 32 = 64 \][/tex]
[tex]\[ 80 - 64 = 16 \][/tex]
So now [tex]\( n = 2 \)[/tex].
Next, let's evaluate the given statements one by one.
1. [tex]\( 80 \div n \geq 32 \)[/tex]:
[tex]\[ 80 \div 2 = 40 \][/tex]
[tex]\[ 40 \geq 32 \][/tex]
This statement is true.
2. [tex]\( 32 \div n \geq 80 \)[/tex]:
[tex]\[ 32 \div 2 = 16 \][/tex]
[tex]\[ 16 \geq 80 \][/tex]
This statement is false.
3. [tex]\( 80 \div n \leq 32 \)[/tex]:
[tex]\[ 80 \div 2 = 40 \][/tex]
[tex]\[ 40 \leq 32 \][/tex]
This statement is false.
4. [tex]\( 100 - 32 \pi \leq 20 \)[/tex]:
Let's consider [tex]\( 32 \pi \approx 100.48 \)[/tex] (using the approximation [tex]\(\pi \approx 3.14\)[/tex]):
[tex]\[ 100 - 32 \times 3.14 \approx 100 - 100.48 = -0.48 \][/tex]
Since [tex]\(-0.48 \leq 20\)[/tex], the statement is true.
Summarizing:
1. [tex]\( 80 \div n \geq 32 \)[/tex]: TRUE
2. [tex]\( 32 \div n \geq 80 \)[/tex]: FALSE
3. [tex]\( 80 \div n \leq 32 \)[/tex]: FALSE
4. [tex]\( 100 - 32 \pi \leq 20 \)[/tex]: TRUE
Therefore, the correct statements are:
- [tex]\( 80 \div n \geq 32 \)[/tex]
- [tex]\( 100 - 32 \pi \leq 20 \)[/tex]
