Question

select the correct answer.

hattie is making fruit baskets, which include apples and bananas, to send to some of her real estate clients. she wants each basket to have at least 12 pieces of fruit, but the fruit should weigh no more than 80 ounces total. on average, each apple weighs 5 ounces, and each banana weighs 7 ounces.

which combination of apples and bananas will meet hatties requirements for each fruit basket?

• 5 apples and 8 bananas
• 14 apples and 2 bananas
• 10 apples and 4 bananas
• 8 apples and 3 bananas

Explanation:

🆕 New Concept Discovered: Systems of Linear Inequalities
Using multiple inequalities to find a shared solution

Step 1: Define the variables and constraints

Let \( a \) be the number of apples and \( b \) be the number of bananas.

We need to satisfy two conditions:

1. Total pieces of fruit: At least 12 pieces.

$$ a + b \ge 12 $$

1. Total weight: No more than 80 ounces. Each apple weighs 5 ounces, and each banana weighs 7 ounces.

$$ 5a + 7b \le 80 $$

Step 2: Test the given options

We will check which combination satisfies both inequalities.

• Option 1: 5 apples and 8 bananas (\( a = 5, b = 8 \))
• Fruit count: \( 5 + 8 = 13 \ge 12 \) (True)
• Weight: \( 5(5) + 7(8) = 25 + 56 = 81 \le 80 \) (False)

• Option 2: 14 apples and 2 bananas (\( a = 14, b = 2 \))
• Fruit count: \( 14 + 2 = 16 \ge 12 \) (True)
• Weight: \( 5(14) + 7(2) = 70 + 14 = 84 \le 80 \) (False)

• Option 3: 10 apples and 4 bananas (\( a = 10, b = 4 \))
• Fruit count: \( 10 + 4 = 14 \ge 12 \) (True)
• Weight: \( 5(10) + 7(4) = 50 + 28 = 78 \le 80 \) (True)

• Option 4: 8 apples and 3 bananas (\( a = 8, b = 3 \))
• Fruit count: \( 8 + 3 = 11 \ge 12 \) (False)

Answer:

10 apples and 4 bananas