Question

1. alison is buying binders for school. small binders cost $3 each, and large binders cost $5 each. if alison needs to buy at least 12 binders and has no more than $45 to spend, what is the maximum number of large binders she can buy?

a. 9
b. 4
c. 5
d. 8

Explanation:

Step1: Define variables

Let $x$ = number of large binders, $y$ = number of small binders.

Step2: Set up constraints

1. Cost constraint: $5x + 3y \leq 45$
2. Quantity constraint: $x + y \geq 12$

Step3: Rearrange quantity constraint

Solve for $y$: $y \geq 12 - x$

Step4: Substitute into cost constraint

Substitute $y = 12 - x$ (minimum small binders to maximize large ones) into cost inequality:
$5x + 3(12 - x) \leq 45$

Step5: Simplify the inequality

$5x + 36 - 3x \leq 45$
$2x + 36 \leq 45$
$2x \leq 9$
$x \leq 4.5$

Step6: Round to valid integer

Since $x$ must be a whole number, the maximum integer $x$ is 4. Verify: $5(4)+3(8)=20+24=44 \leq 45$, and $4+8=12$ which meets the quantity requirement.

Answer:

B. 4