An online store sells small gift boxes for 10 dollars each and large gift boxes for 15 dol… — SAT

An online store sells small gift boxes for $10$ dollars each and large gift boxes for $15$ dollars each. A customer wants to buy a total of at most $10$ boxes. If the customer must spend at least $120$ dollars on the boxes, what is the minimum number of large gift boxes the customer must buy?

Answer: 4 boxes

Answer

The minimum number of large gift boxes the customer must buy is 4.

Using the constraints from the problem, we set up the system of inequalities:

x + y \leq 10

and

10x + 15y \geq 120

. Expressing the first inequality as

x \leq 10 - y

and substituting it into the second gives

10(10 - y) + 15y \geq 120

. Simplifying this results in

100 + 5y \geq 120

, which simplifies to

5y \geq 20

, or

y \geq 4

. Therefore, the minimum number of large gift boxes the customer must buy is

4

Step-by-Step Solution

Define variables for the quantities of each box.

Let

x

be the number of small boxes and

y

be the number of large boxes.

Establishing variables is necessary to translate the word problem into algebraic inequalities.

Write the system of inequalities representing the constraints.

x + y \leq 10

and

10x + 15y \geq 120

The total number of boxes is at most

10

, and the total cost must be at least

120

dollars.

Express

x

in terms of

y

using the first inequality.

x \leq 10 - y

This allows substitution into the second inequality to solve for the target variable

y

Substitute

x \leq 10 - y

into the second inequality and simplify.

100 + 5y \geq 120

Solving the resulting single-variable inequality will determine the possible values of

y

Solve the inequality for

y

y \geq 4

Subtracting

100

and dividing by

5

isolates

y

to show its minimum possible value is

4

Key Concept

Systems of Linear Inequalities in Two Variables

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