An artist sells small prints for 4 dollars each and large prints for 9 dollars each. The a… — SAT

An artist sells small prints for $4$ dollars each and large prints for $9$ dollars each. The artist wants to sell at least $15$ prints in total and earn at least $80$ dollars from these sales. If the artist sells $5$ small prints, what is the minimum number of large prints the artist must sell to meet both conditions?

Answer: 10 prints

Answer

The minimum number of large prints the artist must sell is 10.

Substituting the value of small prints (

x = 5

) into the total count inequality

x + y \geq 15

gives

5 + y \geq 15

, which simplifies to

y \geq 10

. Substituting it into the earnings inequality

4x + 9y \geq 80

yields

20 + 9y \geq 80

, which simplifies to

y \geq 6.67

. To satisfy both conditions, the value of

y

must be at least 10.

Step-by-Step Solution

Set up the inequalities for the system.

Let

x

represent the number of small prints and

y

represent the number of large prints. The constraint for the minimum number of prints is

x + y \geq 15

. The constraint for the minimum earnings is

4x + 9y \geq 80

To represent the given conditions as a system of linear inequalities in two variables.

Substitute the known value of small prints (

x = 5

) into the inequalities.

Substituting

x = 5

into the first inequality gives

5 + y \geq 15

, which simplifies to

y \geq 10

. Substituting

x = 5

into the second inequality gives

4(5) + 9y \geq 80

, which simplifies to

20 + 9y \geq 80

, then

9y \geq 60

, resulting in

y \geq \frac{20}{3} \approx 6.67

To determine the range of values for the number of large prints (

y

) under both constraints.

Determine the minimum integer value for

y

that satisfies both inequalities.

The first constraint requires

y \geq 10

, and the second constraint requires

y \geq 6.67

. Since the number of prints must be an integer and both conditions must be satisfied, the minimum value is

10

To satisfy both inequalities simultaneously with the smallest possible integer value.

Key Concept

Systems of Linear Inequalities in Two Variables

Estimated Time:1m 0s

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