Question

1. which of the following is a member of ( m \times n )?

( m = {20, 30, 55, 60} ); ( n = {30, 55, 65} )
a. ( (20, 20) ) c. ( (55, 60) )
b. ( (30, 30) ) d. ( (30, 60) )

1. marc sold 461 tickets for the school play. student tickets cost $3 and adult tickets cost $4. marcs sales totaled $1624. how many adult tickets and how many student tickets did marc sell?

a. 220 adult, 241 student c. 236 adult, 225 student
b. 225 adult, 236 student d. 241 adult, 220 student

1. if the exchange rate for 1 u.s. dollar is 1.5 canadian dollars, how many u.s. dollars will margie get for 81 canadian dollars?

a. 28 c. 54
b. 52 d. 112

Explanation:

Question 48

Step1: Recall Cartesian Product

The Cartesian product \( M \times N \) consists of all ordered pairs \((m, n)\) where \( m \in M \) and \( n \in N \). Given \( M = \{20, 30, 55, 60\} \) and \( N = \{30, 55, 65\} \).

Step2: Check Each Option

• Option a: \((20, 20)\) – \( 20

otin N \), so invalid.

• Option b: \((30, 30)\) – \( 30 \in M \) and \( 30 \in N \), valid.
• Option c: \((55, 60)\) – \( 60

otin N \), invalid.

• Option d: \((30, 60)\) – \( 60

otin N \), invalid.

Step1: Define Variables

Let \( x \) be adult tickets, \( y \) be student tickets. We have \( x + y = 461 \) (total tickets) and \( 4x + 3y = 1624 \) (total sales).

Step2: Solve the System

From \( x + y = 461 \), we get \( y = 461 - x \). Substitute into the sales equation:
\( 4x + 3(461 - x) = 1624 \)
\( 4x + 1383 - 3x = 1624 \)
\( x + 1383 = 1624 \)
\( x = 1624 - 1383 = 241 \). Then \( y = 461 - 241 = 220 \). So adult tickets \( = 241 \), student tickets \( = 220 \).

Step1: Set Up Proportion

Let \( x \) be U.S. dollars. The exchange rate is \( 1 \) U.S. dollar \( = 1.5 \) Canadian dollars. So \( \frac{1}{1.5} = \frac{x}{81} \).

Step2: Solve for \( x \)

Cross - multiply: \( 1.5x = 81 \). Then \( x=\frac{81}{1.5}=54 \).

Answer:

b. (30, 30)

Question 49