Question

1. shreya and maria are selling cheesecakes for a school fundraiser. customers can buy pecan cheesecakes and apple cheesecakes. shreya sold 12 pecan cheesecakes and 6 apple cheesecakes for a total of $168. maria sold 7 pecan cheesecakes and 2 apple cheesecakes for a total of $68. find the cost each of one pecan cheesecake and one apple cheesecake.

1. darryl and jennifer each improved their yards by planting rose bushes and ivy. they bought their supplies from the same store. darryl spent $57 on 8 rose bushes and 3 pots of ivy. jennifer spent $42 on 1 rose bush and 12 pots of ivy. find the cost of one rose bush and the cost of one pot of ivy.

1. the senior classes at high school a and high school b planned separate trips to the county fair. the senior class at high school a rented and filled 8 vans and 8 buses with 520 students. high school b rented and filled 4 vans and 6 buses with 374 students. every van had the same number of students in it as did the buses. how many students can a van carry? how many students can a bus carry?

1. a boat traveled 105 miles each way downstream and back. the trip downstream took 5 hours. the trip back took 15 hours. find the speed of the boat in still water and the speed of the current.

Explanation:

Problem 61

Step1: Define Variables

Let \( p \) be the cost of one pecan cheesecake and \( a \) be the cost of one apple cheesecake.

Step2: Set Up Equations

From Shreya's sales: \( 12p + 6a = 168 \) (divide by 6: \( 2p + a = 28 \) → \( a = 28 - 2p \))
From Maria's sales: \( 7p + 2a = 68 \)

Step3: Substitute \( a \)

Substitute \( a = 28 - 2p \) into \( 7p + 2a = 68 \):
\( 7p + 2(28 - 2p) = 68 \)
\( 7p + 56 - 4p = 68 \)
\( 3p = 12 \) → \( p = 4 \)

Step4: Find \( a \)

Substitute \( p = 4 \) into \( a = 28 - 2p \):
\( a = 28 - 8 = 20 \)

Step1: Define Variables

Let \( r \) be the cost of one rose bush and \( i \) be the cost of one pot of ivy.

Step2: Set Up Equations

Darryl: \( 8r + 3i = 57 \)
Jennifer: \( r + 12i = 42 \) (solve for \( r \): \( r = 42 - 12i \))

Step3: Substitute \( r \)

Substitute \( r = 42 - 12i \) into \( 8r + 3i = 57 \):
\( 8(42 - 12i) + 3i = 57 \)
\( 336 - 96i + 3i = 57 \)
\( -93i = -279 \) → \( i = 3 \)

Step4: Find \( r \)

Substitute \( i = 3 \) into \( r = 42 - 12i \):
\( r = 42 - 36 = 6 \)

Step1: Define Variables

Let \( v \) be students per van and \( b \) be students per bus.

Step2: Set Up Equations

School A: \( 8v + 8b = 520 \) (divide by 8: \( v + b = 65 \) → \( v = 65 - b \))
School B: \( 4v + 6b = 374 \)

Step3: Substitute \( v \)

Substitute \( v = 65 - b \) into \( 4v + 6b = 374 \):
\( 4(65 - b) + 6b = 374 \)
\( 260 - 4b + 6b = 374 \)
\( 2b = 114 \) → \( b = 57 \)

Step4: Find \( v \)

Substitute \( b = 57 \) into \( v = 65 - b \):
\( v = 65 - 57 = 8 \)

Answer:

Pecan cheesecake: \$4, Apple cheesecake: \$20

Problem 62