Question

5 a college received 300 more applications for its nursing program than for its science program. the number of applications for its nursing program is triple the number of applications for its science program. the number of people admitted to the nursing program is limited to 350 people. how many people were refused admission to the nursing program?
my calculations

6 a game involves accumulating points by collecting precious stones. each emerald is worth 18 points. the table below shows the number of stones collected by each player and how many points they scored. how many points did john get?

name	# of emeralds	# of rubies	# of diamonds	# of points scored
sam	0	4	10	396
john	8	2	4	?

my calculations

7 frank, peter and rob went to the same green house to buy vegetable plants for their gardens. frank bought 3 cucumber plants and 6 tomato plants for $25.50. peter bought 2 cucumber plants and 5 tomato plants for $20.00. if rob buys 1 cucumber plant and 4 tomato plants, how much will he have spent?
my calculations

8 students sell hoodies and bracelets to raise money for their graduation party. the following table shows the number of items sold and the total profit they could make. what profit did they make if they wound up selling 250 hoodies and 200 bracelets?

# of hoodies	# of bracelets	profit
300	250	$1950.00

my calculations

Explanation:

Problem 5

Step1: Define variables

Let \( x \) be the number of applications for the science program. Then the number of applications for the nursing program is \( 3x \).

Step2: Set up equation

We know that the nursing program received 300 more applications than the science program, so \( 3x - x = 300 \).

Step3: Solve for \( x \)

Simplify the equation: \( 2x = 300 \), so \( x = 150 \).

Step4: Find nursing applications

The number of nursing applications is \( 3x = 3\times150 = 450 \).

Step5: Find refused applicants

The number of admitted is 350, so refused is \( 450 - 350 = 100 \).

Step1: Find rubies and diamonds points

First, find the points for rubies and diamonds using Anna and Sam. Let \( r \) be rubies' points, \( d \) be diamonds' points. For Anna: \( 2\times18 + 9r + 3d = 342 \), so \( 36 + 9r + 3d = 342 \), \( 9r + 3d = 306 \), \( 3r + d = 102 \). For Sam: \( 0\times18 + 4r + 10d = 396 \), so \( 4r + 10d = 396 \), \( 2r + 5d = 198 \).

Step2: Solve the system

From \( 3r + d = 102 \), we get \( d = 102 - 3r \). Substitute into \( 2r + 5d = 198 \): \( 2r + 5(102 - 3r) = 198 \), \( 2r + 510 - 15r = 198 \), \( -13r = -312 \), \( r = 24 \). Then \( d = 102 - 3\times24 = 30 \).

Step3: Calculate John's points

John has 8 emeralds, 2 rubies, 4 diamonds. Points: \( 8\times18 + 2\times24 + 4\times30 = 144 + 48 + 120 = 312 \).

Step1: Define variables

Let \( c \) be cucumber plant cost, \( t \) be tomato plant cost.

Step2: Set up equations

Frank: \( 3c + 6t = 25.50 \). Peter: \( 2c + 5t = 20.00 \).

Step3: Solve the system

Multiply Frank's equation by 2: \( 6c + 12t = 51 \). Multiply Peter's equation by 3: \( 6c + 15t = 60 \). Subtract: \( 3t = 9 \), so \( t = 3 \). Substitute \( t = 3 \) into Peter's equation: \( 2c + 15 = 20 \), \( 2c = 5 \), \( c = 2.5 \).

Step4: Calculate Rob's cost

Rob buys 1 cucumber and 4 tomatoes: \( 2.5 + 4\times3 = 2.5 + 12 = 14.5 \).

Answer:

100

Problem 6