9. at an amusement park, 360 visitors rode the roller coaster in 3 hours. write and solve a proportion to find the number of visitors at this rate who will ride the roller coaster in 7 hours.
10. reason abstractly use the table to write a proportion relating the weights on two planets. then find the missing weight. round to the nearest tenth.
a. earth: 90 pounds; venus: ____ pounds
b. mercury: 55 pounds; earth: ____ pounds
c. jupiter: 350 pounds; uranus: ____ pounds
d. venus: 115 pounds; mercury: ____ pounds
(table: weights on different planets, earth weight = 120 pounds; mercury: 45.6, venus: 109.2, uranus: 96, jupiter: 304.8)
11. justify conclusions a powdered drink mix calls for a ratio of powder to water of 1:8. if there are 32 cups of powder, how many total cups of water are needed? explain your reasoning.
12. solve each equation: ( \frac{2}{3} = \frac{18}{x + 5} )
13. ( \frac{x - 4}{10} = \frac{7}{5} )
14. ( \frac{4.5}{17 - x} = \frac{3}{8} )
15. justify conclusions a rectangle has an area of 36 square units. as the length and the width change, what do you know about their product? is the length proportional to the width? justify your reasoning. (table: rectangle, length, width; a: 3,12; b:6,6; c:9,4)
chapter 1 ratios and proportional reasoning

Question

9. at an amusement park, 360 visitors rode the roller coaster in 3 hours. write and solve a proportion to find the number of visitors at this rate who will ride the roller coaster in 7 hours.
10. reason abstractly use the table to write a proportion relating the weights on two planets. then find the missing weight. round to the nearest tenth.
  a. earth: 90 pounds; venus: ____ pounds
  b. mercury: 55 pounds; earth: ____ pounds
  c. jupiter: 350 pounds; uranus: ____ pounds
  d. venus: 115 pounds; mercury: ____ pounds
  (table: weights on different planets, earth weight = 120 pounds; mercury: 45.6, venus: 109.2, uranus: 96, jupiter: 304.8)
11. justify conclusions a powdered drink mix calls for a ratio of powder to water of 1:8. if there are 32 cups of powder, how many total cups of water are needed? explain your reasoning.
12. solve each equation: ( \frac{2}{3} = \frac{18}{x + 5} )
13. ( \frac{x - 4}{10} = \frac{7}{5} )
14. ( \frac{4.5}{17 - x} = \frac{3}{8} )
15. justify conclusions a rectangle has an area of 36 square units. as the length and the width change, what do you know about their product? is the length proportional to the width? justify your reasoning. (table: rectangle, length, width; a: 3,12; b:6,6; c:9,4)
chapter 1 ratios and proportional reasoning

Accepted Answer

### Problem 9
# Explanation:
## Step1: Define the proportion
Let $ x $ be the number of visitors in 7 hours. The rate is constant, so $\frac{360}{3}=\frac{x}{7}$.
## Step2: Solve for $ x $
Cross - multiply: $ 3x = 360\times7 $. Then $ 3x=2520 $, and $ x=\frac{2520}{3}=840 $.
# Answer:
840

### Problem 10a
# Explanation:
## Step1: Find the ratio of Venus to Earth weight
From the table, when Earth weight is 120 pounds, Venus weight is 109.2 pounds. The ratio $ r=\frac{109.2}{120} $.
## Step2: Calculate Venus weight for 90 Earth pounds
Let $ x $ be the Venus weight. Then $\frac{x}{90}=\frac{109.2}{120}$. Cross - multiply: $ 120x = 90\times109.2 $. $ 120x = 9828 $, $ x=\frac{9828}{120}=81.9 $.
# Answer:
81.9

### Problem 10b
# Explanation:
## Step1: Find the ratio of Mercury to Earth weight
From the table, when Earth weight is 120 pounds, Mercury weight is 45.6 pounds. The ratio $ r = \frac{45.6}{120}$.
## Step2: Calculate Earth weight for 55 Mercury pounds
Let $ x $ be the Earth weight. Then $\frac{55}{x}=\frac{45.6}{120}$. Cross - multiply: $ 45.6x=55\times120 $. $ 45.6x = 6600 $, $ x=\frac{6600}{45.6}\approx144.7 $.
# Answer:
144.7

### Problem 10c
# Explanation:
## Step1: Find the ratio of Uranus to Jupiter weight
From the table, when Jupiter weight is 304.8 pounds, Uranus weight is 96 pounds. The ratio $ r=\frac{96}{304.8}$.
## Step2: Calculate Uranus weight for 350 Jupiter pounds
Let $ x $ be the Uranus weight. Then $\frac{x}{350}=\frac{96}{304.8}$. Cross - multiply: $ 304.8x = 350\times96 $. $ 304.8x=33600 $, $ x = \frac{33600}{304.8}\approx110.2 $.
# Answer:
110.2

### Problem 10d
# Explanation:
## Step1: Find the ratio of Mercury to Venus weight
From the table, when Earth weight is 120 pounds, Mercury is 45.6 pounds and Venus is 109.2 pounds. The ratio of Mercury to Venus is $ r=\frac{45.6}{109.2}$.
## Step2: Calculate Mercury weight for 115 Venus pounds
Let $ x $ be the Mercury weight. Then $\frac{x}{115}=\frac{45.6}{109.2}$. Cross - multiply: $ 109.2x=115\times45.6 $. $ 109.2x = 5244 $, $ x=\frac{5244}{109.2}\approx48.0 $.
# Answer:
48.0

### Problem 11
# Explanation:
## Step1: Define the proportion
The ratio of powder to water is $ 1:8 $. Let $ x $ be the cups of water. So $\frac{1}{8}=\frac{32}{x}$.
## Step2: Solve for $ x $
Cross - multiply: $ x = 32\times8=256 $.
# Answer:
256

### Problem 12
# Explanation:
## Step1: Cross - multiply
Given $\frac{2}{3}=\frac{18}{x + 5}$, cross - multiply to get $ 2(x + 5)=3\times18 $.
## Step2: Simplify and solve for $ x $
Expand: $ 2x+10 = 54 $. Subtract 10 from both sides: $ 2x=54 - 10=44 $. Divide by 2: $ x = 22 $.
# Answer:
22

### Problem 13
# Explanation:
## Step1: Cross - multiply
Given $\frac{x - 4}{10}=\frac{7}{5}$, cross - multiply: $ 5(x - 4)=10\times7 $.
## Step2: Simplify and solve for $ x $
Expand: $ 5x-20 = 70 $. Add 20 to both sides: $ 5x=70 + 20 = 90 $. Divide by 5: $ x = 18 $.
# Answer:
18

### Problem 14
# Explanation:
## Step1: Cross - multiply
Given $\frac{4.5}{17 - x}=\frac{3}{8}$, cross - multiply: $ 3(17 - x)=4.5\times8 $.
## Step2: Simplify and solve for $ x $
Expand: $ 51-3x = 36 $. Subtract 51 from both sides: $ - 3x=36 - 51=-15 $. Divide by - 3: $ x = 5 $.
# Answer:
5

### Problem 15
# Explanation:
## Step1: Recall the area formula of a rectangle
The area of a rectangle $ A = l\times w $, where $ l $ is length and $ w $ is width. Given $ A = 36 $, so $ l\times w=36 $.
## Step2: Check proportionality
For two quantities to be proportional, $ l=\ k\times w $ (or $ w = k\times l $) where $ k $ is a constant. But from $ l\times w = 36 $, we can rewrite it as $ l=\frac{36}{w} $, which is an inverse - variation relationship ($ l=\frac{k}{w} $ with $ k = 36 $) rather than a direct - proportion relationship. So the length and width are not proportional because their product is constant (36) and for proportionality, their ratio should be constant.
# Answer:
The product of length and width is always 36 (since $ A=l\times w = 36 $). The length is not proportional to the width because $ l=\frac{36}{w} $ (inverse variation) and not of the form $ l = kw $ (direct variation) where $ k $ is a non - zero constant.

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QUESTION IMAGE

Question

Explanation:

Problem 9

Step1: Define the proportion

Step2: Solve for \( x \)

Step1: Find the ratio of Venus to Earth weight

Step2: Calculate Venus weight for 90 Earth pounds

Step1: Find the ratio of Mercury to Earth weight

Step2: Calculate Earth weight for 55 Mercury pounds

Answer:

Problem 10a