2.1 unit rates
find each unit rate. round to the nearest hundredth, if necessary.
1. $140 for 18 ft²
2. $2.99 for 14 lb
circle the better deal in each pair. then give the unit rate for the better deal.
3. $\frac{$56}{25gal}$ or $\frac{$32.05}{15gal}$
4. $\frac{$160}{5g}$ or $\frac{$315}{9g}$
2.2 constant rates of change
5. the table shows the amount of money tyler earns for mowing lawns. is the relationship a proportional relationship? why or why not?
| number of lawns | 1 | 2 | 3 | 4 |
|--|--|--|--|--|--|
| amount earned ($) | 15 | 30 | 48 | 64 |
6. on a recent day, 8 euros were worth $9 and 24 euros were worth $27. write an equation of the form y = kx to show the relationship between the number of euros and the value in dollars.
____, where y is dollars and x is euros
2.3 proportional relationships and graphs
7. the graph shows the number of servings in different amounts of frozen yogurt listed on a carton. write an equation that gives the number of servings y in x pints.
8. a refreshment stand makes 2 large servings of frozen yogurt from 3 pints. add the line to the graph and write its equation.
9. how can you use rates to determine whether a situation is a proportional relationship?

Question

Accepted Answer

### 1.
# Explanation:
## Step1: Divide total cost by area
To find the unit - rate of cost per square - foot, divide the total cost by the number of square - feet. The formula for the unit rate $r$ is $r=\frac{	ext{total cost}}{	ext{area}}$. Given a cost of $\$140$ for $18$ square - feet, $r = \frac{140}{18}\approx7.78$ dollars per square - foot.
# Answer:
$\$7.78$ per square - foot

### 2.
# Explanation:
## Step1: Divide total cost by weight
To find the unit - rate of cost per pound, divide the total cost by the number of pounds. Using the formula $r=\frac{	ext{total cost}}{	ext{weight}}$, with a cost of $\$2.99$ for $14$ pounds, $r=\frac{2.99}{14}\approx0.21$ dollars per pound.
# Answer:
$\$0.21$ per pound

### 3.
# Explanation:
## Step1: Find unit rate for first option
For the first option $\frac{56}{25}$ dollars per gallon, calculate the unit rate: $r_1=\frac{56}{25}=2.24$ dollars per gallon.
## Step2: Find unit rate for second option
For the second option $\frac{32.05}{15}$ dollars per gallon, calculate the unit rate: $r_2=\frac{32.05}{15}\approx2.14$ dollars per gallon.
## Step3: Compare unit rates
Since $2.14<2.24$, the second option is the better deal.
# Answer:
Circle $\frac{\$32.05}{15	ext{ gal}}$, $\$2.14$ per gallon

### 4.
# Explanation:
## Step1: Find unit rate for first option
For the first option $\frac{160}{5}$ dollars per gram, calculate the unit rate: $r_1 = \frac{160}{5}=32$ dollars per gram.
## Step2: Find unit rate for second option
For the second option $\frac{315}{9}$ dollars per gram, calculate the unit rate: $r_2=\frac{315}{9}=35$ dollars per gram.
## Step3: Compare unit rates
Since $32<35$, the first option is the better deal.
# Answer:
Circle $\frac{\$160}{5	ext{ g}}$, $\$32$ per gram

### 5.
# Explanation:
## Step1: Calculate ratios
For a proportional relationship, the ratio $\frac{	ext{amount earned}}{	ext{number of lawns}}$ should be constant.
For $1$ lawn and $\$15$ earned, the ratio is $\frac{15}{1}=15$.
For $2$ lawns and $\$30$ earned, the ratio is $\frac{30}{2}=15$.
For $3$ lawns and $\$48$ earned, the ratio is $\frac{48}{3}=16$.
For $4$ lawns and $\$64$ earned, the ratio is $\frac{64}{4}=16$.
Since the ratios are not all equal (the first two ratios are $15$ and the last two are $16$), it is not a proportional relationship.
# Answer:
No, the ratios of amount earned to number of lawns are not constant.

### 6.
# Explanation:
## Step1: Find the constant of proportionality $k$
We know that when $x = 8$ euros, $y = 9$ dollars. Using the equation $y=kx$, we can find $k$ by substituting the values: $k=\frac{y}{x}=\frac{9}{8}=1.125$.
## Step2: Write the equation
The equation relating the number of euros $x$ and the value in dollars $y$ is $y = 1.125x$.
# Answer:
$y = 1.125x$

### 7.
# Explanation:
## Step1: Find the constant of proportionality from the graph
From the graph, when $x = 2$ pints, $y = 10$ servings. Using the equation $y=kx$, we find $k=\frac{y}{x}=\frac{10}{2}=5$.
## Step2: Write the equation
The equation that gives the number of servings $y$ in $x$ pints is $y = 5x$.
# Answer:
$y = 5x$

### 8.
# Explanation:
## Step1: Find the constant of proportionality
If $2$ large servings are made from $3$ pints, then the ratio of servings to pints is $\frac{y}{x}=\frac{2}{3}$. The equation of the line is $y=\frac{2}{3}x$.
To add the line to the graph, we can find two points. When $x = 0$, $y = 0$; when $x=3$, $y = 2$. Plot these points and draw a straight line through them.
# Answer:
$y=\frac{2}{3}x$

### 9.
# Explanation:
In a proportional relationship, the ratio (rate) between the two quantities is constant. If you have two related quantities $x$ and $y$, calculate the rate $\frac{y}{x}$ for different pairs of values. If the value of $\frac{y}{x}$ is the same for all pairs, then the situation represents a proportional relationship.
# Answer:
Calculate the ratio (rate) of the two related quantities for different data - points. If the rate is constant, it is a proportional relationship.

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QUESTION IMAGE

Question

Explanation:

1.

Step1: Divide total cost by area

Step1: Divide total cost by weight

Step1: Find unit rate for first option

Step2: Find unit rate for second option

Step3: Compare unit rates

Answer:

2.