1. a hiker descends $2\frac{1}{2}$ miles each day for 4 days. write and solve a numerical expression to find the total change in elevation.
2. the table shows the cost of notebooks:
number of notebooks | 2 | 4 | 6
cost ($) | 3 | 6 | 9
is this a proportional relationship? explain your reasoning.
3. a runner travels $\frac{5}{6}$ mile in $\frac{1}{3}$ hour. find the unit rate in miles per hour.
4. explain how dividing by a fraction is related to multiplying by its reciprocal. use a numeric example.
5. the graph of a proportional relationship passes through (0,0) and (4,10). find the constant of proportionality.
6. write a real - world problem that can be represented by the equation $y = - 3x$. explain the negative sign means.

Question

1. a hiker descends $2\frac{1}{2}$ miles each day for 4 days. write and solve a numerical expression to find the total change in elevation.
2. the table shows the cost of notebooks:
number of notebooks | 2 | 4 | 6
cost ($)            | 3 | 6 | 9
is this a proportional relationship? explain your reasoning.
3. a runner travels $\frac{5}{6}$ mile in $\frac{1}{3}$ hour. find the unit rate in miles per hour.
4. explain how dividing by a fraction is related to multiplying by its reciprocal. use a numeric example.
5. the graph of a proportional relationship passes through (0,0) and (4,10). find the constant of proportionality.
6. write a real - world problem that can be represented by the equation $y = - 3x$. explain the negative sign means.

Accepted Answer

### Problem 1
# Explanation:
## Step1: Convert mixed number to improper fraction
The daily descent is $2\frac{1}{2}$ miles, which is $\frac{5}{2}$ miles.
## Step2: Multiply by number of days
To find total change, multiply daily descent by 4 days: $\frac{5}{2} \times 4$
## Step3: Simplify the expression
$\frac{5}{2} \times 4 = 5\times2 = 10$
# Answer:
The total change in elevation is 10 miles.

### Problem 2
# Brief Explanations:
For a proportional relationship, $\frac{\text{Cost}}{\text{Number of Notebooks}}$ should be constant. Calculate ratios: $\frac{3}{2} = 1.5$, $\frac{6}{4} = 1.5$, $\frac{9}{6} = 1.5$. Since the ratio (unit rate) is constant and the graph passes through $(0,0)$ (implied by proportionality), it is proportional.
# Answer:
Yes, it is a proportional relationship. The ratio of cost to number of notebooks is constant ($1.5$ dollars per notebook), and the relationship passes through $(0,0)$ (if 0 notebooks, cost is 0).

### Problem 3
# Explanation:
## Step1: Recall unit rate formula
Unit rate (miles per hour) is distance divided by time, so $\frac{\frac{5}{6}}{\frac{1}{3}}$
## Step2: Divide by multiplying by reciprocal
Dividing by $\frac{1}{3}$ is multiplying by 3: $\frac{5}{6} \times 3$
## Step3: Simplify the expression
$\frac{5}{6} \times 3=\frac{5}{2} = 2.5$
# Answer:
The unit rate is $\frac{5}{2}$ (or $2.5$) miles per hour.

### Problem 4
# Brief Explanations:
Dividing by a fraction is equivalent to multiplying by its reciprocal. For example, $2 \div \frac{1}{2}$. The reciprocal of $\frac{1}{2}$ is 2, so $2 \div \frac{1}{2}=2\times2 = 4$. This works because dividing by a number is the same as multiplying by its multiplicative inverse (reciprocal).
# Answer:
Dividing by a fraction equals multiplying by its reciprocal. Example: $2\div\frac{1}{2}=2\times2 = 4$.

### Problem 5
# Explanation:
## Step1: Recall constant of proportionality formula
For proportional relationship $y = kx$, $k=\frac{y}{x}$.
## Step2: Use given points
Using $(4,10)$, $k=\frac{10}{4}=\frac{5}{2}=2.5$
# Answer:
The constant of proportionality is $\frac{5}{2}$ (or $2.5$).

### Problem 6
# Brief Explanations:
Example: A car is moving backward (negative direction) at 3 miles per hour. Let $x$ be time in hours, $y$ be position change. $y=-3x$ means position decreases (negative sign) as time increases, representing backward motion.
# Answer:
Real - world problem: A car is moving in reverse at a speed of 3 miles per hour. Let $x$ represent the number of hours the car has been moving in reverse, and $y$ represent the change in the car's position (in miles) from its starting point. The equation $y = - 3x$ models this situation. The negative sign means that the car's position is decreasing (it is moving in the opposite direction of a positive forward - moving reference frame), i.e., it is moving backward.

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QUESTION IMAGE

Question

Explanation:

Problem 1

Step1: Convert mixed number to improper fraction

Step2: Multiply by number of days

Step3: Simplify the expression

Step1: Recall unit rate formula

Step2: Divide by multiplying by reciprocal

Step3: Simplify the expression

Answer:

Problem 2