with rational numbers
directions for multiplying /dividing word problems-
translate the following questions into an appropriate mathematical
expression. show your work to justify your answers and give your
answers in complete sentences, relating it to the given problem.
what ideas indicate a multiplication?
what ideas indicate a division?
1. at noon on friday the temperature was 40°f. four hours later
the temperature was 25°f. on average, what was the
temperature change per hour?
2. a submarine is descending at a rate of $\frac{1}{5}$ meter per minute. if
they continue this maneuver for 12 minutes, where will they be
considering where they started?

Question

Accepted Answer

### Problem 1:
# Explanation:
## Step1: Identify temperature change
The initial temperature is $ 40^\circ\text{F} $, and the final temperature is $ 25^\circ\text{F} $. The temperature change is $ 25 - 40 = -15^\circ\text{F} $ (negative because it decreased).
## Step2: Calculate average change per hour
We divide the total temperature change by the number of hours (4 hours). So the expression is $ \frac{-15}{4} $.
## Step3: Compute the value
$ \frac{-15}{4} = -3.75 $. This means the temperature decreased by an average of $ 3.75^\circ\text{F} $ per hour.

# Answer: The average temperature change per hour was a decrease of $ 3.75^\circ\text{F} $ (or $ -3.75^\circ\text{F} $ per hour).

### Problem 2:
# Explanation:
## Step1: Identify the formula for distance
Distance (or change in position) is rate multiplied by time. The rate is $ \frac{1}{5} $ meter per minute (descending, so we can consider it as a negative rate if we take ascending as positive, but since we just need the position relative to start, we can calculate the distance descended).
## Step2: Set up the multiplication
The time is 12 minutes. So the distance descended is $ \frac{1}{5} \times 12 $.
## Step3: Compute the value
$ \frac{1}{5} \times 12 = \frac{12}{5} = 2.4 $ meters. Since it's descending, the submarine will be 2.4 meters below its starting position.

# Answer: The submarine will be 2.4 meters below its starting position (or at a position of $ -2.4 $ meters relative to the start, if we take the start as 0 and descending as negative direction).

### Ideas for Multiplication:
# Brief Explanations:
- **Repeated Addition**: For example, 3 groups of 5 is $ 3 \times 5 $, which is the same as $ 5 + 5 + 5 $.
- **Rate and Time**: When you have a rate (like speed, temperature change rate, etc.) and a time, you multiply them to get the total change (e.g., distance = speed × time).
- **Scaling or Proportions**: If you need to find a fraction or multiple of a quantity, like "half of 10" ( $ \frac{1}{2} \times 10 $) or "3 times a number".
- **Area/Volume Calculations**: For example, area of a rectangle is length × width, which involves multiplying two dimensions.

# Answer: Ideas indicating multiplication include repeated addition, rate - time relationships, scaling/proportions, and area/volume calculations (e.g., distance = rate×time, finding a fraction of a quantity, area of a rectangle = length×width).

### Ideas for Division:
# Brief Explanations:
- **Sharing Equally**: If you have a total quantity and you want to divide it into equal groups, like dividing 10 candies among 5 kids ( $ 10\div5 $).
- **Finding a Rate**: When you know the total change and the time (or number of groups) and want to find the change per unit (like the temperature change per hour in problem 1, which was total temperature change divided by number of hours).
- **Inverse of Multiplication**: If you know the product and one factor, and want to find the other factor (e.g., if $ 3\times x = 12 $, then $ x = 12\div3 $).
- **Part - Whole Relationships**: If you know a part of a quantity and the fraction it represents, you can divide to find the whole (e.g., if $ \frac{1}{3} $ of a number is 5, then the number is $ 5\div\frac{1}{3} $).

# Answer: Ideas indicating division include equal sharing, finding a rate (per unit), inverse of multiplication (finding a missing factor), and part - whole relationship calculations (e.g., dividing a total into equal groups, total change divided by time to find change per unit, finding the whole when a part and its fraction are known).

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QUESTION IMAGE

Question

Explanation:

Problem 1:

Step1: Identify temperature change

Step2: Calculate average change per hour

Step3: Compute the value

Step1: Identify the formula for distance

Step2: Set up the multiplication

Step3: Compute the value

Answer:

Problem 2: