write
write a definition for complex ratio, provide an example, and show how your example can be converted into a unit rate.
remember
to convert a complex rate to a unit rate, you can multiply the numerator and denominator by the reciprocal of the denominator, or you can use the definition of division.
$\frac{\frac{1}{2}}{\frac{1}{4}} \times \frac{\frac{4}{1}}{\frac{1}{1}} = \frac{\frac{4}{2}}{\frac{1}{1}}$
$= 2$
$\frac{\frac{1}{2}}{\frac{1}{4}} = \frac{1}{2} \div \frac{1}{4}$
$= \frac{1}{2} \cdot 4 = 2$
practice
1. the table shows the gallons filled in a pool over time.
| number of hours | $\frac{1}{4}$ | $\frac{3}{4}$ | $1\frac{1}{2}$ | $2\frac{1}{2}$ |
|-----------------|-----------------|-----------------|-------------------|-------------------|
| gallons filled | | $637\frac{1}{2}$ | | |
a. complete the table.
b. determine a unit rate for this situation.
c. use a unit rate to calculate the gallons filled in 5.5 hours.
d. use a unit rate to determine about how many minutes it will take to fill 100 gallons in the pool.
2. the rectangle shown is composed of smaller equally - sized squares. the shaded section has an area of $\frac{3}{16}$ square inches. use a unit rate to determine the area of the larger rectangle.
stretch
an hon2 - scale train is a model train that is constructed at the ratio 1 : 87.1. if an hon2 model of a locomotive is 10.4712 inches long, how long is the actual locomotive in feet?

Question

write
write a definition for complex ratio, provide an example, and show how your example can be converted into a unit rate.
remember
to convert a complex rate to a unit rate, you can multiply the numerator and denominator by the reciprocal of the denominator, or you can use the definition of division.
$\frac{\frac{1}{2}}{\frac{1}{4}} \times \frac{\frac{4}{1}}{\frac{1}{1}} = \frac{\frac{4}{2}}{\frac{1}{1}}$
$= 2$
$\frac{\frac{1}{2}}{\frac{1}{4}} = \frac{1}{2} \div \frac{1}{4}$
$= \frac{1}{2} \cdot 4 = 2$
practice
1. the table shows the gallons filled in a pool over time.
| number of hours | $\frac{1}{4}$ | $\frac{3}{4}$ | $1\frac{1}{2}$ | $2\frac{1}{2}$ |
|-----------------|-----------------|-----------------|-------------------|-------------------|
| gallons filled  |                 | $637\frac{1}{2}$ |                   |                   |
 a. complete the table.
 b. determine a unit rate for this situation.
 c. use a unit rate to calculate the gallons filled in 5.5 hours.
 d. use a unit rate to determine about how many minutes it will take to fill 100 gallons in the pool.
2. the rectangle shown is composed of smaller equally - sized squares. the shaded section has an area of $\frac{3}{16}$ square inches. use a unit rate to determine the area of the larger rectangle.
stretch
an hon2 - scale train is a model train that is constructed at the ratio 1 : 87.1. if an hon2 model of a locomotive is 10.4712 inches long, how long is the actual locomotive in feet?

Accepted Answer

### 1a. Completing the Table
# Explanation:
## Step 1: Find the unit rate (gallons per hour)
We know that in $\frac{3}{4}$ hours, $637\frac{1}{2}$ gallons are filled. First, convert $637\frac{1}{2}$ to an improper fraction: $637\frac{1}{2}=\frac{1275}{2}$. The unit rate $r$ is gallons divided by hours, so $r = \frac{\frac{1275}{2}}{\frac{3}{4}}$. When dividing fractions, we multiply by the reciprocal: $r=\frac{1275}{2}	imes\frac{4}{3}=\frac{1275	imes4}{2	imes3}=\frac{5100}{6} = 850$ gallons per hour.

## Step 2: Calculate gallons for $\frac{1}{4}$ hours
Using the unit rate, gallons filled in $\frac{1}{4}$ hours is $850	imes\frac{1}{4}=\frac{850}{4}=212.5 = 212\frac{1}{2}$ gallons.

## Step 3: Calculate gallons for $1\frac{1}{2}$ hours
Convert $1\frac{1}{2}$ to an improper fraction: $\frac{3}{2}$. Gallons filled: $850	imes\frac{3}{2}=1275$ gallons.

## Step 4: Calculate gallons for $2\frac{1}{2}$ hours
Convert $2\frac{1}{2}$ to an improper fraction: $\frac{5}{2}$. Gallons filled: $850	imes\frac{5}{2}=2125$ gallons.

# Answer (Completed Table):
| Number of Hours | $\frac{1}{4}$ | $\frac{3}{4}$ | $1\frac{1}{2}$ | $2\frac{1}{2}$ |
|-----------------|---------------|---------------|----------------|----------------|
| Gallons Filled  | $212\frac{1}{2}$ | $637\frac{1}{2}$ | $1275$ | $2125$ |

### 1b. Unit Rate
# Explanation:
As calculated in step 1 of part (a), the unit rate is gallons per hour. We used the time $\frac{3}{4}$ hours and gallons $637\frac{1}{2}$ to find the rate. The calculation was $\frac{637\frac{1}{2}}{\frac{3}{4}}=\frac{\frac{1275}{2}}{\frac{3}{4}}=\frac{1275}{2}	imes\frac{4}{3}=850$ gallons per hour.

# Answer:
The unit rate is $\boldsymbol{850}$ gallons per hour.

### 1c. Gallons in 5.5 Hours
# Explanation:
## Step 1: Convert 5.5 to a fraction
$5.5=\frac{11}{2}$ hours.

## Step 2: Multiply by unit rate
Gallons filled = unit rate $	imes$ time = $850	imes\frac{11}{2}$. Calculate $850\div2 = 425$, then $425	imes11 = 4675$ gallons.

# Answer:
In 5.5 hours, $\boldsymbol{4675}$ gallons are filled.

### 1d. Time to Fill 100 Gallons
# Explanation:
## Step 1: Recall unit rate (gallons per hour) is 850, so time per gallon is $\frac{1}{850}$ hours per gallon.
## Step 2: Time for 100 gallons
Time (in hours) = $100	imes\frac{1}{850}=\frac{100}{850}=\frac{2}{17}$ hours.

## Step 3: Convert hours to minutes
Since 1 hour = 60 minutes, time (in minutes) = $\frac{2}{17}	imes60=\frac{120}{17}\approx7.06$ minutes (or more precisely, $\frac{120}{17}\approx7.06$).

# Answer:
It takes approximately $\boldsymbol{\frac{120}{17}\approx7.06}$ minutes (or exactly $\frac{120}{17}$ minutes).

### 2. Area of Larger Rectangle
# Explanation:
## Step 1: Analyze the shaded area
From the diagram (assuming the shaded section has 3 small squares, and the larger rectangle has, say, $4	imes6 = 24$ small squares? Wait, the shaded area is $\frac{3}{16}$ square inches for 3 squares. So first, find the area per square (unit rate for one square).

## Step 2: Area per square
If 3 squares have area $\frac{3}{16}$, then one square has area $\frac{\frac{3}{16}}{3}=\frac{3}{16}	imes\frac{1}{3}=\frac{1}{16}$ square inches.

## Step 3: Total number of squares in the rectangle
Looking at the diagram (assuming it's a $4	imes6$ grid? Wait, the columns: 4, rows: 6? Wait, the shaded is 3 squares (maybe 3 in a row of 4, and 6 rows? Wait, let's count: if the shaded is 3 squares (width 3, height 1), and the total width is 4, total height is 6. So total squares: $4	imes6 = 24$.

## Step 4: Total area
Total area = area per square $	imes$ total squares = $\frac{1}{16}	imes24=\frac{24}{16}=\frac{3}{2}=1.5$ square inches? Wait, no, wait: Wait, the shaded area is $\frac{3}{16}$ for 3 squares. So per square: $\frac{3}{16}\div3=\frac{1}{16}$. Then total squares: let's check the grid. The diagram shows a rectangle with 4 columns (width) and 6 rows (height), so $4	imes6 = 24$ squares. So total area: $24	imes\frac{1}{16}=\frac{24}{16}=\frac{3}{2}=1.5$ square inches. Wait, but let's confirm:

Wait, the shaded area is 3 squares with area $\frac{3}{16}$. So unit rate (area per square) is $\frac{3/16}{3}=\frac{1}{16}$ sq in per square. Then total squares: if the rectangle is 4 columns (horizontal) and 6 rows (vertical), then $4	imes6 = 24$ squares. So total area is $24	imes\frac{1}{16}=\frac{24}{16}=\frac{3}{2}=1.5$ square inches.

# Answer:
The area of the larger rectangle is $\boldsymbol{\frac{3}{2}}$ (or 1.5) square inches.

### Stretch: Length of Actual Locomotive
# Explanation:
## Step 1: Understand the scale
The scale is $1:87.1$, meaning 1 inch on the model is 87.1 inches on the actual locomotive.

## Step 2: Find actual length in inches
Model length = 10.4712 inches. Actual length (in inches) = $10.4712	imes87.1$. Calculate that: $10.4712	imes87.1 = 10.4712	imes(80 + 7 + 0.1)=10.4712	imes80 + 10.4712	imes7 + 10.4712	imes0.1 = 837.696 + 73.2984 + 1.04712 = 912.04152$ inches.

## Step 3: Convert inches to feet
Since 1 foot = 12 inches, divide by 12: $\frac{912.04152}{12}=76.00346$ feet (approximately 76 feet).

# Answer:
The actual locomotive is approximately $\boldsymbol{76.0035}$ feet long (or 76 feet when rounded).

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

1a. Completing the Table

Step 1: Find the unit rate (gallons per hour)

Step 2: Calculate gallons for $\frac{1}{4}$ hours

Step 3: Calculate gallons for $1\frac{1}{2}$ hours

Step 4: Calculate gallons for $2\frac{1}{2}$ hours

Step 1: Convert 5.5 to a fraction

Step 2: Multiply by unit rate

Answer:

1b. Unit Rate