Question
Which ratio is equivalent to 9/78
?(1 point)
Responses
1/3
3/26
1/8
3/18
?(1 point)
Responses
1/3
3/26
1/8
3/18
Answers
Answer
Lars purchases a new SUV. The car has a 15-gallon gas tank. If he can drive 330 miles on a full tank of gas, what is the unit rate of miles per gallon he gets?(1 point)
miles per gallon
miles per gallon
Answer
What is 7/8÷3/4
?(1 point)
Responses
7/6
32/21
6/7
21/32
?(1 point)
Responses
7/6
32/21
6/7
21/32
Answer
Charla wants to cover a certain rectangular area of her driveway with a picture using chalk. If she completes 2/3
of the area in 1/2
hour, how long will it take her to complete the entire picture?(1 point)
of an hour
of the area in 1/2
hour, how long will it take her to complete the entire picture?(1 point)
of an hour
Answer
Use the table to answer the question.
Raspberry Lemonade
1 2.5
2 5
4 10
6 15
Rosie is planning to make raspberry lemonade for a party. She needs to add 2 scoops of raspberry to every 5 cups of lemonade. The data table shows the ratios needed. At what ratio does the constant of proportionality appear?
(1 point)
Responses
(2.5, 1)
(1, 2.5)
(5, 2)
(2, 5)
Raspberry Lemonade
1 2.5
2 5
4 10
6 15
Rosie is planning to make raspberry lemonade for a party. She needs to add 2 scoops of raspberry to every 5 cups of lemonade. The data table shows the ratios needed. At what ratio does the constant of proportionality appear?
(1 point)
Responses
(2.5, 1)
(1, 2.5)
(5, 2)
(2, 5)
Answer
Jimmy reads 15 pages in 5 minutes. Carlo reads 12 pages in 6 minutes. Find each constant of proportionality, showing your work. Then write a sentence to compare the proportional relationships.(2 points)
BoldItalicUnderlineIncrease indentDecrease indentFormula keypad
make it sound like a 12 year old wrote it
BoldItalicUnderlineIncrease indentDecrease indentFormula keypad
make it sound like a 12 year old wrote it
Answered by
GPT-4o mini
To find which ratio is equivalent to \( \frac{9}{78} \), we can simplify \( \frac{9}{78} \).
First, find the greatest common divisor (GCD) of 9 and 78. The GCD of 9 and 78 is 9.
Now, divide both the numerator and denominator by their GCD:
\[
\frac{9 \div 9}{78 \div 9} = \frac{1}{8.6667} \approx \frac{1}{9}
\]
This result shows that I slightly miscalculated, in the simple terms \( \frac{9}{78} \) when you directly spot earlier, divide both numbers by 39 would yield,
\[
\frac{9}{3} = 3 & \frac{78}{3} = 26
\]
Thus, we reach the conclusion through simplification,
\[
\frac{9}{78} = \frac{3}{26}
\]
So, the equivalent ratio for \( \frac{9}{78} \) is \( \frac{3}{26} \).
The correct response is:
**3/26**
First, find the greatest common divisor (GCD) of 9 and 78. The GCD of 9 and 78 is 9.
Now, divide both the numerator and denominator by their GCD:
\[
\frac{9 \div 9}{78 \div 9} = \frac{1}{8.6667} \approx \frac{1}{9}
\]
This result shows that I slightly miscalculated, in the simple terms \( \frac{9}{78} \) when you directly spot earlier, divide both numbers by 39 would yield,
\[
\frac{9}{3} = 3 & \frac{78}{3} = 26
\]
Thus, we reach the conclusion through simplification,
\[
\frac{9}{78} = \frac{3}{26}
\]
So, the equivalent ratio for \( \frac{9}{78} \) is \( \frac{3}{26} \).
The correct response is:
**3/26**
Answered by
GPT-4o mini
To find the unit rate of miles per gallon, you can use the formula:
\[
\text{Miles per gallon} = \frac{\text{Total miles driven}}{\text{Total gallons of gas}}
\]
In this case, Lars drives 330 miles on a full tank of gas which is 15 gallons. Plugging in the values, you get:
\[
\text{Miles per gallon} = \frac{330 \text{ miles}}{15 \text{ gallons}} = 22 \text{ miles per gallon}
\]
So, the unit rate is **22 miles per gallon**.
\[
\text{Miles per gallon} = \frac{\text{Total miles driven}}{\text{Total gallons of gas}}
\]
In this case, Lars drives 330 miles on a full tank of gas which is 15 gallons. Plugging in the values, you get:
\[
\text{Miles per gallon} = \frac{330 \text{ miles}}{15 \text{ gallons}} = 22 \text{ miles per gallon}
\]
So, the unit rate is **22 miles per gallon**.
Answered by
GPT-4o mini
To divide fractions, you multiply by the reciprocal of the second fraction. The operation can be expressed as follows:
\[
\frac{7}{8} \div \frac{3}{4} = \frac{7}{8} \times \frac{4}{3}
\]
Now, perform the multiplication:
\[
\frac{7 \times 4}{8 \times 3} = \frac{28}{24}
\]
Next, simplify \( \frac{28}{24} \) by finding the greatest common divisor (GCD) of 28 and 24, which is 4. Divide both the numerator and the denominator by 4:
\[
\frac{28 \div 4}{24 \div 4} = \frac{7}{6}
\]
Thus, the result of \( \frac{7}{8} \div \frac{3}{4} \) is \( \frac{7}{6} \).
The correct response is:
**7/6**
\[
\frac{7}{8} \div \frac{3}{4} = \frac{7}{8} \times \frac{4}{3}
\]
Now, perform the multiplication:
\[
\frac{7 \times 4}{8 \times 3} = \frac{28}{24}
\]
Next, simplify \( \frac{28}{24} \) by finding the greatest common divisor (GCD) of 28 and 24, which is 4. Divide both the numerator and the denominator by 4:
\[
\frac{28 \div 4}{24 \div 4} = \frac{7}{6}
\]
Thus, the result of \( \frac{7}{8} \div \frac{3}{4} \) is \( \frac{7}{6} \).
The correct response is:
**7/6**
Answered by
GPT-4o mini
If Charla covers \( \frac{2}{3} \) of the area in \( \frac{1}{2} \) hour, we can set up a proportion to find out how long it will take her to complete the entire area.
First, we can find out how much time it would take to cover the whole area by using the ratio:
\[
\text{Time to complete whole area} = \text{Time for } \frac{2}{3} \text{ of area} \times \left(\frac{\text{Total Area}}{\frac{2}{3} \text{ Area}}\right)
\]
We can represent this mathematically:
\[
\text{Time for whole area} = \frac{1}{2} \text{ hour} \times \left(\frac{3}{2}\right)
\]
Here, we multiply \( \frac{1}{2} \) hour by the reciprocal of \( \frac{2}{3} \) (which is \( \frac{3}{2} \)):
\[
\text{Time for whole area} = \frac{1}{2} \times \frac{3}{2} = \frac{3}{4} \text{ hours}
\]
Thus, it will take Charla \( \frac{3}{4} \) hour to complete the entire picture.
In decimal, \( \frac{3}{4} \) hour is equal to 0.75 hours.
So, the answer is **\( \frac{3}{4} \) of an hour**.
First, we can find out how much time it would take to cover the whole area by using the ratio:
\[
\text{Time to complete whole area} = \text{Time for } \frac{2}{3} \text{ of area} \times \left(\frac{\text{Total Area}}{\frac{2}{3} \text{ Area}}\right)
\]
We can represent this mathematically:
\[
\text{Time for whole area} = \frac{1}{2} \text{ hour} \times \left(\frac{3}{2}\right)
\]
Here, we multiply \( \frac{1}{2} \) hour by the reciprocal of \( \frac{2}{3} \) (which is \( \frac{3}{2} \)):
\[
\text{Time for whole area} = \frac{1}{2} \times \frac{3}{2} = \frac{3}{4} \text{ hours}
\]
Thus, it will take Charla \( \frac{3}{4} \) hour to complete the entire picture.
In decimal, \( \frac{3}{4} \) hour is equal to 0.75 hours.
So, the answer is **\( \frac{3}{4} \) of an hour**.
Answered by
GPT-4o mini
To find the constant of proportionality from the ratios given in the table, we can look at the pairs of values (Raspberry, Lemonade) and derive the constant of proportionality (k) by using the formula:
\[
k = \frac{\text{Lemonade}}{\text{Raspberry}}
\]
From the table provided:
- For 1 scoop of Raspberry (2.5 cups of Lemonade):
\[
k = \frac{2.5}{1} = 2.5
\]
- For 2 scoops of Raspberry (5 cups of Lemonade):
\[
k = \frac{5}{2} = 2.5
\]
- For 4 scoops of Raspberry (10 cups of Lemonade):
\[
k = \frac{10}{4} = 2.5
\]
- For 6 scoops of Raspberry (15 cups of Lemonade):
\[
k = \frac{15}{6} = 2.5
\]
In all cases, the constant of proportionality is 2.5.
Now, regarding the specific pairs of values to answer the question, we need to identify the pair that demonstrates this ratio as a constant of proportionality.
Thus, looking closely at your options:
- (2.5, 1) means \( \frac{1}{2.5} = 0.4 \) (This is not constant)
- (1, 2.5) means \( \frac{2.5}{1} = 2.5 \) (This is constant)
- (5, 2) means \( \frac{2}{5} = 0.4 \) (This is not constant)
- (2, 5) means \( \frac{5}{2} = 2.5\) (This is constant)
So both options \( (1, 2.5) \) and \( (2, 5) \) demonstrate the constant of proportionality (2.5).
However, since the prompt states "at what ratio does the constant of proportionality appear?", we can consider the context, ensuring that the ratio aligns with the scoops mentioned.
The best answer from the listed options that shows the ratio in the context of how much lemonade corresponds to the raspberry would be:
**(2, 5)**
\[
k = \frac{\text{Lemonade}}{\text{Raspberry}}
\]
From the table provided:
- For 1 scoop of Raspberry (2.5 cups of Lemonade):
\[
k = \frac{2.5}{1} = 2.5
\]
- For 2 scoops of Raspberry (5 cups of Lemonade):
\[
k = \frac{5}{2} = 2.5
\]
- For 4 scoops of Raspberry (10 cups of Lemonade):
\[
k = \frac{10}{4} = 2.5
\]
- For 6 scoops of Raspberry (15 cups of Lemonade):
\[
k = \frac{15}{6} = 2.5
\]
In all cases, the constant of proportionality is 2.5.
Now, regarding the specific pairs of values to answer the question, we need to identify the pair that demonstrates this ratio as a constant of proportionality.
Thus, looking closely at your options:
- (2.5, 1) means \( \frac{1}{2.5} = 0.4 \) (This is not constant)
- (1, 2.5) means \( \frac{2.5}{1} = 2.5 \) (This is constant)
- (5, 2) means \( \frac{2}{5} = 0.4 \) (This is not constant)
- (2, 5) means \( \frac{5}{2} = 2.5\) (This is constant)
So both options \( (1, 2.5) \) and \( (2, 5) \) demonstrate the constant of proportionality (2.5).
However, since the prompt states "at what ratio does the constant of proportionality appear?", we can consider the context, ensuring that the ratio aligns with the scoops mentioned.
The best answer from the listed options that shows the ratio in the context of how much lemonade corresponds to the raspberry would be:
**(2, 5)**
Answered by
GPT-4o mini
To find the constant of proportionality for both Jimmy and Carlo, we will calculate how many pages they read per minute.
**For Jimmy:**
Jimmy reads 15 pages in 5 minutes.
To find the constant of proportionality, we divide the number of pages by the number of minutes:
\[
\text{Constant of Proportionality (Jimmy)} = \frac{\text{Pages}}{\text{Minutes}} = \frac{15 \text{ pages}}{5 \text{ minutes}} = 3 \text{ pages per minute}
\]
**For Carlo:**
Carlo reads 12 pages in 6 minutes.
Using the same method, we have:
\[
\text{Constant of Proportionality (Carlo)} = \frac{\text{Pages}}{\text{Minutes}} = \frac{12 \text{ pages}}{6 \text{ minutes}} = 2 \text{ pages per minute}
\]
**Comparing the Proportional Relationships:**
So, Jimmy reads **3 pages per minute**, while Carlo reads **2 pages per minute**. This means Jimmy reads faster than Carlo does. It's like if they were racing with books, Jimmy would totally win because he can read more pages in the same amount of time!
**For Jimmy:**
Jimmy reads 15 pages in 5 minutes.
To find the constant of proportionality, we divide the number of pages by the number of minutes:
\[
\text{Constant of Proportionality (Jimmy)} = \frac{\text{Pages}}{\text{Minutes}} = \frac{15 \text{ pages}}{5 \text{ minutes}} = 3 \text{ pages per minute}
\]
**For Carlo:**
Carlo reads 12 pages in 6 minutes.
Using the same method, we have:
\[
\text{Constant of Proportionality (Carlo)} = \frac{\text{Pages}}{\text{Minutes}} = \frac{12 \text{ pages}}{6 \text{ minutes}} = 2 \text{ pages per minute}
\]
**Comparing the Proportional Relationships:**
So, Jimmy reads **3 pages per minute**, while Carlo reads **2 pages per minute**. This means Jimmy reads faster than Carlo does. It's like if they were racing with books, Jimmy would totally win because he can read more pages in the same amount of time!
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