Val is planning to make fruit punch for his birthday sleepover. He needs to add 2 scoops of fruit punch mix for every 8 cups of water. The data table shows the ratios needed. At what ratio does the constant of proportionality appear?
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1 point
Fruit punch mix water
1 4
2 8
4 16
6 24
The constant of proportionality appears when the ratio of the fruit punch mix to water is the same for all the given pairs.
Looking at the data table, we can see that for every increase of 1 in the fruit punch mix, the water increases by 4 cups. Therefore, the constant of proportionality is 4.
(?,?)
The constant of proportionality is not represented by a specific pair of numbers in the data table. It is the value that remains constant as the ratio of fruit punch mix to water changes. In this case, the constant of proportionality is 4 cups of water for every 1 scoop of fruit punch mix.
Look at the quantities in the data table and determine if they are in a proportional relationship. If they are not in a proportional relationship, select why.
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2 points
# of trees #of oranges
2 30
4 60
5 75
7 105
9 135
12 180
To determine if the quantities in the data table are in a proportional relationship, we need to check if the ratio of the number of trees to the number of oranges remains constant for all the given pairs.
If we calculate the ratios for each pair, we get:
For the first pair (2 trees, 30 oranges), the ratio is 2/30 = 1/15.
For the second pair (4 trees, 60 oranges), the ratio is 4/60 = 1/15.
For the third pair (5 trees, 75 oranges), the ratio is 5/75 = 1/15.
For the fourth pair (7 trees, 105 oranges), the ratio is 7/105 = 1/15.
For the fifth pair (9 trees, 135 oranges), the ratio is 9/135 = 1/15.
For the sixth pair (12 trees, 180 oranges), the ratio is 12/180 = 1/15.
Since the ratio of trees to oranges remains constant as 1/15 for all the given pairs, the quantities are in a proportional relationship.
To find the ratio at which the constant of proportionality appears, you need to determine if there is a consistent relationship between the quantities of fruit punch mix and water.
First, observe the ratio between the quantities of fruit punch mix and water. Divide the quantity of water by the quantity of fruit punch mix for each row in the data table:
For the first row: 4/1 = 4
For the second row: 8/2 = 4
For the third row: 16/4 = 4
For the fourth row: 24/6 = 4
As you can see, the ratios are consistently equal to 4. This means that for every increase in the quantity of fruit punch mix, the quantity of water increases by a factor of 4.
Therefore, the constant of proportionality appears at a ratio of 4:1 between the fruit punch mix and water.