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A 3-pound bag of apples costs $5.25. What is the unit price of a pound of apples?

per pound

At this rate, how many pounds of apples can be purchased for $17.50?

pounds of apples

Let x represent the number of pounds of apples and y the total cost of the apples. What is the equation that represents the direct variation between the total cost of the apples and the number of pounds of apples?

9 answers

To find the unit price of a pound of apples, we start with the cost of the 3-pound bag:

\[ \text{Unit price} = \frac{\text{Total cost}}{\text{Weight in pounds}} = \frac{5.25}{3} = 1.75 \text{ dollars per pound} \]

So, the unit price of a pound of apples is $1.75 per pound.

Next, to find out how many pounds of apples can be purchased for $17.50, we use the unit price:

\[ \text{Number of pounds} = \frac{\text{Total money}}{\text{Unit price}} = \frac{17.50}{1.75} = 10 \text{ pounds} \]

At this rate, 10 pounds of apples can be purchased for $17.50.

Lastly, we can express the direct variation between the total cost \( y \) and the number of pounds of apples \( x \) with the equation:

\[ y = kx \]

Where \( k \) is the unit price. Since the unit price is $1.75, the equation becomes:

\[ y = 1.75x \]

So, the equation that represents the direct variation is \( y = 1.75x \).

A 3-pound bag of apples costs $5.25. What is the unit price of a pound of apples?

✔ $1.75
per pound

At this rate, how many pounds of apples can be purchased for $17.50?

✔ 10
pounds of apples

Let x represent the number of pounds of apples and y the total cost of the apples. What is the equation that represents the direct variation between the total cost of the apples and the number of pounds of apples?

✔ y = 1.75x

What other points are on the line of direct variation through (5, 12)? Check all that apply.

(0, 0)
(2.5, 6)
(3, 10)
(7.5, 18)
(12.5, 24)
(15, 36)

To determine which points are on the line of direct variation represented by \( y = 1.75x \), we will check each point to see if it satisfies the equation.

  1. For (0, 0): \[ y = 1.75(0) = 0 \quad \text{(True)} \] This point is on the line.

  2. For (2.5, 6): \[ y = 1.75(2.5) = 4.375 \quad \text{(False)} \] This point is not on the line.

  3. For (3, 10): \[ y = 1.75(3) = 5.25 \quad \text{(False)} \] This point is not on the line.

  4. For (7.5, 18): \[ y = 1.75(7.5) = 13.125 \quad \text{(False)} \] This point is not on the line.

  5. For (12.5, 24): \[ y = 1.75(12.5) = 21.875 \quad \text{(False)} \] This point is not on the line.

  6. For (15, 36): \[ y = 1.75(15) = 26.25 \quad \text{(False)} \] This point is not on the line.

To summarize, the only point that is on the line of direct variation is (0, 0).

Therefore, the correct answer is:

✔ (0, 0)

that cant be the only one its more than 1

Apologies for the oversight in my previous response. Let's re-evaluate the points to check for which ones satisfy the direct variation equation \( y = 1.75x \).

We need to check if each point satisfies the equation \( y = 1.75x \).

  1. (0, 0): \[ y = 1.75(0) = 0 \quad \text{(True)} \] This point is on the line.

  2. (2.5, 6): \[ y = 1.75(2.5) = 4.375 \quad \text{(False)} \] This point is not on the line.

  3. (3, 10): \[ y = 1.75(3) = 5.25 \quad \text{(False)} \] This point is not on the line.

  4. (7.5, 18): \[ y = 1.75(7.5) = 13.125 \quad \text{(False)} \] This point is not on the line.

  5. (12.5, 24): \[ y = 1.75(12.5) = 21.875 \quad \text{(False)} \] This point is not on the line.

  6. (15, 36): \[ y = 1.75(15) = 26.25 \quad \text{(False)} \] This point is not on the line.

Calculation Correction

It seems like I misunderstood your query regarding these points. Let's calculate again where each \( x \) should yield the expected \( y \) in direct variation, which comes from the equation \( y = 1.75x \).

To find additional valid points, let's take several values of \( x \) and compute \( y \):

  • For \( x = 5 \): \[ y = 1.75(5) = 8.75 \]

  • For \( x = 10 \): \[ y = 1.75(10) = 17.5 \]

Next, we can check which of the given points match either calculated or known values. From our check:

In summary:

  • Valid point: (0, 0) is confirmed.
  • None of the checked options (2.5, 6), (3, 10), (7.5, 18), (12.5, 24), and (15, 36) satisfy the equation \( y = 1.75x \).

Conclusion

The only point from your list that lies on the line is (0, 0).

If I failed to include other valid points from outside the provided list, please provide their coordinates, and I'd be happy to check those!

okay LIAR (15, 36) (7.5, 18) (2.5, 6) AND 0,0

Consider the sequence of numbers: , , 1StartFraction 3 Over 8 EndFraction, StartFraction 3 Over 4 EndFraction, 1 and StartFraction 1 Over 8 EndFraction, 1 and StartFraction 1 Over 8 EndFraction, 1 and StartFraction 7 Over 8 EndFraction. . ., 1 , 1, . . .Which statement is a description of the sequence?The sequence is recursive, where each term is StartFraction 1 Over 4 EndFraction greater than its preceding term.The sequence is recursive and can be represented by the function f(n + 1) = f(n) + f of n plus 1 equals f of n plus StartFraction 3 Over 8 EndFraction..The sequence is arithmetic, where each pair of terms has a constant difference of StartFraction 3 Over 4 EndFraction.The sequence is arithmetic and can be represented by the function f(n + 1) = f(n)f of n plus 1 equals f of n plus left parenthesis StartFraction 3 Over 8 EndFraction right-parenthesis..

Let's analyze the sequence you provided:

  • The sequence starts with the numbers \( \frac{3}{8}, \frac{3}{4}, 1, \frac{1}{8}, 1, \frac{1}{7}, \frac{1}{8}, \dots \)

Listing out the relevant terms based on your description, let's clarify:

  1. \( \frac{3}{8} \)
  2. \( \frac{3}{4} \)
  3. \( 1 \)
  4. \( \frac{1}{8} \)
  5. \( 1 \)
  6. \( \frac{1}{7} \)
  7. \( 1 \)

Let's find each successive term and determine the differences:

  1. From \( \frac{3}{8} \) to \( \frac{3}{4} \): \( \frac{3}{4} - \frac{3}{8} = \frac{6}{8} - \frac{3}{8} = \frac{3}{8} \)

  2. From \( \frac{3}{4} \) to \( 1 \): \( 1 - \frac{3}{4} = \frac{4}{4} - \frac{3}{4} = \frac{1}{4} \)

  3. From \( 1 \) to \( \frac{1}{8} \): \( \frac{1}{8} - 1 = \frac{1}{8} - \frac{8}{8} = -\frac{7}{8} \)

  4. From \( \frac{1}{8} \) to \( 1 \): \( 1 - \frac{1}{8} = \frac{8}{8} - \frac{1}{8} = \frac{7}{8} \)

Looking at these calculations:

  • The differences between consecutive terms are not constant, which indicates that this sequence is not arithmetic.

Next, let's evaluate the descriptions given:

  1. The sequence is recursive, where each term is \( \frac{1}{4} \) greater than its preceding term.

    • This does not hold, as we determined that the differences are not \( \frac{1}{4} \).

  2. The sequence is recursive and can be represented by the function \( f(n + 1) = f(n) + \frac{3}{8} \).

    • This is also incorrect since the differences are not constant, thus not fitting a recursive relation based on a constant increment.

  3. The sequence is arithmetic, where each pair of terms has a constant difference of \( \frac{3}{4} \).

    • This is incorrect because the differences vary between the terms and do not maintain a constant difference.

  4. The sequence is arithmetic and can be represented by the function \( f(n + 1) = f(n) + \left( \frac{3}{8} \right) \).

    • This is also incorrect for the same reason.

Since none of the statements accurately describe the sequence based on the calculated differences, it seems there might be a misunderstanding in the terms provided or the sequence itself.

If the sequence is indeed irregular, it may not fit purely into any of the provided options—there could be a pattern arisen from other terms beyond what is stated. Please review the given options based on a complete view of the sequence, or let me clarify by analyzing specific aspects you're referring to.

The sequence is recursive and can be represented by the function f(n + 1) = f(n) 3/8.

Thank you for your patience, and let's clarify the sequence you provided.

If we consider the sequence:

  • \( \frac{3}{8} \)
  • \( \frac{3}{4} \)
  • \( 1 \)
  • \( \frac{1}{8} \)
  • \( 1 \)
  • \( \frac{1}{7} \)
  • \( 1 \)

To determine the nature of the sequence based on your provided conclusion:

  1. Assuming the function \( f(n+1) = f(n) + \frac{3}{8} \):

    • This would imply that each term is derived from its preceding term plus \( \frac{3}{8} \).

  2. Let’s validate this assumption:

    • Start with \( f(1) = \frac{3}{8} \)
    • If \( f(n) = \frac{3}{8} \), then: \( f(2) = f(1) + \frac{3}{8} = \frac{3}{8} + \frac{3}{8} = \frac{6}{8} = \frac{3}{4} \) (Correct)
    • Next: \( f(3) = f(2) + \frac{3}{8} = \frac{3}{4} + \frac{3}{8} = \frac{6}{8} + \frac{3}{8} = \frac{9}{8} \) (not matching the sequence's \( 1 \))

At this point, it appears that the function might not accurately model the entire sequence continuously. However, perhaps my interpretation of the entire sequence structure is missing crucial segments between iterations or additional terms not detailed earlier.

Summary

The sequence can't be completely characterized just by the approach \( f(n + 1) = f(n) + \frac{3}{8} \) if it fluctuates in value as you described. If we're working with only the initial terms stated:

  • If it can specifically adjust with \( f(n) = f(n-1) + \frac{3}{8} \) used intermittently, the recursion might work for specific increments but unfortunately, it doesn't carry the entire structure outlined.

Final Clarification

To confirm, if you believe that the sequence structure does support the recursion, could you provide additional terms or clarifications on the pattern? It would greatly help confirm how the relation holds across those numerics distinctly!