Identify Constant of Proportionality - 7th Grade Math

1. What is the constant of proportionality $k$ on a graph of $y$ versus $x$ for a proportional relationship?

   $k$ is the slope: $k = \frac{\Delta y}{\Delta x}$. The slope represents the constant of proportionality in graphs.

2. What is the unit rate $k$ if $12$ apples cost $18 and cost is proportional to number of apples?

   $k = \$1.50$ per apple. Unit price = $\frac{\text{total cost}}{\text{quantity}} = \frac{18}{12} = 1.50$.

3. What is the constant of proportionality $k$ in the equation $y = -2x$?

   $k = -2$. In $y = kx$, $k$ is the coefficient multiplying $x$.

4. What is the constant of proportionality $k$ if $y$ is proportional to $x$ and $y$ increases by $8$ when $x$ increases by $2$?

   $k = \frac{8}{2} = 4$. Rate of change = $\frac{\Delta y}{\Delta x} = \frac{8}{2} = 4$.

5. What is the constant of proportionality $k$ for a line through $(0, 0)$ and $(10, 4)$ in $y = kx$?

   $k = \frac{4}{10} = \frac{2}{5}$. Calculate $k = \frac{y}{x} = \frac{4}{10} = \frac{2}{5}$.

6. What must be true about the point $(0,0)$ on the graph if a relationship is proportional?

   The graph must pass through $(0,0)$

7. What is the constant of proportionality $k$ in the equation $y = kx$ if $y$ is $2.4$ when $x$ is $3$?

   $k = \frac{2.4}{3} = 0.8$. Substitute the given $x$ and $y$ values to solve for $k$.

8. What is the unit rate (constant of proportionality) if you travel $180$ miles in $3$ hours?

   $k = \frac{180}{3} = 60$ miles per hour. Divide total distance by time to find speed (miles per hour).

9. What is the constant of proportionality $k$ in the table where $x = 2$ gives $y = 11$ and $y = kx$?

   $k = \frac{11}{2}$. Divide $y$ by $x$ to find $k$: $k = \frac{11}{2}$.

10. On a graph of a proportional relationship $y = kx$, what does the constant $k$ represent?

    $k$ is the slope (rise per $1$ unit run)

11. What is the formula for the constant of proportionality in $y = kx$ using any nonzero point $(x,y)$?

    $k = \frac{y}{x}$

12. What is the constant of proportionality $k$ in the equation $y = \frac{3}{5}x$?

    $k = \frac{3}{5}$. The coefficient of $x$ in $y = kx$ form is the constant of proportionality.

13. What is the constant of proportionality $k$ in the equation $y = 4x$?

    $k = 4$. In $y = kx$ form, the coefficient of $x$ is the constant of proportionality.

14. Identify $k$ for the proportional relationship $y$ varies directly with $x$ and $y = 18$ when $x = 6$.

    $k = 3$. Find $k$ by dividing: $k = \frac{y}{x} = \frac{18}{6} = 3$.

15. Identify the constant of proportionality $k$ if the graph includes the point $(5, -15)$ and passes through $(0, 0)$.

    $k = \frac{-15}{5} = -3$. The slope of a line through $(0,0)$ equals $\frac{y}{x}$ for any point.

16. Which expression gives the constant of proportionality for a proportional table of $x$ and $y$ values?

    $k = \frac{y}{x}$. For proportional relationships, $k$ equals $y$ divided by $x$.

17. Identify the unit rate $k$ if a recipe uses $\frac{3}{4}$ cup of sugar for $2$ batches.

    $k = \frac{\frac{3}{4}}{2} = \frac{3}{8}$ cup per batch. Divide amount of sugar by number of batches for sugar per batch.

18. What is the constant of proportionality $k$ in the equation $y = 0.75x$?

    $k = 0.75$. The coefficient of $x$ in the equation $y = kx$ is the constant $k$.

19. What is the constant of proportionality $k$ in the equation $y = 0.6x$?

    $k = 0.6$. The constant $k$ is the coefficient of $x$ in the equation.

20. Identify $k$ for the proportional relationship $y = kx$ when $x = 6$ gives $y = 15$.

    $k = \frac{15}{6} = 2.5$. Substitute values into $k = \frac{y}{x}$ to find the constant.

21. Identify $k$ in a proportional table where the pair is $(x, y) = (5, -20)$ and $y = kx$.

    $k = -4$. Calculate $k = \frac{y}{x} = \frac{-20}{5} = -4$.

22. Identify $k$ if $45 buys $6$ tickets and total cost is proportional to number of tickets.

    $k = \$7.50$ per ticket. Price per ticket = $\frac{45}{6} = 7.50$ dollars.

23. What is the constant of proportionality $k$ in the equation $y = 5x$?

    $k = 5$. In $y = kx$, the coefficient of $x$ is the constant of proportionality.

24. Identify $k$ if $y$ is proportional to $x$ and the point $(4, 14)$ is on the graph of $y = kx$.

    $k = \frac{14}{4} = \frac{7}{2}$. Substitute the point into $y = kx$: $14 = k(4)$, so $k = \frac{14}{4}$.

25. What is the constant of proportionality $k$ for the table where $x=4$ gives $y=10$?

    $k = \frac{10}{4} = 2.5$. Calculate $k = \frac{y}{x}$ using any corresponding pair from the table.

26. Identify the unit rate $k$ if $\frac{5}{6}$ pound of cheese costs $4.

    $k = \frac{4}{\frac{5}{6}} = \frac{24}{5} = 4.8$ dollars per pound. Divide price by weight to find dollars per pound.

27. Identify the constant of proportionality $k$ if the graph includes the point $(3, 12)$ and passes through $(0, 0)$.

    $k = \frac{12}{3} = 4$. For a line through origin, $k$ equals slope $= \frac{y}{x}$ at any point.

28. What is the constant of proportionality $k$ for a graph of $y = kx$ that passes through $(0, 0)$ and $(3, 9)$?

    $k = 3$. Use the non-origin point: $k = \frac{9}{3} = 3$.

29. Which value is the constant of proportionality for $y = kx$ if the relationship includes the point $(7, -28)$?

    $k = -4$. Solve for $k$: $-28 = k(7)$, so $k = \frac{-28}{7} = -4$.

30. What is the constant of proportionality $k$ in the equation $y = kx$ if $y$ is $9$ when $x$ is $1$?

    $k = 9$. Substitute given values into $y = kx$ and solve for $k$.

31. What is the constant of proportionality $k$ for the table where $x=1$ gives $y=7$?

    $k = 7$. For proportional relationships, $k = \frac{y}{x}$ for any pair.

32. Identify the constant of proportionality $k$ in this table: $(x,y)=(1,7),(2,14),(3,21)$.

    $k = 7$. Each $y$-value divided by its $x$-value equals $7$.

33. What is the constant of proportionality $k$ in the equation $y = -2x$?

    $k = -2$. In $y = kx$, the coefficient of $x$ (including sign) is $k$.

34. Identify the unit rate $k$ if $x = 9$ corresponds to $y = 12$ in a proportional relationship $y = kx$.

    $k = \frac{12}{9} = \frac{4}{3}$. The unit rate is $k = \frac{y}{x} = \frac{12}{9} = \frac{4}{3}$.

35. Identify $k$ for the proportional relationship $y = kx$ when $x = 8$ gives $y = -12$.

    $k = \frac{-12}{8} = -1.5$. Use $k = \frac{y}{x}$ with the given values to find the constant.

36. What is the unit rate (constant of proportionality) if $5$ notebooks cost $12.50?

    $k = \frac{12.50}{5} = 2.50$ dollars per notebook. Divide total cost by number of items to find cost per item.

37. What is the constant of proportionality $k$ in the equation $y = \frac{x}{8}$?

    $k = \frac{1}{8}$. Rewrite as $y = \frac{1}{8}x$ to identify $k = \frac{1}{8}$.

38. Identify $k$ in this table: $(x,y)=(2,3),(6,9),(10,15)$ for $y = kx$.

    $k = \frac{3}{2}$. For each pair, $\frac{y}{x} = \frac{3}{2}$.

39. What is the constant of proportionality $k$ for the table where $x=2$ gives $y=-9$?

    $k = \frac{-9}{2} = -4.5$. Find $k$ by dividing $y$ by $x$ for any table entry.

40. What is the unit rate (constant of proportionality) if a car travels $180$ miles in $3$ hours at a constant speed?

    $k = 60$ miles per hour. Unit rate = $\frac{\text{distance}}{\text{time}} = \frac{180}{3} = 60$ mph.

41. What is the constant of proportionality $k$ in the equation $y = \frac{3}{4}x$?

    $k = \frac{3}{4}$. The coefficient of $x$ in $y = kx$ form is the constant $k$.

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