In the equation $$P = 150t$$, the constant of proportionality is 150, representing the growth rate of 150 bacteria per hour. The initial population of 50 bacteria is separate information that doesn't change the proportional growth rate shown in the equation. Choice A uses only the initial population. Choice B subtracts 150 - 50. Choice D adds 150 + 50.

When you encounter a problem about constant rates, you're working with proportional relationships where the rate of change stays the same throughout. The key is finding the rate per unit of time.

To find the constant rate in gallons per minute, you need to calculate the total gallons divided by the total time. In the first 4 minutes, 18 gallons flow in. In the next 6 minutes, 27 gallons flow in. This gives you a total of $$18 + 27 = 45$$ gallons over a total time of $$4 + 6 = 10$$ minutes.

The constant rate is $$\frac{45 \text{ gallons}}{10 \text{ minutes}} = 4.5$$ gallons per minute. You can verify this works: in 4 minutes at 4.5 gallons/minute, you get $$4 \times 4.5 = 18$$ gallons. In 6 minutes, you get $$6 \times 4.5 = 27$$ gallons. Perfect!

Answer choice A ($$6.0$$ gallons per minute) likely comes from incorrectly dividing 27 by 4 or 18 by 3, mixing up the time periods. Answer choice B ($$4.0$$ gallons per minute) might result from rounding 4.5 down or dividing 24 by 6 through some calculation error. Answer choice C ($$5.5$$ gallons per minute) could come from finding the average of individual rates (4.5 and 6.75) rather than using total gallons over total time.

Remember: for constant rate problems, always use total amount divided by total time. Don't average separate rates or use individual time periods incorrectly.

In the equation $$d = 8.5h$$, the constant of proportionality is the coefficient 8.5, representing $8.50 earned per hour. The $12 transportation cost is additional information that doesn't affect the original proportional relationship. Choice B uses only the transportation cost. Choice C incorrectly adds 8.5 + 12. Choice D multiplies 8.5 × 12 - 6.

The constant of proportionality is the ratio of white paint to blue paint: $$\frac{15}{6} = \frac{5}{2} = 2.5$$ ounces of white paint per ounce of blue paint. Choice A gives the ratio of blue to white (reciprocal). Choice C adds the quantities instead of finding their ratio. Choice D presents the unreduced fraction in the wrong order (blue to white).

When you encounter problems about proportional relationships, you're looking for a constant rate that describes how one quantity changes with respect to another. Here, you need to find how many inches the spring stretches per pound of weight.

The problem tells you that for every 3 pounds added, the spring stretches 2.5 inches. To find the constant of proportionality (inches per pound), you need to create a ratio and simplify it:

$$\frac{2.5 \text{ inches}}{3 \text{ pounds}} = \frac{2.5}{3}$$

Convert 2.5 to a fraction: $$2.5 = \frac{5}{2}$$

So you have: $$\frac{\frac{5}{2}}{3} = \frac{5}{2} \times \frac{1}{3} = \frac{5}{6}$$ inches per pound

This matches answer choice D.

Let's examine why the other answers are wrong: Choice A gives $$\frac{3}{2}$$, which incorrectly puts pounds in the numerator instead of inches. Choice B gives $$\frac{6}{5}$$, which appears to flip the correct fraction. Choice C gives $$\frac{2}{3}$$, which seems to use 2 instead of 2.5 in the calculation, possibly from converting 2.5 incorrectly or misreading the problem.

Study tip: For proportional relationships, always set up your ratio with the units you want in the answer. If you need "inches per pound," put inches in the numerator and pounds in the denominator. Then simplify the fraction completely. Double-check by asking: "Does this unit rate make sense given the original relationship?"

This question tests identifying the constant of proportionality k (unit rate, slope) from tables (y/x ratio), graphs (slope or (1,k) point), equations (coefficient of x), or verbal descriptions. A proportional relationship y=kx has constant k equal to: (1) ratio y/x for any point (14/2=7, 28/4=7, k=7), (2) slope of graph (rise/run through origin), (3) coefficient of x in equation (y=7x → k=7), (4) unit rate stated ("7 dollars per item" → k=7); the point (1,k) is special because when x=1, y=k (so a graph passing through (1,7) has k=7 directly readable—unit rate at one unit of x). For example, a table showing x:2,4,6 y:14,28,42 allows calculating ratios 14/2=7, 28/4=7, 42/6=7 (all equal k=7), or a graph through (0,0) and (1,7) has k=7 from the point or slope=7/1=7, or equation y=7x shows k=7 directly, or "costs $7 per item" states k=7. Here, the verbal description states 3 cups of flour per batch, so k=3 cups per batch in y=3x. Common errors include inverting to 1/3, using multiples like 6 or 9, or misinterpreting "per batch" as x instead of the rate. From verbal descriptions, the stated rate is k ("3 meters per second" → k=3 m/s); to find k from a table, pick any (x,y) pair, calculate k=y/x, verify with others. Special point (1,k): proportional graphs pass through (1,k) where k is constant—makes k directly readable; not proportional if y-intercept ≠0, no k value.

This question tests identifying the constant of proportionality k (unit rate, slope) from tables (y/x ratio), graphs (slope or (1,k) point), equations (coefficient of x), or verbal descriptions. A proportional relationship y=kx has constant k equal to: (1) ratio y/x for any point (14/2=7, 28/4=7, k=7), (2) slope of graph (rise/run through origin), (3) coefficient of x in equation (y=7x → k=7), (4) unit rate stated ("7 dollars per item" → k=7); the point (1,k) is special because when x=1, y=k (so a graph passing through (1,7) has k=7 directly readable—unit rate at one unit of x). For example, a table showing x:2,4,6 y:14,28,42 allows calculating ratios 14/2=7, 28/4=7, 42/6=7 (all equal k=7), or a graph through (0,0) and (1,7) has k=7 from the point or slope=7/1=7, or equation y=7x shows k=7 directly, or "costs $7 per item" states k=7. Here, the graph passes through the origin and (1,3), so k=3 directly from the y-coordinate at x=1, or slope=3/1=3. Common errors include using the inverse like 1/3, confusing with other points, or thinking the slope is 4 if misreading. To find k from a graph, use slope=rise/run through the origin, or read y at x=1 giving (1,k) where k is that y-value; from an equation y=kx, k is the coefficient. Proportional graphs pass through (1,k) where k is the constant—making k directly readable (no calculation needed, just read the y-coordinate at x=1); mistakes include calculating ratios wrong (x/y not y/x) or reading the graph at the wrong point.

This question tests identifying the constant of proportionality k (unit rate, slope) from tables (y/x ratio), graphs (slope or (1,k) point), equations (coefficient of x), or verbal descriptions. A proportional relationship y=kx has constant k equal to: (1) ratio y/x for any point (14/2=7, 28/4=7, k=7), (2) slope of graph (rise/run through origin), (3) coefficient of x in equation (y=7x → k=7), (4) unit rate stated ("7 dollars per item" → k=7); the point (1,k) is special because when x=1, y=k (so a graph passing through (1,7) has k=7 directly readable—unit rate at one unit of x). For example, a table showing x:2,4,6 y:14,28,42 allows calculating ratios 14/2=7, 28/4=7, 42/6=7 (all equal k=7), or a graph through (0,0) and (1,7) has k=7 from the point or slope=7/1=7, or equation y=7x shows k=7 directly, or "costs $7 per item" states k=7. Relationship 1 has k=6 from y=6x; Relationship 2 has k=4 from table ratios (4/1=4, 8/2=4, 12/3=4), so Relationship 1 has greater k since 6>4. Common errors include miscalculating table ratios (e.g., x/y=1/4), thinking they are equal, or inverting. To find k, from equation it's the coefficient, from table calculate y/x and verify; special point (1,k) for graphs. Mistakes: assuming not enough info when data is given, or confusing with non-proportional cases.

This skill tests identifying the constant of proportionality k (unit rate, slope) from tables (y/x ratio), graphs (slope or (1,k) point), equations (coefficient of x), or verbal descriptions. Proportional relationship $y=kx$ has constant k equal to: (1) ratio y/x for any point ($14/2=7$, $28/4=7$, $k=7$), (2) slope of graph ($rac{\text{rise}}{\text{run}}$ through origin), (3) coefficient of x in equation ($y=7x \to k=7$), (4) unit rate stated ("7 dollars per item" $\to k=7$). Point (1,k) special: when x=1, y=k (so graph passing through (1,7) has k=7 directly readable—unit rate at one unit of x). In this verbal description, the car travels 65 miles per gallon, so with y miles and x gallons, k=65 as the stated unit rate in $y=65x$. Common mistakes include inverting to 1/65 (A), using decimals like 0.65 (C) or multiplying unnecessarily to 650 (D). Finding k: from table (pick any (x,y) pair, calculate k=y/x, verify with other pairs—should all equal), from graph (slope=\frac{\text{rise}}{\text{run}}$ through origin, or read y at x=1 giving (1,k), k is that y-value), from equation y=kx (k is coefficient: $y=7x \to k=7$), from verbal (stated rate is k: "3 meters per second" $\to k=3$ m/s). Not proportional: if y-intercept≠0 (line misses origin), no constant of proportionality exists ($y=mx+b$ with b≠0 is linear but not proportional, no k value).

This question tests identifying the constant of proportionality k (unit rate, slope) from tables (y/x ratio), graphs (slope or (1,k) point), equations (coefficient of x), or verbal descriptions. Proportional relationship y=kx has constant k equal to: (1) ratio y/x for any point (14/2=7, 28/4=7, k=7), (2) slope of graph (rise/run through origin), (3) coefficient of x in equation (y=7x → k=7), (4) unit rate stated ("7 dollars per item" → k=7). Point (1,k) special: when x=1, y=k (so graph passing through (1,7) has k=7 directly readable—unit rate at one unit of x). For example, table showing x:2,4,6 y:14,28,42, calculate ratios 14/2=7, 28/4=7, 42/6=7 (all equal k=7), or graph through (0,0) and (1,7) has k=7 from point or slope=7/1=7, or equation y=7x shows k=7 directly, or "costs $7 per item" states k=7. In the equation y=4.5x, k is the coefficient 4.5. Errors include mistaking it for 1/4.5 (inverting) or confusing with non-proportional forms like y=4.5x+something. From an equation y=kx, k is simply the coefficient of x; if there's a y-intercept (b≠0), it's not proportional and has no k.