Question

is the plant growth shown in the table below proportional? explain why or why not.

days	plant height (mm)
1	\\(\frac{2}{3}\\)
2	\\(\frac{4}{3}\\)

Explanation:

Step1: Recall proportionality rule

For two quantities (days \(x\) and plant height \(y\)) to be proportional, the ratio \(\frac{y}{x}\) must be constant for all pairs of values.

Step2: Calculate ratio for first pair

For \(x = \frac{1}{2}\) and \(y=\frac{1}{3}\), the ratio \(\frac{y}{x}=\frac{\frac{1}{3}}{\frac{1}{2}}=\frac{1}{3}\times\frac{2}{1}=\frac{2}{3}\).

Step3: Calculate ratio for second pair

For \(x = 1\) and \(y=\frac{2}{3}\), the ratio \(\frac{y}{x}=\frac{\frac{2}{3}}{1}=\frac{2}{3}\).

Step4: Calculate ratio for third pair

For \(x = 2\) and \(y=\frac{4}{3}\), the ratio \(\frac{y}{x}=\frac{\frac{4}{3}}{2}=\frac{4}{3}\times\frac{1}{2}=\frac{2}{3}\).

Answer:

Yes, the plant growth is proportional. Because the ratio of plant height to days (\(\frac{\text{Plant Height}}{\text{Days}}\)) is constant (\(\frac{2}{3}\)) for all given pairs of days and plant height.