Proportion, percentage increase and decrease

The original value is 50 cm and the new value is 60 cm. We need to find the percentage increase.

$\text{Increase}=60-50=10\text{ cm}$

$\text{Percentage increase}=\frac{\text{increase}}{\text{original}}\times 100=\frac{10}{50}\times 100=20\%$

The plant's height increased by 20%. Note that we always divide by the original value (50), not the new value.

We need the percentage increase from 50 cm to 60 cm. The increase is $60-50=10$ cm, and we divide by the original value of 50.

10% would come from $\frac{10}{100}\times 100$, using the wrong denominator of 100. This is incorrect.

16.7% comes from $\frac{10}{60}\times 100$, which uses the new value as the denominator instead of the original. This is the most common mistake.

25% would require an increase of 12.5 cm from 50 cm, giving a new value of 62.5 cm. That's not what happened here.

20% is correct: $\frac{10}{50}\times 100=20\%$, using the original value of 50 as the denominator.

Two quantities are directly proportional if $y=kx$, where $k$ is the constant of proportionality. When $x=0$, $y=0$.

Because the equation is $y=kx$, this is a linear (straight line) equation. Since it has no added constant, the line must pass through the origin (0, 0).

A straight line passing through the origin is the only graph that correctly represents a directly proportional relationship.

Direct proportion means $y=kx$. This means two things: the graph must be straight (linear) and it must pass through (0, 0).

A curved line represents a non-linear relationship (like $y=k{x}^2$), not direct proportion. Eliminate this option.

A horizontal line means $y$ stays constant regardless of $x$, so the quantities are not related proportionally at all.

A straight line not through the origin would have the form $y=kx+c$ (where $c\ne 0$). This is a linear relationship but not a directly proportional one.

A straight line passing through the origin satisfies both conditions of direct proportion: it is linear and $y=0$ when $x=0$.