Direct & Inverse Proportion Flashcards

All 13 cards in this deck

What equation do you write when $y$ is directly proportional to $x$?

$y = kx$, where $k$ is the constant of proportionality.

What are the steps to answer a question on proportion to a root?

e.g. $y \propto \sqrt{x}$, $y = 10$ when $x = 4$; find $y$ when $x = 25$

1. Write the equation: $y = k\sqrt{x}$
2. Substitute the known pair to find $k$: $10 = k \times 2$, so $k = 5$
3. Substitute the new value: $y = 5 \times 5 = 25$

On a graph of $y$ against $x$ showing direct proportion, what shape is the graph and what does its gradient give?

A straight line through the origin; the gradient is the constant of proportionality $k$.
e.g. a line through $(4,5)$ gives $k = 1.25$.

True or false? If $y$ goes up by the same amount every time $x$ goes up by 2, then $y$ must be directly proportional to $x$.

False. $y \div x$ must be the same for every pair (equivalently, the graph must pass through the origin).

What are the steps of the unitary method for a direct proportion problem?

e.g. 4 identical boxes of seed cover $60$ m$^2$; what area do 7 boxes cover?

1. Divide to find the amount for one: $60 \div 4 = 15$ m$^2$ per box
2. Multiply by the number wanted: $15 \times 7 = 105$ m$^2$

What equation do you write when $y$ is inversely proportional to $x$?

$y = \dfrac{k}{x}$, where $k$ is the constant of proportionality.

True or false? "$T$ is inversely proportional to $r$" means the same as "$T$ is proportional to $\frac{1}{r}$".

True. Both give $T = \dfrac{k}{r}$.

What are the steps to find $x$ from an inverse square proportion?

e.g. $y$ is inversely proportional to $x^2$, $y = 9$ when $x = 1$; find the positive value of $x$ when $y = 4$

1. Write the equation: $y = \dfrac{k}{x^2}$
2. Substitute to find $k$: $k = 9 \times 1^2 = 9$
3. Substitute $y = 4$ and take the positive root: $x^2 = \frac{9}{4}$, so $x = 1.5$

What does the graph of $y$ against $x$ look like when $y$ is inversely proportional to $x$ (for $x > 0$)?

A smooth curve falling from left to right, getting closer to but never touching either axis.

What are the steps to solve a 'workers and time' inverse proportion problem?

e.g. 5 identical pumps fill a tank in 14 hours; how long do 4 pumps take?

1. Multiply to find the total work: $5 \times 14 = 70$ pump-hours
2. Divide by the new number: $70 \div 4 = 17.5$ hours

$y$ is inversely proportional to the square of $x$. What happens to $y$ when $x$ is multiplied by 3?

$y$ is divided by 9 (multiplied by $\frac{1}{9}$).

How do you write "$X$ is inversely proportional to $Y$" using the proportionality symbol $\propto$?

e.g. the time $T$ taken is inversely proportional to the number of workers $n$

$X \propto \dfrac{1}{Y}$

e.g. $T \propto \dfrac{1}{n}$

True or false? If $y$ is inversely proportional to $x$, then plotting $y$ against $\frac{1}{x}$ gives a straight line through the origin.

True. Inverse proportion to $x$ is direct proportion to $\frac{1}{x}$, so $y = k \times \frac{1}{x}$ has gradient $k$ through the origin.