Step by step
The question
Question: y varies inversely with x.
y = 12 when x = 3.
Find y when x = 9.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Write the rule
Varies inversely means y is a number divided by x.
When x gets bigger, y gets smaller.
We call the number k: y = k over x.
k stays the same for every pair of numbers.
Books call it the constant of variation.
2 Find k
We put in y = 12 and x = 3.
k/3 means k divided by 3.
We want just k.
To remove divided by 3, we multiply by 3.
We do it on both sides, so they stay equal.
Divided by 3, then times 3: they cancel.
Cross them out.
12 times 3 is 36.
So k = 36.
3 Write the equation
We put 36 in place of k.
So the equation is y = 36 over x.
4 Find y
We put in x = 9.
Careful!
Inverse is not direct.
x got bigger, from 3 to 9.
So y must get smaller.
The right way: y = 36/9.
36 divided by 9 is 4.
So y = 4.
5 Check
Let's check.
In inverse variation, x times y is always k.
That is how we found k.
First pair: 3 times 12 is 36.
New pair: 9 times 4 is 36 too.
The same k.
So the answer is right!
6 Pick the choice
Our answer is y = 4.
That is choice A.
7 Why not the others
B is 18.
Wrong: 12 + 6: added the change in x.
C is 36.
Wrong: 12 × 3: that is direct variation; inverse means y goes down.
That is the Careful mistake from before.
D is 108.
Wrong: 12 × 9: multiplied by the new x.
Let's look back.
First: write the rule.
Then: find k.
Then: write the equation.
Then: find y.
Then: check.
Then: pick the choice.
Last: why not the others.