Step by step
The question
Question: d varies directly with t.
d = 120 when t = 2.
Find d when t = 5.
1 Write the rule
Varies directly means d is always the same number times t.
When t gets bigger, d gets bigger too.
We call the number k: d = kt.
k stays the same for every pair of numbers.
Books call it the constant of variation.
2 Find k
We put in d = 120 and t = 2.
k times 2 is written 2k.
2k means 2 times k.
We want just k.
To remove times 2, we divide by 2.
We do it on both sides, so they stay equal.
Careful!
k is d divided by t: 120/2.
Not 2/120.
The d number goes on top.
120/2 on the left.
2k/2 on the right.
2 divided by 2 is 1.
So just k is left.
120 divided by 2 is 60.
So k = 60.
3 Write the equation
We put 60 in place of k.
So the equation is d = 60t.
4 Find d
We put in t = 5.
60 times 5 is 300.
So d = 300.
5 Check
Let's check.
In direct variation, d divided by t is always k.
First pair: 120/2 is 60.
New pair: 300/5 is 60 too.
The same k.
So the answer is right!
Let's look back.
First: write the rule.
Then: find k.
Then: write the equation.
Then: find d.
Last: check.