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I think the formula is messing people up on this. I have found that using words, rather than variable names, to be helpful when writing out the problem.

So we have two kangaroos starting at different locations, and jumping forward at different distances. If we want to know where any kangaroo is at any given time, there is an intuitive equation for that:

such that K = Kangaroo Position, y = Number of Jumps, v = Distance Per Jump, and x = Starting Position.

That almost looks like an equation your teacher went over in algebra that one time you were dozing off: y = mx + b. I know we are talking about kangaroos here, but in the background we are really just checking to see when two lines intersect.

If we have two kangaroos and we want to know when (or if) they will intersect, given their Starting Position and Distance per Jump, the only thing left to solve for is Number of Jumps.

The kangaroos crossing paths essentially means that Kangaroo Position is equal for both kangaroos. Remember the equation up above K = yv + x? Now that we have two kangaroos, we need to have 2 different equations, and need to determine which value of y (Number of Jumps) can be plugged in to make them equal. So now we have something like this:

(y*v1)+x1=(y*v2)+x2

We need to get y on one side of the equation, so we will begin reducing it down.

(x1-x2)=(y*v2)-(y*v1)

(x1-x2)=y(v2-v1)

(x1-x2)/(v2-v1)=y

Luckily, the problem statement gives us the Starting Position and Distance per Jump (x1, x2, v1, and v2) for each kangaroo. When we plug in these numbers it will tell us how many jumps it would take for the kangaroos to end up in the same spot. But not so fast! We can do a a little work up front to check if the kangaroo that is starting in front is moving faster than the kangaroo in the rear i.e we need to see if Distance Per Jump for the kangaroo in front is larger than the one in the rear. If so, then the one in the back will never catch up. Before we even attempt find an intersection we need to ensure that v2 < v1 is true. If this evalutes to false then we are done and the lines will not intersect at any point in the future. If the kangaroos started going the other direction then that would be a different story.

Anyway, so we plugged the numbers in and we are ready to see how many jumps it will take. At this point there are two scenarios that will occur:

you got a whole number greater than zero

you got a fractional number greater than zero

In scenario one, this means that after y jumps, the kangaroos will be in the same spot.

In scenario two, the kangaroos will intersect, but they will be in the air. Kinda cool, but not what we wanted.

Now we get to the part that seems to mess with peoples heads: the dreaded % operator. Keep reading, and you will see the solution to this problem.

SPOILER ALERT

The code below is how we validate that the point of intersection is a whole number.

(x1-x2)%(v2-v1)==0

The % operator returns the remainder of dividing two numbers. Lets look at an example:

x1=0x2=4v1=3v2=2(0-4)/(2-3)=>(-4/-1)=>4=y

There is no remainder here. So (-4 % -1) == 0 and the kangaroos will intersect after 4 jumps.

So to put it all together, we need to check the Distance per Jump of the kangaroo in front is less than the one in the rear, and that the equation above gives us a positive integer.

## Number Line Jumps

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I think the formula is messing people up on this. I have found that using words, rather than variable names, to be helpful when writing out the problem.

So we have two kangaroos starting at different locations, and jumping forward at different distances. If we want to know where any kangaroo is at any given time, there is an intuitive equation for that:

We could also write this as:

such that K =

Kangaroo Position, y =Number of Jumps, v =Distance Per Jump, and x =Starting Position.That almost looks like an equation your teacher went over in algebra that one time you were dozing off:

`y = mx + b`

. I know we are talking about kangaroos here, but in the background we are really just checking to see when two lines intersect.If we have two kangaroos and we want to know when (or if) they will intersect, given their

Starting PositionandDistance per Jump, the only thing left to solve for isNumber of Jumps.The kangaroos crossing paths essentially means that

Kangaroo Positionis equal for both kangaroos. Remember the equation up above`K = yv + x`

? Now that we have two kangaroos, we need to have 2 different equations, and need to determine which value of y (Number of Jumps) can be plugged in to make them equal. So now we have something like this:We need to get

`y`

on one side of the equation, so we will begin reducing it down.Luckily, the problem statement gives us the

Starting PositionandDistance per Jump(`x1`

,`x2`

,`v1`

, and`v2`

) for each kangaroo. When we plug in these numbers it will tell us how many jumps it would take for the kangaroos to end up in the same spot. But not so fast! We can do a a little work up front to check if the kangaroo that is starting in front is moving faster than the kangaroo in the rear i.e we need to see ifDistance Per Jumpfor the kangaroo in front is larger than the one in the rear. If so, then the one in the back will never catch up. Before we even attempt find an intersection we need to ensure that`v2 < v1`

is`true`

. If this evalutes to`false`

then we are done and the lines will not intersect at any point in the future. If the kangaroos started going the other direction then that would be a different story.Anyway, so we plugged the numbers in and we are ready to see how many jumps it will take. At this point there are two scenarios that will occur:

In scenario one, this means that after

`y`

jumps, the kangaroos will be in the same spot.In scenario two, the kangaroos will intersect, but they will be in the air. Kinda cool, but not what we wanted.

Now we get to the part that seems to mess with peoples heads: the dreaded

`%`

operator. Keep reading, and you will see the solution to this problem.## SPOILER ALERT

The code below is how we validate that the point of intersection is a whole number.

The

`%`

operator returns the remainder of dividing two numbers. Lets look at an example:There is no remainder here. So

`(-4 % -1) == 0`

and the kangaroos will intersect after 4 jumps.So to put it all together, we need to check the

Distance per Jumpof the kangaroo in front is less than the one in the rear, and that the equation above gives us a positive integer.