Question

When they are both racing on hoverboards, Victoria is 3 times as fast as her brother Max. When she is on foot, she is 3 times slower than Max on his hoverboard. They took off on hoverboards at the same time, but after 12 minutes, Victoria’s hoverboard broke and she immediately started to run. If the race was a tie, how long, in minutes, did it last from start to finish?

Answer 1

The time will be 48 minutes

Explanation:

Let r be Max's rate of speed and t be the total time.

Using the equation d = rt for this situation

Victoria is 3 times as fast as Max so her part gives us the expression 3r(12)

For the remaining part of the race, Victoria is 3 times slower than Max. Lets represent it as 1/3r. Since we do not know the amount of time she travels this way so we will represent this as 1/3r(t-12)

Together the distance on hoverboard and the distance on foot can be represented by d=3r(12)+1/3r(t-12).

So we have

3r(12)+1/3r(t-12) = rt

Simplifying, we have

36r+1/3r(t)-1/3r(12) = rt

36r + 1/3rt - 12/3r = rt

36r + 1/3rt - 4r = rt

Combining all the like terms,

32r + 1/3rt = rt

Subtract 1/3rt from each side:

32r + 1/3rt - 1/3rt = rt - 1/3rt

32r = 2/3rt

Divide both sides by r we get

32r/r = (2/3rt)/r

32 = 2/3t

Divide both sides by 2/3 we get

32 ÷ 2/3 = 2/3t ÷ 2/3

32 ÷ 2/3 = t

32/1 × 3/2 = t

96/2 = t

48 = t

Answer 2

Race completed in 48 minutes.

Explanation:

Let the speed of Max on hoverboard is = x

Then as per question speed of Victoria on hoverboard = 3x

Now it has been given in the question that speed of Victoria on foot is 1/3 of the speed of Max on hoverboard that will be = x/3

Now we will form the equation.

As we know the formula speed = distance/time

Let the time taken by both to complete the race be t minutes.

Distance covered by Victoria in 12 minutes + Distance covered by Victoria on foot = distance covered by Max on hoverboard

Then the equation will be

12(3x)+(1/3)x(t-12)=xt

`36x+(xt)/(3)-4x=xt`

`x(36+(t)/(3)-4)=xt`

`36+(t)/(3)-4=t`

`32=t-(t)/(3)`

`(2t)/(3)=32`

`t=((32)3)/(2)=48`

So 48 minutes it took to complete the race.