Question

Josh and Karen live 20 miles apart and at exactly 2 pm, both decide to ride their bikes towards each other's houses so that they can meet on the path somewhere in between. Josh rides at an average speed of 15 miles per hour and Karen rides at an average speed of 10 miles per hour. If Josh takes a 10 minute break after he starts to look at some birds while Karen rides the whole time, how many miles from Karen's house will they meet?

Answer 1

They will meet at a distance 9 miles from Karen's house.

Explanation:

Let the distance covered by Josh be
`S_(1)` and the distance covered by Karen be
`S_(2)`

The distance between Josh and Karen's houses is 20 miles. That is, the total distance both of them will cover is 20 miles.

Hence,
`S_(1) + S_(2) = 20 miles`

From the question, if Josh takes a 10 minute break after he starts while Karen rides the whole time, this means

If Karen rides for a total of
`t` mins, then Josh would spend
`t-10` mins for the journey

Also, Josh rides at an average speed of 15 miles per hour

and Karen rides at an average speed of 10 miles per hour

Now, for Josh

Distance =
`S_(1)`

Speed = 15 mile/hour (Covert to mile/minute); (NOTE: 1 hour = 60 mins)

Speed= 0.25 mile/min

Time spent =
`t - 10` mins

From

`Speed =(Distance)/(Time)`

Distance = Speed × Time

Distance = 0.25 mile/min × (t-10) min

`S_(1)` = 0.25(t-10) mile

`S_(1)` = 0.25t - 2.5

t = (
`S_(1)` + 2.5) / 0.25 ....... (1)

For Karen,

Distance =
`S_(2)`

Speed = 10 mile/hour (Covert to mile/minute);

Speed=
`(1)/(6)` mile/min

Time spent = t

`Speed =(Distance)/(Time)`

Distance = Speed × Time

Then,
`S_(2)` =
`(1)/(6)` mile/min × t min

`S_(2)` =
`(1)/(6)t` mile

t = 6
`S_(2)` ......... (2)

Equating equations (1) and (2), we get

(
`S_(1)` + 2.5) / 0.25 = 6
`S_(2)`

Then,

(
`S_(1)` + 2.5) = 1.5
`S_(2)`

Recall that

`S_(1) + S_(2) = 20 miles`

Then,
`S_(1) = 20 - S_(2)`

∴
`20 - S_(2)` + 2.5 = 1.5
`S_(2)`

2.5
`S_(2)` = 22.5

`S_(2)` = 22.5 / 2.5

`S_(2)` = 9 miles

This means Karen covers a total distance of 9 miles

Hence, they will meet at a distance 9 miles from Karen's house.