Question

X takes 3 hrs more than Y to walk 30 km. But, if X doubles his pace, he is ahead of Y by 1.5 hrs. What is their speed of walking?

Answer 1

`v_x = 10/3`, `v_y = 5`

Explanation:

Assuming that both `X` and `Y` walk at constant speeds `v_x` and `v_y`, the time they need to walk `30`km is given by `t = 30/v`. The first sentence translates into

`30/v_x = 30/v_y+3 \qquad \qquad (1)`

In fact, the time it takes to `X` is `30/v_x`, and it is `3` more than the time it takes to `Y`, which is `30/v_y`.

As for the second sentence, if `X` doubles his speed he will pass from `v_x` to `2v_x`, and this time his time is `1.5` less than `Y`'s, thus translating into

`30/(2v_x) = 30/v_y-1.5 \qquad \qquad (2)`

We can solve the system by subtracting `(1)-(2)`, so that `Y`'s time simplifies: we have

`30/(v_x)-30/(2v_x) = (30/v_y+3)-(30/v_y-1.5)`

which can be written as

`cancel(30)^15/(cancel(2)v_x) = 15/v_x = 4.5`

solving for `v_x`, we have `v_x = 15/4.5 = 10/3`

Substitute this value for `v_x` in `(1)` to obtain `v_y`:

`30/(10/3) = 30/v_y+3 \iff 30*3/10 = 30/v_y+3 \iff 9=30/v_y+3`

which leads to `30/v_y = 6` and thus `v_y = 5`

Answer 2

`"v"_"X" = 10/3\ "kmph"` and `"v"_"Y" = "5 kmph"`

Explanation:

Speed (`"v"`), distance (`"d"`) and time (`"t"`) are related as

`"v" = "d"/"t"`

For first case

`"v"_"X" = 30/"t"_"X" color(white)(..)` and `color(white)(..) "v"_"Y" = 30/"t"_"Y"`

It’s given that `"t"_"X" = "t"_"Y" + 3`

So, `"v"_"X" = 30/("t"_"Y" + 3) color(white)(...)` ——(1)

For second case

“X doubles his pace” means X doubles his speed. So speed of X now is `"2v"_"X"`

“X is ahead of Y by 1.5 hrs” means now X takes 1.5 hrs less than Y to travel same distance.
So, time tiken by X to travel 30 km is now `"t"_"Y" - 1.5`

`"2v"_"X" = 30/("t"_"Y" - 1.5)`

Substitute `"v"_"X"` from first equation

`2 × cancel(30)/("t"_"Y" + 3) = cancel(30)/("t"_"Y" - 1.5)`

`2/("t"_"Y" + 3) = 1/("t"_"Y" - 1.5)`

`"2t"_"Y" - 3.0 = "t"_"Y" + 3`

`color(blue)("t"_"Y" = 6)`

• `"v"_"X" = 30/"t"_"X" = 30/("t"_"Y" + 3) = 30/(6 + 3) = 30/9 = 10/3`

• `"v"_"Y" = 30/"t"_"Y" = 30/6 = 5`