Alan and Sumei are 9km apart on a straight road and they are walking uniformly towards each other.If they start walking at the same time,they will meet in 1 hour 30 minutes.If Sumei starts walking 30 minutes after Alan,they will meet after Sumei has walked 1 hour 12 minutes.Find the speed of Alan and the speed of Sumei in km/h.

Please try using the substitution or elimination method as I am currently learning about it.Thanks

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Let the speed of Alan be ‘a’ km/hour and that of Sumei, ‘s’ km/hour.

When they approach towards each other, the relative speed will be (a+s).

They will meet after the combined distance travelled by them becomes equal to the total distance (i.e. 9 km).

So, both of them beginning at the same time, the time taken to cover 9 km will be `(9)/(a+s)`

By the first condition, `(9)/(a+s) = 3/2` ----(i)

or, `(a+s)=18/3 = 6`

Again, if Sumei starts 1/2 an hour after Alan, distance covered by Alan before Sumei starts is thus, a/2. So, remaining distance = `9-a/2 = (18-a)/(2)`

This time too, the relative speed will be the same, i.e. (a+s) km/hr. So, time taken to cover the remaining distance `= (18-a)/(2(a+s))` By the second condition,

`(18-a)/(2(a+s)) = 6/5` ----(ii)

Substituting the value of (a+s) from equation (i), into equation (ii), we get,

`(18-a)/(2*6)=6/5`

or, 5(18-a)= 72

or, (18-a) = 72/5

or, `a = 18 - 72/5 = (90-72)/5=18/5 = 3 3/5` ` `

To evaluate s, substituting the value of a in equation (i), we get,

`9/((18/5)+s) = 3/2`

or, `18/5 +s = 6`

or, `s = (6 - 18/5) = 12/5 = 2 2/5`

**Hence the speed of Alan is 18/5 km/hr. and the speed of Sumei is 12/5 km/hr.**

Let S = the speed of Sumei

Let A = the speed of Alan

Let d = the distance between them

The distance between them at any time t is equal to the starting distance subtracted by the distance Sumei has travelled subtracted by the distance Alan has travelled.

Distance is equal to speed multiplied by time; therefore, the distance that Suemi has traveled = St, and the distance Alan has traveled = At

Therefore, the equation for the distance between them is: d = 9 – St – At

At t = 1.5, d=0

0 = 9 - 1.5S - 1.5t

1.5S+1.5A = 9

If tS = 1.2 then tA = 1.2+0.5 = 1.7, when d = 0

0 = 9 - 1.2S - 1.7A

1.2S + 1.7A = 9

Using the elimination method:

1.5S + 1.5A = 9

- 1.2S + 1.7A = 9

---------------------

In order to eliminate the S term, they must have the same numerical coefficient. In order to achieve that, multiply the first equation by 1.2 and the second equation by 1.5:

1.2*(1.5S + 1.5A) = 1.2*9

1.8S + 1.8A = 10.8

1.5*(1.2S+1.7A) = 1.5*9

1.8S + 2.55A = 13.5

Thefore:

1.8S + 1.8A = 10.8

- 1.8S + 2.55A = 13.5

----------------------

0S - 0.75A = -2.7

A = -2.7/-0.75 = 3.6 km/h

Substitute this value into either equation to find S:

1.5S + 1.5(3.6) = 9

1.5S = 9 – 5.4

S = 3.6/1.5 = 2.4 km/h

Alternatively, using the substitution method, rearrange either equation in terms of S:

1.5S = 9 – 1.5A

S = 6 – A

Substitute this equation into the second equation:

1.2(6-A) + 1.7A = 9

7.2 – 1.2A + 1.7A = 9

0.5A = 1.8

A = 1.8/0.5 = 3.6 km/h

Substitute the value of A back into the first equation:

S = 6 – 3.6 = 2.4 km/h

Therefore, Sumei’s speed is 2.4 km/h

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