A car A starts from a point P towards another point Q. Another car B also starts from P some time after car A and overtakes it after covering 40% of the distance PQ. On reaching Q, car B turns back and meets car A when car A has covered 3/4th of the distance PQ. The ratio of speed of car A to that of car B is:

Let's denote the speed of car A as Va and the speed of car B as Vb.

Step 1: When car B overtakes car A, they have both covered the same distance, which is 40% of the total distance from P to Q. Let's denote the total distance from P to Q as D. So, the distance they both covered is 0.4D.

Step 2: The time it took for car A to cover this distance is 0.4D/Va. The time it took for car B to cover the same distance is less, because it started later. Let's denote this time difference as T. So, the time it took for car B to cover the 0.4D distance is 0.4D/Va - T.

Step 3: From this, we can write the equation 0.4D/Va = 0.4D/Vb + T.

Step 4: When car B reaches Q, it turns back and meets car A at a point where car A has covered 3/4 of the distance. This means car B has covered the total distance D plus an additional 1/4D, so 1.25D in total.

Step 5: The time it took for car B to cover this distance is 1.25D/Vb. In the same time, car A has covered 3/4D, so the time is also 0.75D/Va.

Step 6: From this, we can write the equation 1.25D/Vb = 0.75D/Va.

Step 7: Solving these two equations (from step 3 and step 6) for Va/Vb gives us the ratio of the speed of car A to that of car B.

Step 8: From the equation in step 3, we can express T as T = 0.4D/Va - 0.4D/Vb.

Step 9: Substituting T from step 8 into the equation from step 6 gives us 1.25D/Vb = 0.75D/Va + 0.4D/Va - 0.4D/Vb.

Step 10: Simplifying this equation gives us Va/Vb = 5/3.

So, the ratio of the speed of car A to that of car B is 5:3.