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Try this beautiful Number Theory problem from the AMC 2019 Problem 16. You may use sequential hints to solve the problem.

Qiang drives 15 miles at an average speed of 30 miles per hour. How many additional miles will he have to drive at 55 miles per hour to average 50 miles per hour for the entire trip?

(A) 45

(B) 62

(C) 90

(D) 110

(E) 135

Algebra

Value

Average Speed

But try the problem first...

Answer: is (D) 110

Source

AMC 8, 2019, Problem 16

First hint

Among the options, there is only one option which is divisible by 55 and that is 110.

That option tells the travel hour is 2.

Second Hint

Qiang drives 15 miles at an average speed of 30 miles per hour.

And we know, Average speed = Total Distance/Total Time

So, by the formula,

Final Step

In this case, If we consider the whole journey, Total Distance is **(110+15)=125**

And as Qiang has to drive at 50 miles per hour for the entire trip, and as Average speed = Total Distance/Total Time ,

As both are same , our answer 110 is established.

Cheenta Numerates Program for AMC - AIME