Imogene's car travels 294 mi averaging a certain speed. if the car had gone 7 mph faster, the trip would have taken 1 hour less. find the average speed

 42 mph

Step-by-step explanation:

If you start with the assumption that the answer is an integer, you can solve a problem like this by looking at the factors of 294.

  294 = 2·3·7² = 6·49 = 7·42

At 42 mph, the 294-mile trip took ...

   time = distance/speed = 294 mi/(42 mi/h) = 7 h

At a speed 7 mph faster, the 294-mile trip took ...

   (294 mi)/(49 mi/h) = 6 h . . . . . 1 hour less

The average speed of Imogene's car for the 294-mile trip was 42 miles per hour.

Alternative solution

If you let s represent Imogene's speed, you can use the above time and distance relationship to write an equation relating the trip times:

  294/s = 294/(s+7) +1

Multiplying by s(s+7), we get ...

  294(s+7) = 294s +s(s+7)

  2058 = s^2 +7s . . . . . . . . . . . subtract 294s

We can complete the square by adding (7/2)^2 = 12.25 to both sides

  2070.25 = s^2 +7s +12.25 = (s +3.5)^2

  ±45.5 = s +3.5 . . . . . take the square root; next subtract 3.5

  -3.5 +45.5 = s = 42 . . . . use the positive solution

Imogene's average speed was 42 mph.

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step-by-step explanation: