MathDB

15

Part of 2011 AMC 10

Problems(2)

Length of trip

Source: 2011 AMC A Problem 15

6/25/2011
Roy bought a new battery-gasoline hybrid car. On a trip the car ran exclusively on its battery for the first 40 miles, then ran exclusively on gasoline for the rest of the trip, using gasoline at a rate of 0.02 gallons per mile. On the whole trip he averaged 55 miles per gallon. How long was the trip in miles?
<spanclass=′latex−bold′>(A)</span> 140<spanclass=′latex−bold′>(B)</span> 240<spanclass=′latex−bold′>(C)</span> 440<spanclass=′latex−bold′>(D)</span> 640<spanclass=′latex−bold′>(E)</span> 840<span class='latex-bold'>(A)</span>\,140 \qquad<span class='latex-bold'>(B)</span>\,240 \qquad<span class='latex-bold'>(C)</span>\,440 \qquad<span class='latex-bold'>(D)</span>\,640 \qquad<span class='latex-bold'>(E)</span>\,840
AMC
&quot;Averaged with&quot; operation

Source: AMC 10 2011 b Problem 15

2/24/2011
Let @ denote the "averaged with" operation: aa @ bb = a+b2\frac{a+b}{2}. Which of the following distributive laws hold for all numbers x,yx,y and zz?
I. x @ (y+z) = (x @ y) + (x @ z) II. x + (y @ z) = (x + y) @ (x + z) III. x @ (y @ z) = (x @ y) @ (x @ z)
<spanclass=′latex−bold′>(A)</span> I only<spanclass=′latex−bold′>(B)</span> II only<spanclass=′latex−bold′>(C)</span> III only<spanclass=′latex−bold′>(D)</span> I and III only<spanclass=′latex−bold′>(E)</span> II and III only <span class='latex-bold'>(A)</span>\ \text{I only} \qquad <span class='latex-bold'>(B)</span>\ \text{II only} \qquad <span class='latex-bold'>(C)</span>\ \text{III only} \qquad <span class='latex-bold'>(D)</span>\ \text{I and III only} \qquad <span class='latex-bold'>(E)</span>\ \text{II and III only}
AMC