Problem 1
<h3>Answer: 48</h3>
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Work Shown:
The set of single digits that are even numbers is {0,2,4,6,8}. There are 5 items here.
The first selection cannot be 0 since a number like 048 is not a three digit number (instead 048 becomes 48 which is a two digit number)
So we have {2,4,6,8} to pick from. There are 4 choices here.
Once we pick a value from that list, we have 5-1 = 4 items left to pick from. For instance, let's say we picked 2 as the first digit. So {0,2,4,6,8} would become {0,4,6,8}
After we pick the second digit, we then have 4-1 = 3 digits to pick from
Overall, there are 4*4*3 = 48 different three digit numbers that can be formed using only even digits, and we cannot repeat a digit.
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Problem 2
<h3>Answer: 6 minutes</h3>
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Work Shown:
400 m = 400/1000 = 0.4 km
800 m = 800/1000 = 0.8 km
The train is 0.4 km long. The tunnel is 0.8 km long.
In order the the train's caboose to clear the tunnel (the very back side of the caboose not be in/under the tunnel), the front of the train's engine must travel 0.8+0.4 = 1.2 km. Any shorter and the train will still be in the tunnel somehow. I recommend drawing out a diagram to see why this is the case.
Since it needs to travel 1.2 km, and its speed is 12 km/hr, this means...
distance = rate*time
d = r*t
1.2 = 12*t
12t = 1.2
t = 1.2/12
t = 0.1
This is the amount of time it takes, in hours, to travel 1.2 km
Multiply by 60 to convert form hours to minutes
0.1 hrs = (0.1 hrs)*(60 min/1 hr) = (0.1*60) min = 6 minutes
0.1 hrs = 6 minutes
The train will take 6 minutes to clear the tunnel.
