Each day for four days, Linda traveled for one hour at a speed that resulted in her traveling one mile in an integer number of minutes. Each day after the first, her speed decreased so that the number of minutes to travel one mile increased by minutes over the preceding day. Each of the four days, her distance traveled was also an integer number of miles. What was the total number of miles for the four trips?
Solution
It is well known that . In the question, we want distance. From the question, we have that the time is minutes or hour. By the equation derived from , we have , so the speed is mile per minutes. Because we want the distance, we multiply the time and speed together yielding . The minutes cancel out, so now we have as our distance for the first day. The distance for the following days are:
dfrac{60}{x},dfrac{60}{x+5},dfrac{60}{x+10},dfrac{60}{x+15}.
We know that are all factors of , therefore, because the factors have to be in an arithmetic sequence with the common difference being and is the only solution.
dfrac{60}{5}+dfrac{60}{10}+dfrac{60}{15}+dfrac{60}{20}=12+6+4+3=boxed{textbf{(C)} 25}.
== Video Solution by OmegaLearn ==
https://youtu.be/XixU0JZ5FLk?t=394
~ pi_is_3.14
~Hipparachus
Toothpicks are used to make a grid that is 60 toothpicks long and 32 toothpicks wide. How many toothpicks are used altogether?
[figure]
Solution
There are 61 vertical columns with a length of 32 toothpicks, and there are 33 horizontal rows with a length of 60 toothpicks, because 32 and 60 are the number of intervals. You can verify this by trying a smaller case, i.e. a 3 times 4 grid of toothpicks, with 3 times 3 and 2
times 4.
Thus, our answer is 61cdot 32 + 33 cdot 60 = boxed{textbf{(E)} 3932}.
~Note by Theraccoon: The person who posted this answer did not include their name. Minor edit by ~NXC
After Euclid High School’s last basketball game, it was determined that of the team’s points were scored by Alexa and were scored by Brittany. Chelsea scored points. None of the other team members scored more than points. What was the total number of points scored by the other team members?
Solution
Given the information above, we start with the equation , where is the total number of points scored and is the number of points scored by the remaining 7 team members, we can simplify to obtain the Diophantine equation , or . Since is necessarily divisible by 28, let where and divide by 28 to obtain . Then, it is easy to see () is the only candidate remaining, giving .
-scrabbler94
~Minor Edits by Wrenmath
The faces of a cube are painted in six different colors: red , white , green , brown , aqua , and purple . Three views of the cube are shown below. What is the color of the face opposite the aqua face?
[figure]
Solution
is on the top, and is on the side, and is on the right side. That means that (image ) is on the left side. From the third image, you know that must be on the bottom since is sideways. That leaves us with the back, so the back must be . The front is opposite of the back, so the answer is .