1. A door has a lock with a 5-digit keypad. To unlock the door, a person must enter a passcode consisting of three different digits, with each digit entered in the correct order. What is the probability that a randomly chosen passcode consisting of three different digits unlocks the door?
I know that it should be 1/n, but I never seemed to figure out he n. How do you find the total number of possibilities for this probability question?
2.A cyclist travels away from his house at an average speed of 40 km per hour. He immediately returns to the house by the same route at an average speed of 60 km per hour. What is the cyclist's average speed for the round trip in km per hour?
How do I find it when it asks for "average speed"?
3. The average cost of b books in a customer's online shopping cart is $24. The average cost of d DVD's in the same customer's online shopping cart is $16. If the average cost of the b books and d DVD's is $19, what is the value of d/b?
This question just makes no sense, and confuses me for some reason.
4. If the average(arithmetic mean)of four different positive odd integers is 80, what is the greatest possible value of one of the integers?
How do I approach these questions?
#1. I can't see where any n comes in at all. The chance of guessing the 3 digits in order is 1/5 * 1/4 * 1/3
That is, since there are 5P3=60 ways of picking 3 digits, and only one is correct, that's 1/60 chance of getting it right.
#2
avg speed = totaldistance/totaltime
total distance = 2d
total time = d/40 + d/60
avg speed is thus
2d/(d/40+d/60) = 48 km/hr
#3
total cost of books: 24b
total cost of dvds: 16d
total average cost: (24b+16d)/(b+d) = 19
So,
24b+16d = 19b+19d
5b = 3d
d/b = 5/3
#4
the greatest value of the largest can be achieved when the other three have the smallest total.
The sum of all four numbers is 4*80 = 320.
So, if the smallest three numbers are 1,3,5 then that leaves 320-9=311 for the largest one.
For #1, what does 5P3 mean? What is the P? And so for these type of questions its always 1/total number choices, times 1/total number of choices minus 1, etc for however many digits sought after for the question?
For #3, average cost of b books is $24 means that one book is $24 right? That was the only part I didn't get.. maybe is the language issue for me.
1. To find the total number of possibilities for the passcode, we need to determine the number of choices for each digit.
Since the passcode consists of three different digits, the first digit can be chosen from 0 to 9 (excluding the other two digits), so there are 10 choices.
After the first digit is chosen, the second digit can be chosen from 0 to 9 (excluding the first digit and the third digit), so there are 9 choices.
Similarly, the third digit can be chosen from 0 to 9 (excluding the first two digits), resulting in 8 choices.
To find the total number of possibilities, you multiply the number of choices for each digit together: 10 x 9 x 8 = 720.
Therefore, there are 720 possible passcodes.
The probability of randomly choosing the correct passcode is 1 out of the total number of possibilities, so the probability would be 1/720.
2. To find the average speed for the round trip, you need to consider the total distance traveled and the total time taken.
Let's assume the distance between the cyclist's house and his destination is 'd.'
When the cyclist travels away from his house at an average speed of 40 km/h, the time taken is d/40.
When the cyclist returns to his house at an average speed of 60 km/h, the time taken is d/60.
To calculate the average speed, we need to find the total distance traveled and the total time taken for the round trip.
The total distance traveled is 2d (going to the destination and then returning to the house).
The total time taken is (d/40) + (d/60).
Average speed is defined as total distance divided by total time.
Average speed = (2d) / [(d/40) + (d/60)].
Simplifying this expression, we get:
Average speed = (2d) / [(3d + 2d)/120] = (2d * 120) / (5d) = 240/5 = 48 km/h.
Therefore, the cyclist's average speed for the round trip is 48 km/h.
3. Let's break down the problem step by step:
The average cost of b books is given as $24. This means the total cost of b books is b * $24.
The average cost of d DVDs is given as $16. This means the total cost of d DVDs is d * $16.
The average cost of b books and d DVDs together is given as $19. This means the total cost of b books and d DVDs is (b * $24) + (d * $16).
Since the average cost is calculated by dividing the total cost by the number of items, we can write the equation:
[(b * $24) + (d * $16)] / (b + d) = $19.
Now, we can solve this equation to find the value of d/b.
4. To approach this question, let's first understand what an average (arithmetic mean) is.
The average of a set of numbers is obtained by summing all the numbers and then dividing the sum by the total number of values.
In this case, we have four different positive odd integers with an average of 80.
Let's assume the integers are a, b, c, and d.
Since they are positive odd integers, we can represent them as a = 2x + 1, b = 2y + 1, c = 2z + 1, and d = 2w + 1, where x, y, z, and w are positive integers.
We know that the average of these four numbers is 80, so we can write the equation:
(a + b + c + d) / 4 = 80.
Substituting the expressions for a, b, c, and d, we have:
[(2x + 1) + (2y + 1) + (2z + 1) + (2w + 1)] / 4 = 80.
Simplifying this equation by combining like terms, we get:
(2x + 2y + 2z + 2w + 4) / 4 = 80.
Cancelling out the common factors, we have:
2x + 2y + 2z + 2w + 4 = 320.
Dividing both sides by 2, we get:
x + y + z + w + 2 = 160.
Now, let's analyze the problem. We need to find the greatest possible value of one of the integers. Since the integers are positive odd numbers, we should try to maximize each value.
To maximize the sum (x + y + z + w), we can assign x, y, z, and w as 39, 39, 40, and 40 respectively.
Substituting these values into the equation, we have:
39 + 39 + 40 + 40 + 2 = 160.
Therefore, the greatest possible value of one of the integers is 40.