1) A man has 2 pennies, 3 nickels, 1 dime, and 2 quarters. How many different sums of money can he make using one or more of these 8 coins?
a) 8
b) 12
c) 47
d) 77
2) A train left Albany for Buffalo, a distance of 290 miles, at 10:10 am. The train was scheduled to reach Buffalo at 3:45 pm. If the average rate of the train on this trip was 50 miles per hour, it arrived in Buffalo
a) about 5 minutes ahead of schedule
b) on time
c) about 5 minutes late
d) about 13 minutes later
3) more than a quarter of an hour late
He can use the 2 pennies in 3 ways, take none, take 1 or take 2
He can use the 3 nickels in 4 ways, take none, take 1, etc
the dime in 2 ways,
and the quarters in 3 ways
total number of ways = (3)(4)(2)(3) = 72
but that includes taking none of any coins,
so number of ways to use the coins is 71
Here is a little silly computer program that finds and lists all the possible sumes
10 for p = 0 to 2
20 for n = 0 to 3
30 for d = 0 to 1
40 for q = 0 to 2
50 sum = p+5*n+10*d+25*q
60 print sum,
70 next q
80 next d
90 next n
100 next p
>run
0 25 50 10 35 60 5 30 55 15 40 65 10 35 60 20
45 70 15 40 65 25 50 75 1 26 51 11 36 61 6 31
56 16 41 66 11 36 61 21 46 71 16 41 66 26 51 76
2 27 52 12 37 62 7 32 57 17 42 67 12 37 62 22
47 72 17 42 67 27 52 77
Notice there are duplications,
e.g. picking 2 nickels or 1 dime yields the same sum
All you have to do is eliminate all duplications, and not count the 0
I think there are 47 or them
1) To find the number of different sums of money the man can make using the 8 coins, we can use the concept of combinations.
First, let's consider each type of coin individually:
- There are 2 choices for the number of pennies he can use (0 or 1).
- There are 4 choices for the number of nickels he can use (0, 1, 2, or 3).
- There are 2 choices for the number of quarters he can use (0 or 1).
To find the total number of combinations, we multiply the number of choices for each coin together:
2 (pennies) x 4 (nickels) x 2 (quarters) = 16 combinations.
However, this calculation only covers the combinations using 1 coin each. We also need to consider combinations using multiple coins.
To do this, we can use the concept of binary representation. We can think of each coin as a digit in a binary number, where 0 represents not using the coin, and 1 represents using the coin.
Since there are 8 coins in total, we can think of each combination as an 8-bit binary number. In binary representation, there are 2^8 = 256 possible combinations.
However, we need to subtract 1 from this count because 00000000 (no coins used) is not a valid combination. So the total number of valid combinations using one or more coins is 256 - 1 = 255.
Therefore, the correct answer is not listed among the given options.
2) To find out when the train arrived in Buffalo, we can use the formula: time = distance / rate.
The distance between Albany and Buffalo is 290 miles, and the average rate of the train is 50 miles per hour.
Using the formula, the time taken by the train is:
time = 290 miles / 50 miles per hour = 5.8 hours.
Since the departure time was 10:10 am and the time taken is 5.8 hours, we need to add 5 hours and 48 minutes to the departure time.
10:10 am + 5 hours = 3:10 pm.
3:10 pm + 48 minutes = 3:58 pm.
Therefore, the train arrived in Buffalo at 3:58 pm.
Comparing the arrival time (3:58 pm) with the scheduled arrival time (3:45 pm), we can see that the train arrived about 13 minutes later.
Therefore, the correct answer is (d) about 13 minutes later.
3) If the train arrives more than a quarter of an hour (15 minutes) late, it would mean the train arrived after 4:00 pm.
Since the scheduled arrival time was 3:45 pm and the train arrived after 4:00 pm, the correct answer is that the train arrived more than a quarter of an hour late.
Therefore, the correct answer is (3) more than a quarter of an hour late.