Daily Test ~ {ERG}

[LoneRanger]

20 June 2008:

THE SOLUTION:

Number 1. Here’s why:
58 x 13 = 754; 27 x 13 = 351; 65 x 13 = 845; 17 x 13 = 221.



An angel materializes on your doorstep and speaks.

“I am thinking of two whole numbers that represent the number of apples and grapes in the Garden of Eden. The product of these numbers
is 1,000 times larger than their sum. What are the two numbers?”

You frown. “Wait, that is not enough information.”

The angel nods. “You are right. There is more than one solution, and you have 24 hours to give me any solution. If you are correct, you will live the rest of your life in bliss.”

[LoneRanger]

21 June 2008:

THE SOLUTION:

We need to search for numbers such that (a + b) x 1000 = a x b. One such answer is 1,250 and 5,000.

There are twenty-five possible unique solutions for positive integers. The largest number of possible apples or grapes is 1,001,000.

Here are all the (a, b) solutions: (1,001,000; 1,001), (501,000; 1,002),
(251,000; 1,004), (201,000; 1,005), (126,000; 1,008), (101,000; ,1010), (63,500; 1,016), (51,000; 1,020), (41,000; 1,025), (32,250;
1,032), (26,000; 1,040), (21,000; 1,050), (16,625; 1,064), (13,500; 1,080), (11,000; 1,100), (9,000; 1,125), (7,250; 1,160), (6,000; ,1200), (5,000; 1,250), (4,125; 1,320), (3,500; 1,400), (3,000; 1,500), (2,600; 1,625), (2,250; 1,800), and (2,000; 2,000).



Captain Kirk’s starship leaves Earth for Mars. Captain Eck’s starship leaves from Mars for Earth. Their ships start at the same time and travel at uniform speeds, but one is faster than the other. After meeting and passing, Kirk requires 17 hours and Eck requires 10 hours to complete the journey. Approximately what total time did each starship require for its interplanetary journey? Assume stationary planets.

[LoneRanger]

22 June 2008:

THE SOLUTION:

Kirk requires 30 hours. Eck requires 23 hours.

Let Kirk’s ship travel at speed u, and Eck’s travel at speed v, in arbitrary units. Let the time of their meeting be t hours. We know that distance is time x speed.

Thus, we have the following distances covered:

Kirk, before meeting: tu

Kirk, after meeting: 17u

Eck, before meeting: tv

Eck, after meeting: 10v

Note that the distance that Kirk travels before meeting Eck is the same as the distance that Eck travels after meeting Kirk. Thus, we then have tu = 10v and 17u = tv. We can manipulate the formulas to solve for t. For
example, t = 10v/u and u = tv/17. Thus t = 170v/tv, which means t^ 2 = 170, and t is approximately equal to 13.

Thus Kirk’s ship requires 17 + 13 hours = 30 hours, and Eck requires 10 + 13 = 23 hours.



Insert the consecutive numbers 1 through 11 in the 11 empty cells. Each number in the gray cells is the sum of the adjacent empty cells (right, left, up and down) that touch the gray cell.

[LoneRanger]

23 June 2008:

THE SOLUTION:

Missing numbers. Here is one solution. Are there others?

We believe that there are 32 different solutions for this problem. If we randomly scrambled the digits 1 through 11 to fill the white cells, the probability of producing a correct answer is 32/11! = 32/39,916,800, which simplifies to 1/1,247,400-a roughly one in a million shot.



An old car starts at a city in New Jersey and travels at a speed of 21 miles per hour toward New York.

After reaching New York, the car returns, traveling over exactly the same distance, at only 3 miles per hour. What is the car’s average speed over
the entire journey? (Assume that the car turned around instantly once it reached New York.)

[LoneRanger]

24 June 2008:

THE SOLUTION:

Let the distance between the city and New York be D miles. We know that
time equals distance divided by velocity. This means that it takes D/21 hours to travel to New York and D/3 hours to return. The total time for the trip is D/21 + D/3 = 8D/21 hours to travel 2D miles, or 4/21 hours to travel 1 mile. Therefore, the average speed is 21/4 or 5.25 miles an hour.



Four intelligent gorillas in the forests of Africa are holding an election to determine who should be king of the gorillas. At the recent election, a total of 8,888 votes were received for the four candidate gorillas, the winner beating its opponents by 888, 88, and 8 votes, respectively. How many votes did the successful gorilla receive?

[LoneRanger]

25 June 2008:

THE SOLUTION:

The successful gorilla received 2,468 votes. Let the winner receive w votes. Then the other three candidates received w - 888, w - 88, and w - 8 votes. So the total number of votes is 4w - (888 + 88 + 8), which we know equals 8,888.

Hence, w = (8,888 + 888 + 88 + 8)/4 = 2,468.




Noah is a candy store owner. He has 20 pounds of cashews, costing $3.55 a pound. And he has peanuts that cost $2.50 a pound. How many pounds of peanuts would Noah have to mix with all the cashews to get a mixture that costs $3.20 per pound?

[LoneRanger]

26 June 2008:

THE SOLUTION:

He’d have to mix 10 pounds of peanuts. Let y be the pounds of peanuts Noah would have to mix with all the cashews. The cost of the peanuts is 2.50 x y. Thus, the total weight, w, of the final mix is 20 + y. The equation to solve is:

3.55 x 20 + 2.50 x y = 3.20 y (20 + y),
or 71 + 2.5y = 64 + 3.2y,
or 7y = 0.7y,
or y = 10 pounds of peanuts



Mike collects lifelike models of human hands. He wants to put them on shelves in his
den. His first thought is to put each one on a shelf, but when he tries this, two hands have to share one shelf.

Next, Mike tries to place the hands on the shelves so that every shelf contains two hands, but when he tries this, one shelf is left empty.


How many hands does Mike own? How many shelves is he using?

[LoneRanger]

27 June 2008:

THE SOLUTION:

He owns 4 hands and uses 3 shelves. Let H represent the number of hands and S the number of shelves. We know that: H - 1 = S (all the hands except one had a shelf), and H/2 = S - 1 (half the number of hands filled all the shelves but 1). Solving for
H and S, we get 4 hands and 3 shelves.




Ages. I have five wonderful friends, one of whom is quite
young. The sum of all their ages is 109. If I add pairs of ages, I get the following:

How old are my friends?

[LoneRanger]

28 June 2008:

THE SOLUTION:

Teja: 5; Danielle: 11; Nick: 21; Pete: 31; Mark: 41.



Let n be a positive integer. Can both n + 3 and n^2 + 3 be perfect cubes? (A perfect cube is a number that is the cube of an integer, like 2^3 = 8 or 3^3 = 27.).

It would seem as if I gave you too little information to solve this, and that is why the problem is so fascinating. How can you answer this problem with such meager information and without guessing?

[LoneRanger]

29 June 2008:

THE SOLUTION:

No, they cannot be perfect cubes.

Perfect cubes grow very rapidly: 1, 8, 27, 64, 125, . . . Thus, there are no perfect cubes that differ by 8 for positive n. Therefore, n + 3 and n^2 + 3 cannot both be perfect cubes.



I have four pets: an iguana, a cat, a bird, and a large ape. Today, I loaned all the pets to friends who enjoy the company of pets. The iguana is returned every 6 days. The cat comes home every 4 days. The bird comes home every 3 days and the ape every 7 days. The pets are always returned at noon and stay with me for an hour during my lunch break so that I can enjoy their company before loaning them out again. In how many days will all my pets be together again in my cozy home?

[LoneRanger]

30 June 2008:

THE SOLUTION:

In 84 days; 84 is the lowest common multiple of 6, 4, 3, and 7. The lowest common multiple (LCM) of a group of numbers is the smallest number that is a multiple of each number in the group. One way of finding the LCM is to list the multiples of each number, and then pick the smallest number that is common to all numbers in the group. You can learn about other ways of finding the LCM on the World Wide Web.



Swap two numbers in the numerator with two numbers in the denominator to form a fraction equaling 1⁄3.

1630/4542

If you get this fast enough try swapping three numbers

[LoneRanger]

01 July 2008:

THE SOLUTION:

1,534/4,602 = 1/3. How long did you take to solve this? What method is the most efficient to use when solving problems of this kind? Do you think there is more than one answer? Recently, I found four different answers: 1354/4062, 1534/4602, 0454/1362, and 0544/1632. However, the last two solutions involve swapping 3 numbers, not 2.



Donald Trumpet died, leaving a peculiar will. His will states that he will leave one million dollars to be split between his son William and his daughter Hillary. Hillary, his favorite child, gets four times the amount of William. If Hillary takes less than 30 seconds to determine how much William will get, the money is distributed immediately; Can you help her? What did William get?

[LoneRanger]

02 July 2008:

THE SOLUTION:

William got $200,000. ($200,000 + $800,000 = $1,000,000.)



Three huge rubies and two emeralds weigh 32 pounds. Four rubies and three emeralds together weigh 44 pounds. Assume that the three rubies have an identical weight, as do the emeralds. What is the weight of two rubies and one emerald?

[LoneRanger]

03 July 2008:

THE SOLUTION:

The combined weight is 20 pounds. Let R = the weight of one ruby and E = the weight of one emerald. We know that 3R + 2E = 32 and 4R + 3E = 44, which means that the weight of a single ruby is 8 pounds, and the weight of a single emerald is 4 pounds. So, 2R + E = 20 pounds.

Perhaps a more elegant solution is to note that removing one ruby and one emerald
(4R + 3E -> 3R + 2E) reduces the weight by 12 pounds (44 to 32), so doing the same again (3R + 2E -> 2R + E) reduces by the same amount, to 20 pounds. Using this logic, there is no need to calculate the individual weight of each stone.



A crow and an eagle are gazing at 100 worms. The crow says, “Because I am smaller than you, you will get six times the number of worms I will get. If you tell me how many worms you will get in this fine deal, I will build a beautiful nest for you. If you are not that smart, I will get all the worms.”

The eagle said, “It’s a deal!” How many worms did the eagle get? (Hint: You may have to cut some of the worms to be fair and accurate.)

[LoneRanger]

04 July 2008:

THE SOLUTION:

The crow gets 14 and 2/7 worms (approximately 14.2857 worms). The eagle gets 85 and 5/7 worms (85.7143 worms).


Substitute four different digits for A, B, C, and D to make the following mathematical expression correct. (AB is a two-digit number, and DAC is a three-digit number. A, B, C, and D are single-digit numbers.)

(AB + A) x C = DAC