As you can see, it's from Car Talk -- the NPR show with Click and Clack. They are actually MIT grads. And the key point to the puzzle is also discussed by the physicist Feynman in his collection of autobiographiphical stories "Surely You are Joking, Mr. Feynamn". If you want the hint, scroll down...
Consider the story from the reference frame of the hat moving with the current.
Thanks for answering my rate question, everyone! You guys are superduper helpful. I have another problem that bugs me:
Dick spent $7 in order to buy baseballs and tennis balls. If baseballs are 70 cents each and tennis balls are 60 cents each, what is the greatest number of tennis balls that Dick could have bought?
It is 3 mph! And the solution is on the Car Talk website.
As for your other question, key fact is that he spent EXACTLY $7 (which the SAT would probably have made more clear). So start looking at what happens if he bought say 12, then 11, then 10 tennis balls, looking at how much money he has left over. If it isn't an exact multiple of $.70, it's wrong...
Wow, I can't believe i got that rowboat question right! It looked so confusing to me, but I guess all this rate-distance-time help I'm getting from all of you is really working.
Ohhhhhh, I just thought that he had to spend $7 or less. This time I tried solving the problem by doing 0.7b*0.6t=7, where b is baseballs and t is tennis balls. I just did trial-and-error until I got the answer of 7 tennis balls, which is correct.
Hi! Could you please help me with this question? Here is a link:
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Thanks!
The shaded figure is a quadrilateral, and so the 4 angles of this quadrilateral add up to 360 degrees. Since x + y = 80, the other two angles add up to 360 - 80 = 280. Therefore, one of those angles is 280/2 = 140.
The total number of degrees in a polygon with n sides is T = 180(n - 2). So we have that 140n = 180(n - 2) = 180n - 360. Now just solve this equation for n:
140n = 180n - 360
360 = 40n
9 = n.
So the answer is choice (B).
Alternative: An exterior angle to the polygon is 180 - 140 = 40. Since the sum of the measures of the exteror angles of a polygon (1 at each vertex) is 360, n = 360/40 = 9, choice (B).
The Acme Plumbing Company will send a team of 3 plumbers to work on a certain job. The company has 4 experienced plumbers and 4 trainees. If a team consists of 1 experienced plumber and 2 trainees, how many different such teams are possible?
Answer:24
I need an algebraic method to solving this and why can the answer not be 48?
There are 4 ways to choose 1 experienced plumber from 4. There are 6 ways to choose 2 trainees from 4. The counting principle now says to multiply the numbers of possibilities: 4*6 = 24.
Notes:
(1) You can do the 2 computations formally as 4C1 = 4, and 4C2 = 6. Just use the nCr button in your calculator.
(2) You can also compute these combinations just by listing. For the first one:
O***
*O**
**O*
***O
In the list above, O represents the experienced plumber selected, and * represents that that plumber is not selected.
For the second one:
OO**
O*O*
O**O
*OO*
*O*O
**OO
In practice you can form these lists without any writing just by holding up 4 fingers on one hand, and using the appropriate number of fingers on your other hand to use as pointers for counting. For the first computation you would use 1 finger to count. For the second computation you would use 2 fingers to count.
A store charges $28 for a certain type of sweater. This price is 40 percent more than the amount it costs the store to buy one of these sweaters. At an end-of-season sale, store employees can purchase any remaining sweaters at 30 percent off the store's cost. How much would it cost an employee to purchase a sweater of this type at this sale?
A) $8.40 $14.00
C) $19.60
D) $20.00
E) $25.20
The answer is B but how would you solve this algebraically?
• 3, 5, -5,...
The first term in the sequence of numbers shown above is 3. Each even-numbered term is 2 more than the previous term, after the first, is -1 times the previous term. For example, the second term is 3 + 2, and the third term is (-1) x 5. What is the 55th term of the sequence?
(A) -5
(B) -3
(C) -1
(D) 3
(E) 5
• Esther drove to work in the morning at an average speed of 45 miles per hour. She returned home in the evening along the same route and averaged 30 miles per hour. If Esther spent a total of one hour commuting to and from work, how many miles did Esther drive to work in the morning?
Second one: We use the simple formula distance = rate * time. Let’s put the given information into the following chart.
home to work d 45 d/45
work to home d 30 d/30
The first column above is the distance, the second is the rate, and the third is the time.
Note that although we don’t know either distance, we do know that they are the same, so we can call them both “d.”
Also, since distance = rate * time, we have that time = distance/rate. We use this to get the first two entries in column three. The total time is given in the question. So we have
In particular every 4th term is -3. Since 52 is divisible by 4, the 52nd term is -3. Now just count from there. The 53rd term is 3, the 54th term is 5, and the 55th term is -5, choice (A).
Remark: I call this a "remainders in disguise" problem. You can get the answer quickly by computing the remainder when 55 is divided by 4. Note that 4 goes into 55 thirteen times with a remainder of 3. Therefore the 55th term of the sequence is the same as the 3rd term of the sequence.
Important note: You cannot get a remainder by dividing in your calculator. You must perform the division by hand!
^ DrSteve -- I agree with your solution to this remainder problem. But I would add one note: you CAN get remainders with your calculator. It just takes an extra step...
Do the division. If it comes out evenly, then of course the remainder is zero.
If it does not come out evenly, then your answer has an integer part and a decimal part. (The decimal part is NOT the remainder...you have one more step to do.)
Multiply the decimal part by the divisor. That's your remainder.
In this case: 55/4 = 13.75
.75 times 4 = 3
So the calcutor does tell you the remainder after all!
Also, there are a couple of calculators out there with built-in remainder functions.
^ draw a line from the center of the square face to one of its vertices. This has length m*sqrt(2)/2 since it is half the diagonal. A right triangle is formed with legs m*sqrt(2)/2 and h. Its hypotenuse is m (since e=m). Using the Pythagorean Theorem, m^2/2+h^2=m^2, so h^2=m^2/2 and h=m/sqrt(2), which is A.
Since the area is 4000, xy=4000. The length of the rope needed is x+x+x+x+y=4x+y. The four x's are the lengths of the vertical separators. using xy=4000 we get x=4000/y. Plugging this in to the above equation we get y+1600/y, which is B.
Replies to: Math help center
Row, Row, Row Your Boat | Car Talk
As you can see, it's from Car Talk -- the NPR show with Click and Clack. They are actually MIT grads. And the key point to the puzzle is also discussed by the physicist Feynman in his collection of autobiographiphical stories "Surely You are Joking, Mr. Feynamn". If you want the hint, scroll down...
Consider the story from the reference frame of the hat moving with the current.
Thanks for answering my rate question, everyone! You guys are superduper helpful. I have another problem that bugs me:
Dick spent $7 in order to buy baseballs and tennis balls. If baseballs are 70 cents each and tennis balls are 60 cents each, what is the greatest number of tennis balls that Dick could have bought?
As for your other question, key fact is that he spent EXACTLY $7 (which the SAT would probably have made more clear). So start looking at what happens if he bought say 12, then 11, then 10 tennis balls, looking at how much money he has left over. If it isn't an exact multiple of $.70, it's wrong...
Ohhhhhh, I just thought that he had to spend $7 or less. This time I tried solving the problem by doing 0.7b*0.6t=7, where b is baseballs and t is tennis balls. I just did trial-and-error until I got the answer of 7 tennis balls, which is correct.
Is there a formulaic way to do this?
[url=http://****.com/view.php?pic=35a191s&s=6]Image - **** - Free Image Hosting, Photo Sharing & Video Hosting[/url]
Thanks!
The shaded figure is a quadrilateral, and so the 4 angles of this quadrilateral add up to 360 degrees. Since x + y = 80, the other two angles add up to 360 - 80 = 280. Therefore, one of those angles is 280/2 = 140.
The total number of degrees in a polygon with n sides is T = 180(n - 2). So we have that 140n = 180(n - 2) = 180n - 360. Now just solve this equation for n:
140n = 180n - 360
360 = 40n
9 = n.
So the answer is choice (B).
Alternative: An exterior angle to the polygon is 180 - 140 = 40. Since the sum of the measures of the exteror angles of a polygon (1 at each vertex) is 360, n = 360/40 = 9, choice (B).
Answer:24
I need an algebraic method to solving this and why can the answer not be 48?
Notes:
(1) You can do the 2 computations formally as 4C1 = 4, and 4C2 = 6. Just use the nCr button in your calculator.
(2) You can also compute these combinations just by listing. For the first one:
O***
*O**
**O*
***O
In the list above, O represents the experienced plumber selected, and * represents that that plumber is not selected.
For the second one:
OO**
O*O*
O**O
*OO*
*O*O
**OO
In practice you can form these lists without any writing just by holding up 4 fingers on one hand, and using the appropriate number of fingers on your other hand to use as pointers for counting. For the first computation you would use 1 finger to count. For the second computation you would use 2 fingers to count.
A) $8.40
$14.00
C) $19.60
D) $20.00
E) $25.20
The answer is B but how would you solve this algebraically?
20=x
.7 times 20 = 14
• 3, 5, -5,...
The first term in the sequence of numbers shown above is 3. Each even-numbered term is 2 more than the previous term, after the first, is -1 times the previous term. For example, the second term is 3 + 2, and the third term is (-1) x 5. What is the 55th term of the sequence?
(A) -5
(B) -3
(C) -1
(D) 3
(E) 5
• Esther drove to work in the morning at an average speed of 45 miles per hour. She returned home in the evening along the same route and averaged 30 miles per hour. If Esther spent a total of one hour commuting to and from work, how many miles did Esther drive to work in the morning?
home to work d 45 d/45
work to home d 30 d/30
The first column above is the distance, the second is the rate, and the third is the time.
Note that although we don’t know either distance, we do know that they are the same, so we can call them both “d.”
Also, since distance = rate * time, we have that time = distance/rate. We use this to get the first two entries in column three. The total time is given in the question. So we have
d/45 + d/30 = 1
30d + 45d = 30*45
75d = 30*45
d = 30*45/75 = 18
average speed for round trip = 2(speed 1)(speed 2)/(speed + speed 2) = 2(45)(30)/(45 + 30) = 36.
So 2d = 36(1), and thus d = 18.
The sequence is 3, 5, -5, -3, 3, 5, -5, -3, ...
In particular every 4th term is -3. Since 52 is divisible by 4, the 52nd term is -3. Now just count from there. The 53rd term is 3, the 54th term is 5, and the 55th term is -5, choice (A).
Remark: I call this a "remainders in disguise" problem. You can get the answer quickly by computing the remainder when 55 is divided by 4. Note that 4 goes into 55 thirteen times with a remainder of 3. Therefore the 55th term of the sequence is the same as the 3rd term of the sequence.
Important note: You cannot get a remainder by dividing in your calculator. You must perform the division by hand!
Do the division. If it comes out evenly, then of course the remainder is zero.
If it does not come out evenly, then your answer has an integer part and a decimal part. (The decimal part is NOT the remainder...you have one more step to do.)
Multiply the decimal part by the divisor. That's your remainder.
In this case: 55/4 = 13.75
.75 times 4 = 3
So the calcutor does tell you the remainder after all!
Also, there are a couple of calculators out there with built-in remainder functions.
TI89: Enter remain(55,4)
TI30-II (I think)... Enter 55 [shift] [divide] 4
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