## Teeter-Totter Balance Puzzle

A girl wanted to play on the seesaw. However, she couldn’t find another child to play with her, so she tied some books to one end of the plank to balance her weight at the other.

She just balanced against 16 books, when these were fixed to the short end of the plank, but if she fixed them to the long end of the plank, she only needed 11 as balance.

Now, what was the girl’s weight, if a book weighs equal to a three-quarter book and three-quarters of a pound? (Assume all books are uniform and weigh the same)

*[submitted answers to show 1-2 days)*

Lath| Profile December 5th, 2007 - 2:23 pmAbout 39.8 pounds

foger1979| Profile December 5th, 2007 - 4:52 pmThe girl weighs 33 pounds.

Book=3/4 book + .75 pounds

1/4 Book=.75 pounds

Book=3 lbs.

Now, with that info-

A girl wanted to play on the seesaw….so she tied some books to one end of the plank to balance her weight at the other. If she fixed them to the long end of the plank, she only needed 11 as balance.

I am making the assumption about the distance of “the long end of the plank.” Since she only tied two bundles of books onto the seesaw and the other set was at a shorter distance, the 11 books must be the ones “she tied…to one end of the plank.”

11 books x 3 lbs = 33 lbs

Filoso| Profile December 5th, 2007 - 7:19 pmIf she sat opposite to where the middle would be between the Short- and long end then I get 17.8 Pounds.

Doubt this is correct though.

Carpenter| Profile December 5th, 2007 - 9:59 pmI came up with 3 lbs for each book and an inconsistent system with 48 = 33! :x

Filoso| Profile December 6th, 2007 - 1:05 amUh Actually.

Book is 3 Pounds.

So she is about 33 lbs.

Shawn| PUZZLE GRANDMASTER | Profile December 6th, 2007 - 2:25 pmThis is a poorly worded puzzle.

Short end vs. long end of a seesaw plank? Both ends are the same.

” a book weighs equal to a three-quarter book and three-quarters of a pound?” What does this mean? A book weighs equal to a three-quarter book makes no sense.

falwan| Profile December 7th, 2007 - 10:44 pm48 lb.

RK| Profile December 8th, 2007 - 10:33 amwill leave this up for another day or 2 since only 1 person got the correct answer so far

falwan| Profile December 9th, 2007 - 3:27 amSir,

Can you recommend other “Puzzle Sites” for me as excellent as “Smartkit”??

I realy admire your site.It’s one of the best.I think.

Thanks a million…

Falwan

suineg| PUZZLE MASTER | Profile December 10th, 2007 - 9:12 amok, i think this can be a way of solving it

Explanation:

1 book = 3/4 book + 3/4 pound–> 1/4 book =3/4 pound

1 book= 3 pound

If the girls is alone has to be sit in the short end of the teeteer, because she has to grab something in order to balance without falling, this is an assumption i guess, so she is equal to 16 books in weight.

girl= 16 books—> girl= 16*3= 48 pounds

Shawn| PUZZLE GRANDMASTER | Profile December 10th, 2007 - 10:15 amOkay, okay.

A book weighs equal to a 3/4 book and 3/4 of a pound.

B=3/4*B + 3/4

B=3 pounds/book

Because she first “just balanced” with 16 books at the short end of the plank, let’s assume that she also seats herself at the short end on her side of the plank. She would be most likely to make her initial attempt at balancing by putting the books in the same spot on the plank (on the opposite side) as she was going to be sitting

She must then weigh = 48 pounds (16*3)

When she slides the books out to the long end of the plank, and then returns and sits on the short end of her side of the plank, only 11 books are required. This must have taken a lot of experimentation and patience for a child who we must assume, at 48 pounds, is 6-7 years old.

(GIRL)*(SHORT-END LENGTH) = (BOOKS)*(LONG-END-LENGTH)

(LONG-END LENGTH)/(SHORT-END LENGTH) = 48/33 = 1.4545

So the distance from the fulcrum to the long-end is 1.4545 times the distance from the fulcrum to the short end, and these distances can be any amount that meets the ratio.

Also, we have to note that if she originally balanced against the 16 books at the short-end of the plank while she herself was sitting on the long-end of her side of the plank (poor depth perception? outside-the-box mentality? precocious insolence?) the values will be different. In that case she will weigh only 33 pounds (11*3). This would mean she was likely 3-4 years old at the most, and much less likely to undertake the initiative in the first place. The ratio of long-end to short-end remains at 1.4545.

We also haven’t accounted for the weight of the string used to tie the books to the plank.

Milo Zachary| Profile December 10th, 2007 - 1:34 pmIf a book weighs 3/4 of a book plus 3/4 of a pound, then 1/4 of a book weighs 3/4 pounds, so a whole book weighs 3 pounds. Taking the average of her 2 different balance weights, she must weigh 13 1/2 books, or 40 1/2 pounds.

suineg| PUZZLE MASTER | Profile December 10th, 2007 - 4:50 pmmaybe one answer could be the mean between 33 pounds and 48 pounds, but i think it would be wrong, because if you can tell the girl is stuck in the short end of the teeteer, you certanly can not say she is exactly between the long end and the short end.

for the statement in the problems, she can be sitted in the long end or in the short end, but for the drawing you can deduce she was sitted in the short end.

RK| Profile December 10th, 2007 - 8:48 pmHi guys- this problem comes from a puzzle book almost a century old, so that’s why the wording is a little strange.

The correct answer they give is 39.8 (as Lath notes above).

RK| Profile December 10th, 2007 - 10:12 pmEquation 1: (weight of girl) x (long length of fulcrum)=(short length of fulcrum)x(48)

Equation 2weight of girl) x (short length of fulcrum)= (long length of fulcrum) x 33

suineg| PUZZLE MASTER | Profile December 10th, 2007 - 10:50 pmok… so 48 pounds if she is in the short end, but assume she is sitting in the same short end only that she balance with the books on the long end

48-33=15 pounds of difference, the distance goes from 3 to 1 against weight, so 5 units of distance less, i think you lose 1 unit of distance because of the center of gravity 5 books, 3 persons

the only approach i see is that the center of gravity is moved a little bit, because there can be 3 persons and 5 books in line so thats the proportion still did get the answer, i think too many assumptions i have made already

suineg| PUZZLE MASTER | Profile December 11th, 2007 - 9:04 amwow RK, nice, but I still do not now why you used the formula of “torque” assuming that when the books were in the long end of the right or left side, the girl is sitted in the short end of the left or right and viceversa.

there has to be an explanation but i had not been able to figure that out

suineg| PUZZLE MASTER | Profile December 11th, 2007 - 9:12 amIts like you had to keep the same distance between the books and the girl, the only difference is the distance between the center of the teeter and the girl and the books, however i think the problems do not state that, the distance between could be more or less, but cool.

Lath| Profile December 11th, 2007 - 11:32 amThe books weigh 3 pounds as someone worked out above.

The pivot is not in the center of the plank.

To balance, the (weight * distance) on one side of the pivot must equal the (weight * distance) on the other side, (as seen in RK’s equations above). Thus a smaller weight further away can balance a smaller weight closer to the pivot. This is why an adult and a child can use a teeter totter if the adult sits closer to the center than the child.

So if we say that:

S = length of short end

L = length of long end

W = weight of girl, we have the two equations that RK gave:

(1) W * L = 48 * S

(2) W * S = 33 * L

If we want to find W, we have to eliminate the other two variables.

So, convert both equations to have (L / S) on one side.

The first equation becomes:

L / S = 48 / W

The second equation becomes:

W / 33 = L / S

Since 48 / W equals L / S and W / 33 equals L / S, we can set them equal to each other, giving:

W / 33 = 48 / W

Multiply both sides by W and 33 to get:

W * W = 1584

Take the square root to get

W = 39.8 lbs. (approximately)

Lath| Profile December 11th, 2007 - 11:45 amTo correct one sentence in my previous comment, in should have said,

“Thus a smaller weight further away can balance a larger weight closer to the pivot.”

Lath| Profile December 11th, 2007 - 11:51 amAlso, we can then solve for L / S:

L / S = 48 / W

L / S = 48 / 39.8

L / S = 1.206

So, for example the long end might be 35 inches, and the short end about 29 inches.

Carpenter| Profile December 11th, 2007 - 6:42 pmIsn’t the system of equations………

Equation 1: (weight of girl) x (long length of fulcrum)=(short length of fulcrum)x(48)

Equation 2weight of girl) x (short length of fulcrum)= (long length of fulcrum) x 33

w*long = 48 * short => w = 48*short/long

w*short = 33*long => w = 33*long/short

48=33?

………

inconsistent?

How have we arrived at 39.8lbs?

RK| Profile December 11th, 2007 - 8:34 pmHi Suineg & Carpenter- check out Lath’s more detailed explanation, he does a nice job

suineg| PUZZLE MASTER | Profile December 11th, 2007 - 10:53 pmyes, Lath does a great explanation indeed, the center of mass moves relatively, thats why one distance its a function of the other distance, this one was tricky, but i learn about torque

Carpenter| Profile December 12th, 2007 - 6:35 pmYes…I see now. :x Excellent explanation Lath!

RK| Profile December 13th, 2007 - 2:03 pmHi there, Falwan- glad to hear you like the site

i’m not aware of another site like smartkit, though

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