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More ways to get Braingle...

Change for a Dollar

Math brain teasers require computations to solve.

 

Puzzle ID:#113
Fun:*** (2.26)
Difficulty:*** (2.63)
Category:Math
Submitted By:duckrocket**
Corrected By:TRILLIONAIRE

 

 

 



Assuming you have enough coins of 1, 5, 10, 25 and 50 cents, how many ways are there to make change for a dollar?

Open Calculator

Hint

There are more than 200.
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Comments

Nike01
Aug 24, 2001

wrong...192
Mogmatt16Ava*
May 02, 2002

explane please
dewtell*us
Jul 25, 2002

I get 292. Let N(a, c) be the number of ways
to represent an amount a using coins no larger than
value c. Then N(a, 1) = 1 (there is only one
way to represent any amount using only pennies), and N(0, c) = 1 (there
is only one way to represent an amount of zero, no matter what coins are available).
N(5n, 5) = n+1 (you can use anywhere from 0 to n nickels,
and then have to make the rest up in pennies).
N(10n, 10) = (n+1)^2 (the sum of N(0, 5), N(10,5), N(20, 5), ...,
N(10n, 5), which is the sum of the first n+1 odd numbers - it's
an old fact that this sum yields the perfect squares).
N(10n+5, 5) = (n+1)*(n+2).

Now go top down. We want N(100, 50). N(100, 50) = N(0, 25) + N(50, 25) + N(100, 25),
representing the use of 2, 1, and 0 fifty-cent pieces, respectively.
N(50, 25) = N(0, 10) + N(25, 10) + N(50, 10).
N(100, 25) = N(0, 10) + N(25, 10) + N(50, 10) + N(75, 10) + N(100, 10).
Using our previous facts about N(a, 10), we find that
N(100, 25) = 1 + 12 + 36 + 72 + 121 = 242; N(50, 25) = 1 + 12 + 36 = 49;
and finally, N(100, 50) = 1 + 49 + 242 = 292.
dewtell*us
Jul 25, 2002

Oops - reference to N(10n+5, 5) in the
previous comment should be N(10n+5, 10)
instead.
othbrady
Mar 07, 2003

good 1
GTregay*us*
Jan 21, 2005

I didn\'t take the time to follow the logic on this one - created an Excel spreedsheet, and it is indeed 292!
darthforman*
Apr 19, 2005



My head hurts.

It's the right answer but I don't really like the math ones. It takes too much work.

tdc_tmc
Aug 28, 2005

woooooooow!!!!! too much for me
hidentreasure**
Nov 29, 2005

Techer told me a while back. good teaser.
irar1792Aus*
Dec 31, 2005

AAAAAAAAAAAAAAAAAAggggggggghhhhhhhhhhhhh
blackmarket69**
Mar 08, 2006

coins are shiny!!
i like things that r shiny
MethlosAgb*
Mar 12, 2006

it would be the same if you used GB pence
spazzyjazzy93Aus*
Apr 05, 2006

i got that but only because i read that fact on a snapple cap
righteouschick2*
Apr 08, 2006

who has time for that stuff?¿?
banzai*
Jul 03, 2006

my answer was 'alot"!
hsbanana90*us
Aug 20, 2006

that was hard. it would take forever to figure that out, unless you knew how to do the calculator functions.
josty*us*
Aug 27, 2006

easy but long
ari_1Aus*
Sep 04, 2006

wooooah .....umm i got that wrong!!
teasermaster**
Oct 13, 2006

Way to tough for me!
kristybubbaAus*
Dec 06, 2006

people call me the smart goddess of math but i aint that smart
it hurts my head
kristybubbaAus*
Dec 06, 2006

people call me the smart goddess of math but i aint that smart
it hurts my head
QTkateus*
Jan 15, 2007

um yeah well i agree with evryone else that was 2 hard foe my lil head
bgil7604Aus
Feb 23, 2007

Wow, you must have WAY too much time on your hands if you know that answer!
BrockAus*
Jul 30, 2007

nice 1
BrockAus*
Jul 30, 2007

nice 1
SuperDude512*
Jul 31, 2007

wow i jus realized how bad i am at math, NICE TEASER KID!
XosyaraihaAus*
Aug 08, 2007

they tried to ask that to my math class... and didn't succeed. glad SOMEONE could get the answer

thumbs up
srpwuzhereAus*
Nov 23, 2008

I watched a Spongebob Squarepants episode where Spongebob was talking about change for $1 and $5. hehe.
javaguru*us*
Jan 11, 2009

I did it similar to dewtell, slightly less formally.

The real break is between using dimes and smaller and using quarters and larger. So I broke the problem into making 100, 75, 50, 25 or 0 cents from dimes and smaller.

To make 100 cents from dimes and smaller start with 10 dimes and replace from 0 to 10 dimes with two 5s. Each 5 can be made from either a nickel or 5 pennies, so there is 1 way to replace 0 dimes, 3 ways to replace 1 dime, 5 ways for 2 dimes... giving the sum of the first 11 odd numbers, which as dewtell points out is 11^2 = 121. So making 100 cents with only dimes or less is 121.

75 cents has a 5 and 7 10s. From 0 to 7 10s can be replaced with two 5s. The extra 5 means that there is one extra way to make each value (because the first 5 can either be a nickel or 5 pennies). This gives 8^2 + 8 = 72 ways to do this. There is one way to make the 25 so there are 72 ways with 75 cents made from dimes and smaller.

50 cents is 6^2 = 36 times the two ways to make the 50 (50, 25 x 2) = 72.

25 cents is 3^2 + 3 = 12 times the two ways to make 75 (3x25, 50+25) = 2 x 12 = 24.

There are 3 ways to make a dollar with no dimes and pennies (2x50, 50 + 2x25, 4x25) = 3.

121 + 72 + 72 + 24 + 3 = 292

Jessi-watau*
Oct 20, 2011

urgh.... math makes my head hurt soooo bad



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