Gummy Worms Sale

Number Theory Level 2

Calvin was at his local grocery store, and saw that they had gummy worms for sale. To satisfy his foreseeable sugar craving, he bought 72 bags of gummy worms. Upon exiting the store, he noticed that the ink on the receipt was smudged, and he could see the price as $\$ \_ 3.4\_$ . What is the price (in cents) of 1 bag of gummy worms?

Details and assumptions

A bag of gummy worms cost a whole number of cents.

The tens digit of the dollars is missing, and the units digit of the cents is missing. The receipt was for less that $100.

The answer is 102.

7 solutions

Ghanashyam Chakravarthi
Nov 7, 2013

A better method would be as follows:

let the cost per bag be x cents.

i have indicated a blank using -

we have 72x cents=$-3.4-

let k = -34- =72x

k should be divisible by 8, so the second blank is 4. (since 344 is a multiple of 8)

k should be divisible by 9, so the first blank is 7. (since sum of digits must be a multiple of 9)

Thus the total is 7344 cents thus x= 102 cents!!

Moderator note:

This is a comment written by Ghanashyam. As it is insightful and explains a neat approach to the problem, I have converted it into a solution.

Calvin sir, where in the solution is $k$ ? It is not used anywhere, but still there is "k should be divisible by 8". Please specify!

Akshat Jain - 7 years, 7 months ago

It is $x$ or $k$ whatever-same thing. :)

A Brilliant Member - 7 years, 7 months ago

Thanks for pointing that I should define my variables.

$k$ was used to denote the term -34-. I accidentally removed that when I was editing the question.

Calvin Lin Staff - 7 years, 7 months ago

Okay! :)

Akshat Jain - 7 years, 7 months ago

sir also in the solution, it should read as "sum of digits must be a multiple of 9" and not 8

Ghanashyam Chakravarthi - 7 years, 7 months ago

Andre Francesco
Nov 6, 2013

There is only one multiple of 72 that has 34 in it that is under 10000. It must be under 10000 because it is given that it is under $100. With 7344 being a multiple of 72. $73.44 is the final bill. Because each bag is an equal price you can do 7344/72 which equals 102.

Ajmal Siddiqui
Nov 3, 2013

First of all, converting the price into cents, we get 34 cents. Now let x be the price of one bag. Calvin got 72 bags. So 72x = 34 cents. Now we need to find a multiple of 72 which has 3 and 4 as its middle digits. By trail and error, this can be found to be 102, which is the answer. Thus, each bag costs 102 cents

A better method would be as follows:

let the cost per bag be $x.(i have indicated a blank using -)

we have 72x=$-3.4-

let -3.4-=k (for ease of working only)

k should be divisible by 8, so the second blank is 4. k should be divisible by 9, so the first blank is 7. we have k=73.44 thus x=$1.02 or 102 cents!!

Ghanashyam Chakravarthi - 7 years, 7 months ago

Good observation! I've converted this into an actual solution, so that others can learn from it.

Calvin Lin Staff - 7 years, 7 months ago

what is the problem with 23.46,33.42,63.42 and many more as the price of 72 bags

Pratyush Agarwal - 7 years, 7 months ago

The three amounts that you gave are not valid, as they do not satisfy the condition given in the question, which says-

A bag of gummy worms cost a whole number of cents.

If you divide the numbers given by you into 72 gummy bears, you will not end up with the price of one gummy bear as a whole number. So, the random numbers you generated are invalid.

Hope that helped, Pratyush!

Akshat Jain - 7 years, 7 months ago

i thought $1= 100cents

Vishnu M - 7 years, 7 months ago

Shreyas Shastry
Mar 11, 2014

1$ = 100 cents

Calvin bought 72 bags,

72x=-34- cents, where x is the cost of one bag of gummy worms.

By trial and error method,

72*102= 7344

Hence one bag of gummy worms cost 102 cents

Leonardo Kumala
Nov 8, 2013

First, we know that the price of 72 bags of gummy worms is $ 3.4

If x = the tens digit of the dollar and y is the units digit of the cents, the price will be $x3.4y

x34y could be divided by 72, so x34y could be divided by 8 and 9

8 is 2^3 so 34y could be divided by 8, so y is 4

x+3+4+4 could be divided by 9 so x is 7

7344 divided by 72 is 102

Bryan Asuncion
Nov 8, 2013

72 x 102 = 7344

Arnav Shringi
Nov 6, 2013

1$ = 100 cents

Calvin bought 72 bags,

72x= 34 cents , where x is the cost of one bag of gummy worms.

By trial and error method,

                           72x102= 7344

Hence one bag of gummy worms cost 102 cents

Gummy Worms Sale

The answer is 102.

7 solutions

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