A bag of 216 caramels is divided equally among a group of children. If

Question

A bag of 216 caramels is divided equally among a group of children. If 6 children were added to the group, each child would receive 3 fewer caramels. How many children are in the group?

A) 36
B) 24
C) 20
D) 18
E) 12

Archit3110 · Answer

from given info we know
216/x - 216/x+6 = 3
3x^2+18x-1296=0
x^2+6x-432=0
x^2+24x-18x-432=0
solve for x we get x= 18
option D

Shikhar22 · Answer

hi archit. Just curious how you got the last breakdown of 24 and 18. Took me a lot of time

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rsrighosh · Answer

it took a lot of time to bring the equation from  x^2 -6x-432  to  x^2 -18x+24x-432

Is there any simpler way to do this problem?

Fdambro294 · Answer

Common question set up seen in the official guide in several different scenarios involving rates

216 are divided evenly among X children

216/X = k

Where k = positive integer quotient

If 6 more ppl are added to the group, then each child receives -3 fewer candies

216 / (X + 6) = (k - 3)

Thus

(216 / X) - (216 / X + 6) = 3

(1 / X) - (1 / X + 6) = 1 / 72

At this point, rather than running through the quadratic equation, it’s easier to plug in the answer choices

D = 18

(1 / 18) - (1 / 24) = (24 - 18) / (18 * 24) = (6) / (18 * 24) = 1 / 72

Matches

Answer D 18

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AbRathod · Answer

In backsolving, I prefer to start with B and D, My goal is to get difference of 3 caramels from the two scenarios. (Acc. to given conditions)

B = Original 24..(216/24 = 9 Caramels) ; New 30..(216/30 = Not Possible) Hence Eliminate
D = Original 18..(216/18 = 12 Caramels) ; New 24..(216/24 = 9 Caramels) Difference is 3 caramels, hence our answer!

Kaichi · Answer

This is a very tedious question from the forum quiz.
It will be hard to finish <2 minutes unless you get the equation immediately.

216/x = y children - (1)
216/x-3 = y+6 children - (2)

Sub (1) into (2)
216/x-3 = 216/x + 6
216/x-3 - 216/x = 6
..... some manipulation later......
216(3)/x(x-3) = 6
108 = x^2 - 3x
0 = (x-12)(x+9)

Sub x = 12 into (1)
Y = 18.

JROYENG97 · Answer

Here is a relatively quick approach:

K: number of children
C: number of caramels

216 = K * C

Given the provided information, we can express the difference between the number of caramels before and after as:
216/K - 216/(K+6) = 3

As the sum of digits of 216 is divisible by 3, we know 216 is divisible by 3 and so we can simplify as:

72/k - 72/(K+6) = 1, rearranging the expression we get 72*6 = K(K+6).

As we know K is an integer, we can prime factorize 72*6 to figure out how to express it in the form K(K+6), which yields:

18*24 = K(K+6)

.: K = 18

The answer is D.

suntfacilis · Answer

I found an easier way to get to the answer without going through the quadratic equation.

1) Prime 216 = 6 * 9 * 4 =  3 * 3 * 3 * 2 * 2 * 2 
2) Both the original and new group sizes must divide 216 evenly (i.e., result in whole numbers of caramels per child)

Let's go over the answer choices and see which are NOT factors of 216.

A) 36 = 6 * 6  - can construct from primes of 216, keep 
B) 24 = 6 * 4  - can construct from primes of 216, keep  
C) 20 = 4* 5  -- not a factor as we don't have 5, eliminate 
D) 18 = 6 * 3  - can construct from primes of 216, keep  
E) 12 = 3 * 4  - can construct from primes of 216, keep

Now let's add 6 to each of the remaining options and see if  that  number is a factor of 216:
 A) 36 + 6 = 42 = 6 * 7  -- not a factor as we don't have 7, eliminate  
B) 24 + 6 = 30 = 6 * 5  -- not a factor as we don't have 5, eliminate  
C) ELIMINATED
D) 18 + 6 = 24 = 6 * 4  - can construct from primes of 216, keep  
E) 12 + 6 = 18 = 6 * 3   - can construct from primes of 216, keep

Now we are only left with 2 choices: 18 and 12.

Let's consider 18 first:
By looking at primes of 216, let's divide 216 by 18. I do it by crossing off the numbers needed to get 18:   3 * 3    * 3 *    2    * 2 * 2 --> 12 is the number of candy kids will get initially 
Add 6 and let's do the same for 24:   3 *   3 *  3 *    2     * 2     * 2    --> 9 is the number of candy each kid will get after adding 6 kids.  
 12 - 9 = 3 --> sufficient, but let's check the other choice

Doing the same for 12:
216 / 12 =   3   * 3  * 3 *  2     * 2    * 2 = 18 
216 / (12+6) = 12 (from above)
18 - 12 = 6, not our answer

Thus, our answer is D

A bag of 216 caramels is divided equally among a group of children. If

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A bag of 216 caramels is divided equally among a group of children. If

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

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