John purchased large bottles of water for $2 each and small

Question

John purchased large bottles of water for $2 each and small bottles of water for $1.50 each. What percent of the bottles purchased were small bottles?

(1) John spent $33 on the bottles of water
(2) The average price of bottles purchased was $1.65

Bunuel · Accepted Answer

John purchased large bottles of water for $2 each and small bottles of water for $1.50 each. What percent of the bottles purchased were small bottles?

Say John purchased x large bottles and y small bottles.

(1) John spent $33 on the bottles of water -->  2x+1.5y=33  -->  4x+3y=66 . Several integer solutions possible to satisfy this equation, for example  x=15  and  y=2  OR  x=12  and  y=6 . Not sufficient.

(2) The average price of bottles purchased was $1.65 -->  \frac{2x+1.5y}{x+y}=1.65  -->  2x+1.5y=1.65x+1.65y  -->  0.35x=0.15y  -->  \frac{y}{x}=\frac{35}{15} , we have the ratio, which is sufficient to get the percentage.

Just to illustrate  \frac{y}{x+y}=\frac{35}{15+35}=\frac{70}{100} .

Answer: B.

kuttingchai · Answer

got B

(A)

2(l) + 1.5(s) = 33

not sufficient

(B)

/ (l+s) = 1.65

2l + 1.5s = 1.65 (l+s)

2l - 1.65l = 1.65s - 1.5s

0.35(l) = 0.15 (s)

35/15 = s/l

% of small bottles

=35/ (35+15)*100
= 70%

sufficient

- bookmarking for future reference

fameatop · Answer

Hi Bunuel,

There is small error in one of the calculations.
 x=15  and    y=3   
should be
 x=15  and    y=2

Kindly correct me if i am wrong.

Bunuel · Answer

Typo edited. Thank you. +1.

Javoni · Answer

Let x be the number of large bottles of water and y be the number of small bottles of water, from the question stem we get:
2*x+1.5*y=33, thus 1 is INSUFFICIENT.
Mowing to 2 condition:
(2*x+1.5*y)/(x+y) = 1.65 ------>>>> 2x+1.5y = 1.65x+1.65y, from here we easily get that 7x=3y, OR x = 3y/7 now we now x we can easily find the ratio of y in total of bottles: y/(y+3y/7) = 7/10 or 70%

Please correct me, if I went awry. Actually we do not need the solution, as it is data sufficiency  so 2 is SUFFICIENT

EvaJager · Answer

Dealing with Statement (2): remember weighted average (also used when dealing with mixture problems).

If we have  N_1  numbers with average  A_1 , and  N_2  numbers with average  A_2 , the final average being A, then the differences between the final average and the initial averages are inversely proportional to the two numbers of numbers (assume  A_1>A_2 ):

(A_1-A)N_1=(A-A_2)N_2  or  \frac{A_1-A}{A-A_2}=\frac{N_2}{N_1} .

This follows from the equality  \frac{N_1A_1+N_2A_2}{N_1+N_2}=A.

In our case we know  A, A_1,A_2  so we can find the ratio  \frac{N_2}{N_1}  and then, obviously  \frac{N_2}{N_1+N_2} .

mvictor · Answer

1. small bottles can be 2, and 15 large or 6 small and 12 large. 1 insufficient.
2. 2L+3S/L+S = 1.65
we are given proportions. thus, we can solve the question.
2L+1.5S=1.65L + 1.65S
0.35L=0.15S.
multiply by 100
35L=15S.
S/L = 35/15

S/S+L = 35/15+35 = 35/50 or 7/10. 70%

rohit8865 · Answer

(1)Not suff.
(2) I will use weighted avg. method here
W1/W2=(A2-Avg.)/(Avg.-A2)
W1=No. of large bottles
W2=No. of large bottles
A1=2
A2=1.5
Avg.=1.65
W1/W2=3/7
%age of small bottles= 7/(3+7)------>7/10----70%
Ans B

JeffTargetTestPrep · Answer

We are given that John purchased large bottles of water for $2 each and small bottles of water for $1.50 each. We can let s = the number of small bottles purchased, and b = the number of larger bottles purchased. We need to determine what percentage of the bottles purchased were small bottles, i.e. the value of s/(s+b) x 100.

Statement One Alone:

John spent $33 on the bottles of water.

Using the information in statement one, we can create the following equation:

2b + 1.5s = 33

We can multiply the entire equation by 2 and we have:

4b + 3s = 66

4b = 66 - 3s

4b = 3(22 - s)

b =  /4

Since b must be an integer, 3(22 - s) must be a multiple of 4.

3(22 - s) is a multiple of 4 when s = 2, 6, 10, 14, or 18.

Since we have multiple values of s, we will also have multiple values of b, and thus we do not have enough information to answer the question.

Statement Two Alone:

The average price of bottles purchased was $1.65.

Using the information in statement two, we can create the following equation:

1.65 = (2b + 1.5s)/(b + s)

1.65(b + s) = 2b + 1.5s

165(b + s) = 200b + 150s

165b + 165s = 200b + 150s

15s = 35b

3s = 7b

(3/7)s = b

We can now determine the value of s/(s+b) x 100 by substituting (3/7)s for b:

s/  x 100

1/(10/7) x 100

7/10 x 100 = 70

So the small bottles account for 70 percent of the bottles purchased. We have answered the question.

Answer: B

Kritisood · Answer

hey  Bunuel   VeritasKarishma  could you please assist me with st 1.
when i take it as follows:

2x+1.5y=33
20x+15y=330

for this equation, only two valid solutions are there (as per the exchange rate method)
x=0 and y=22 
now adding 15 and subtracting 20 to get another sol:
x=15 and y=2
i cant subtract or add further.

no other sol is valid as per me, hence i had marked A as sufficient. but when i convert the same into 4x+3y=66 then ofc many sols are valid. could pls help me with what I am doing wrong?

KarishmaB · Answer

The equation needs to be in lowest terms to get all the possible solutions. When it is a multiple, you lose out on solutions.
Note what happens when you make it 40x + 15y = 660 and try to solve for x and y.

Kritisood · Answer

of course! you cover this in the blog as well. my mistake. thanks a lot for the clarification.

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John purchased large bottles of water for $2 each and small

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