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A certain customer at a health food store purchased organic bananas at

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A certain customer at a health food store purchased organic bananas at  [#permalink]

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A certain customer at a health food store purchased organic bananas at a price of $.70 each, and conventional bananas at a price of $.60 each. How many total bananas did the customer purchase, if he purchased both organic and conventional bananas?

(1) In total, the customer spent $5.60 on bananas
(2) The customer purchased conventional and organic bananas in the ratio of 7:2 respectively

 

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Re: A certain customer at a health food store purchased organic bananas at  [#permalink]

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New post 15 Jul 2019, 07:10
2
Let no. of organic bananas be x and no. of conventional bananas be y
0.7x +0.6y = 5.6 --- statement 1 (insufficient) 2 solutions possibles x=2, y=7 & x=8, y=0

y/x = 7/2 --- statement 2 (alone insufficient)

Together sufficient ( 2 variables, 2 equations)
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A certain customer at a health food store purchased organic bananas at  [#permalink]

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New post Updated on: 16 Jul 2019, 05:20
1
A certain customer at a health food store purchased organic bananas at a price of $.70 each, and conventional bananas at a price of $.60 each. How many total bananas did the customer purchase, if he purchased both organic and conventional bananas?

(1) In total, the customer spent $5.60 on bananas
(2) The customer purchased conventional and organic bananas in the ratio of 7:2 respectively

Given: A certain customer at a health food store purchased organic bananas at a price of $.70 each, and conventional bananas at a price of $.60 each
Asked: How many total bananas did the customer purchase, if he purchased both organic and conventional bananas?

(1) In total, the customer spent $5.60 on bananas
.7x + .6y = 5.6
7x + 6y = 56
\(x = \frac{56-6y}{7}\)
and x & y are both integers
(56/7,0), (50/7,1), (44/7,2), (38/7,3), (32/7,4),(26/7,5), (20/7,6), (14/7=2,7), (8/7,8), (2/7,9) are possible solution but x&y need to be integers
It gives x = 2 & y=7 (2,7) as the only solution
Total bananas = x+y = 2+7=9
SUFFICIENT

(2) The customer purchased conventional and organic bananas in the ratio of 7:2 respectively
Let us assume that conventional bananas purchased = 7x & organic bananas purchased = 2x
7x *.6 + 2x * .7 = total amount spent
Since any $ amount is not mentioned, value of x can not be derived.
e.g 7 + 2 =9 is valid
But 14 + 4 =18 is also valid
NOT SUFFICIENT

IMO A

Originally posted by Kinshook on 15 Jul 2019, 07:15.
Last edited by Kinshook on 16 Jul 2019, 05:20, edited 2 times in total.
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Re: A certain customer at a health food store purchased organic bananas at  [#permalink]

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New post 15 Jul 2019, 08:02
1
Let the number of organic bananas purchased be \(a\).
Let the number of conventional bananas purchased be \(b\)

Question: \(a + b ?\)

(1) In total, the customer spent $5.60 on bananas
Given \(0.7a+0.6b=5.6\) --> \(7a+6b=56\).
After several iterations, only two following integer pairs of (a,b) satisfy the above equation: (8,0) and (2,7).
As we are told that the customer purchased both organic and conventional bananas, the first pair is not applicable.
Therefore, \(a=2, b=7\) and \(a+b=9\)
(1) is SUFFICIENT

(2) The customer purchased conventional and organic bananas in the ratio of 7:2 respectively
Given \(a:b=2:7\), an infinite number of integer pairs of (a,b) can satisfy the equation: (2,7), (4,14), (6,21), etc.
Therefore, there are infinite possibilities of what the outcome of \(a + b\) is.
(2) is NOT SUFFICIENT

Answer is (A)
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A certain customer at a health food store purchased organic bananas at  [#permalink]

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New post Updated on: 15 Jul 2019, 19:12
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We can get rid of decimals to ease our calculation. Organic bananas = \($7\). Conventional bananas = \($6\). What is the total number of bananas if the customer purchased BOTH?

ST1. In total, the customer spent $56 (get rid of decimals) on bananas. How many organic bananas or conventional bananas we need so that the total sum is $56? Can the ratio of two bananas be different for $56? The easiest way is to check manually:

What if we have 1 organic banana? If so, \(56-7=49\) must be divisible by 6, is it? Not, so out.
What if we have 2 organic bananas? If so, \(56-14=42\) must be divisible by 6, is it? Yes, the ratio of bananas can be \(2:7\)

We need to make sure that we don't have other ratios so that ST1 can be sufficient.

When 3 organic: \(56-21=35\) not divisible by 6
When 4 organic: \(56-28=28\) not divisible by 6
When 5 organic: \(56-35=21\) not divisible by 6
When 6 organic: \(56-42=14\) not divisible by 6
When 7 organic: \(56-49=7\) not divisible by 6
When 8 organic: Not possible because we need both bananas.

Hence the ratio must be \(2:7\) and she must have purchased overall \(9\) bananas. Sufficient

ST2. The customer purchased conventional and organic bananas in the ratio of 7:2 respectively. We already know that from ST1. ST2 alone doesn't give us enough information. The only thing we know for sure is that the total number must be divisible by 9. It can be 9, 18, 27, etc. Insufficient

Hence A
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Originally posted by JonShukhrat on 15 Jul 2019, 08:04.
Last edited by JonShukhrat on 15 Jul 2019, 19:12, edited 1 time in total.
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Re: A certain customer at a health food store purchased organic bananas at  [#permalink]

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New post 15 Jul 2019, 08:23
1
price of organic banana = 0.70
price of conventional banana = 0.60

he purchased both bananas (given)

How many total bananas did the customer purchase?


STATEMENT (1) In total, the customer spent $5.60 on bananas

let the customer buy x(organic banana) and y (conventional banana)

then 0.70x+0.60y = 5.60
7x+6y = 56
7x = 56-6y
x = 8-\(\frac{6y}{7}\) (1)
now y cannot be fraction since y is number of banana so y must be integer and multiple of 7
only y=7 satisfy the equation (1) (other values will give negative values of x which cant be possible AND we cant take y=0 since customer buy both banana)
so from here we get
conventional banana =7
organic banana = 2 (solving equation 1 for y=7 we get x=2)
Total 9 bananas
SUFFICIENT

STATEMENT (2) The customer purchased conventional and organic bananas in the ratio of 7:2 respectively
let conventional banana = 7x
organic banana = 2x
total banana = 9x
total banana can be = 9,18,27........(since x can be 1,2,3,4,5.......)
so INSUFFICIENT

A is the answer
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Re: A certain customer at a health food store purchased organic bananas at  [#permalink]

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New post 15 Jul 2019, 08:28
1
A certain customer at a health food store purchased organic bananas at a price of $.70 each, and conventional bananas at a price of $.60 each. How many total bananas did the customer purchase, if he purchased both organic and conventional bananas?

(1) In total, the customer spent $5.60 on bananas
(2) The customer purchased conventional and organic bananas in the ratio of 7:2 respectively

Solution:

Question Stem Analysis:

We have been provided per unit with the prices of organic and conventional bananas, we can take X as the number of organic bananas and Y as the number of conventional bananas.
From above, we can formulate the equation : 0.70X + 0.60Y= Total price of bananas , we need to ind out X+ Y.

Statement One Alone:

From statement one, we can derive that 0.70X + 0.60Y = 5.60 ,i.e 7X + 6Y =56 (multiplying 10 from both sides)
we can try different values for X & Y , their total must be 56. Only X=2 & Y = 7 satisfies this equation, so we have a value for X+Y i.e 9.
Hence Statement One alone is Sufficient. Eliminate C & E.

Statement Two alone:

we get a ratio between X & Y, Y=\(\frac{7}{2}\) X,
However, without any further information about X & Y , or the total cost, we cannot derive at the value of X+ Y.
Hence statement two alone is insufficient.

Answer is A.
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Re: A certain customer at a health food store purchased organic bananas at  [#permalink]

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New post 15 Jul 2019, 10:16
1
A certain customer at a health food store purchased organic bananas at a price of $.70 each, and conventional bananas at a price of $.60 each. How many total bananas did the customer purchase, if he purchased both organic and conventional bananas?

Let O be number of organic bananas and C be number of conventional bananas.

We know that O and C are both positive integers because number of banans cannot be fractional or negative. Also, it is given that he purchased both, so they are not equal to 0.

Total amount = O*0.7 + C*0.6

We have to find O + C

(1) In total, the customer spent $5.60 on bananas

0.7*O + 0.6*C = 5.6

O has to be 7 or lower ....because if O was 8, total cost would be more than 5.6$ (minimum value of C is 1)
Similarly, C has to be 8 or lower... because if C was 9, total cost would be more than 5.6$ (minimum value of O is 1)

Possible values of O - 7,6,5,4,3,2,1
Possile values of C - 8,7,6,5,4,3,2,1

Lets try all possible values of O to see which combinations allow Total = 5.6

O ____C____ Total
7 ____1____ 5.5..... O=7 is not possible. If O is 7, C =1 is closest to bringning total to 5.6. No value of C will bring total to exactly 5.6.
6 ____2____ 5.4..... O=6 is not possible. If O is 6, C =2 is closest to bringning total to 5.6. No value of C will bring total to exactly 5.6.
5 ____3____ 5.3..... O=5 is not possible. If O is 5, C =3 is closest to bringning total to 5.6. No value of C will bring total to exactly 5.6.
4 ____4____ 5.2..... O=4 is not possible. If O is 4, C =4 is closest to bringning total to 5.6. No value of C will bring total to exactly 5.6.
3 ____5____ 5.1..... O=3 is not possible. If O is 3, C =5 is closest to bringning total to 5.6. No value of C will bring total to exactly 5.6.
2 ____7____ 5.6..... CORRECT
1 ____8____ 5.5..... O=1 is not possible. If O is 1, C =8 is closest to bringning total to 5.6. No value of C will bring total to exactly 5.6.

There is only 1 combination of positive integers that brings total to 5.6$

So O =2, and C = 7,

(1) IS SUFFICIENT


(2) The customer purchased conventional and organic bananas in the ratio of 7:2 respectively

C/O = 7/2

C and O could be 7 and 2 respectively

C and O could be 14 and 4 respectively

Cannot narrow down to 1 value

(2) IS NOT SUFFICIENT


ANSWER: A - (1) Alone is SUFFICIENT
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Re: A certain customer at a health food store purchased organic bananas at  [#permalink]

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New post 15 Jul 2019, 11:14
1
Let number of organic bananas that were purchased= a, where a>0
Number of organic bananas that were purchased= b, where b>0

Total number of bananas = a+b

Statement 1
Total money spent by the customer= 0.7a+0.6b=5.60
or 7a+6b=56
6b=56-7a

As 56-7a=7*(8-a) is divisible by 7, hence 6b or b must be divisible by 7.
b can't be 0, and if b is greater than 10, a will be negative.
Hence only value b can take is 7.
a=(56-42)/7=2
Total number of bananas= 2+7=9
Sufficient

Statement 2
a:b=7:2
(a,b)= (7, 2), (14, 4), (21, 6) and so on
Total number of bananas can be 9, 18, 27....9k.
Insufficient

IMO A
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Re: A certain customer at a health food store purchased organic bananas at  [#permalink]

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New post 15 Jul 2019, 11:32
1
"A certain customer at a health food store purchased organic bananas at a price of $.70 each, and conventional bananas at a price of $.60 each. How many total bananas did the customer purchase, if he purchased both organic and conventional bananas?

(1) In total, the customer spent $5.60 on bananas
(2) The customer purchased conventional and organic bananas in the ratio of 7:2 respectively"


In this type of questions we should beware of tempting answer C. It is clear that both statements together are sufficient, because 7*0,6 + 2*0,7 = 5,6.
But what if the first statement alone is sufficient? It's clear that the second statement is not sufficient, because the customer could purchase 9 bananas or 18 bananas, as well as 27 bananas or more.

We know that the customer purchased at least one organic (O) and one conventional (C) banana. And from the first statement we know that 0,6*C + 0,7*O = 5,6, where both C and O are positive integers.
Let's try to figure out whether there is a possibility that positive integer C is not equals 7, while the statement 0,6*C + 0,7*O = 5,6 (where both C and O are positive integers) is valid.

If C = 1, then 0,7*O (O - positive integer) must be equals 5. There is no such integer O.
If C = 2, then 0,7*O (O - positive integer) must be equals 4,4. There is no such integer O.
Ans so on...

There is no such possibility and C must be equals 7, otherwise the equation 0,6*C + 0,7*O = 5,6 is not valid.
So from the first statement we can unhesitatingly conclude that the customer purchased 9 bananas: 7 conventional and 2 organic.

The answer is A
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Re: A certain customer at a health food store purchased organic bananas at  [#permalink]

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New post 15 Jul 2019, 11:57
1
Price per organic banana=0.7 and price per conventional banana=0.6. Let x and y represent organic and conventional bananas respectively, while t represent total bananas bought.
Total cost of bananas = 0.7x+0.6y
x+y=t
From statement 1: we know that 0.7x+0.6y=5.6
This also means 7x+6y=56
Now if we can find a combination of two non-zero integers for x and y such that 7x+6y=56, then t=x+y
Non zero because the question says categorically that the customer purchased both bananas.

A good place to look is to take a look at the the factors of 56 which are: {1,2,4,7,8,14,28,56}

From the set, when x=2, and y=7, we can tell that the total bananas bought equals 9, since 0.7(2)+0.6(7)=5.6

The only other option available is if the customer bought only organic bananas and no conventional bananas. But from the question, the customer bought both, hence we can disregard this possibility.

Statement 1 alone is sufficient.

Statement 2: x:y=7:2
This means that x=(7/9)*t
and y=(2/9)*t
This is clearly insufficient since the ratio x:y=7:2 can be expressed in many forms such as 14:4, 21:6, 28:8 with corresponding t equals 9,18,27,36. Hence statement 2 on its own is insufficient.

The correct answer therefore is A.

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A certain customer at a health food store purchased organic bananas at  [#permalink]

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New post 15 Jul 2019, 14:36
1
Question: A certain customer at a health food store purchased organic bananas at a price of $.70 each, and conventional bananas at a price of $.60 each. How many total bananas did the customer purchase, if he purchased both organic and conventional bananas?

(1) In total, the customer spent $5.60 on bananas. SUFFICIENT
- The question states that both organic and conventional bananas were purchased, therefore neither quantity can be equal to zero.
- To make things easier, instead of working with .70, .60, or 5.60 change your values to whole numbers 7, 6, 56 respectively.
- Factors of 56 are as follows (1,2,4,7,8,14,28,56).
- 7 goes into four of 56's factors, (7, 14, 28, and 56).
- Organic quantities of (7, 14, 28, and 56) can be compared to multiples of Conventional.
- Any multiple of 6 that goes evenly into 56 after one of the four organic quantities (7, 14, 28, and 56), is correct.
\[
\begin{matrix}
Organic & + & Conventional & = & Total Cost &y = ? & Correct? & x+y \\
7x & + & 6y & = & 56 & & & \\
7(1) & + & 6y & = & 56 & y = 8.166 & Incorrect & \\
7(2) & + & 6y & = & 56 & y = 7 & Correct & 9\\
7(4) & + & 6y & = & 56 & y = 4.66 & Incorrect & \\
7(8) & + & 6y & = & 56 & y = 0 & Incorrect & \\
\end{matrix}
\]

(2) The customer purchased conventional and organic bananas in the ratio of 7:2 respectively. INSUFFICIENT
- Ratio multiples create multiple answer possibilities.
\[
\begin{matrix}
Ratio & Organic & + & Conventional & = & Total \\
y:x & 7x & + & 6y & = & x + y \\
7:2 & 7(2) & + & 6(7) & = & 9&\\
14:4 & 7(4) & + & 6(14) & = & 18\\
21:6 & 7(6) & + & 6(21) & = & 27\\
\end{matrix}
\]

Correct Answer: A
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Re: A certain customer at a health food store purchased organic bananas at  [#permalink]

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New post 15 Jul 2019, 16:52
1
A certain customer at a health food store purchased organic bananas at a price of $.70 each, and conventional bananas at a price of $.60 each. How many total bananas did the customer purchase, if he purchased both organic and conventional bananas?

(1) In total, the customer spent $5.60 on bananas
(2) The customer purchased conventional and organic bananas in the ratio of 7:2 respectively

Number of organic bananas = A
Price of organic bananas = $.7
Number of conventional bananas= B
Price of conventional bananas = $.6

customer bought both type of bananas so total bananas bought by customer = A+B


As per option, total money spent by customer on bananas is $5.6

=> .7A + .6B = 5.6
=> 7A + 6B =56
=> 6B= 56-7A
=> 2*3= 56-7A
=> Pls notice that coefficient of A is prime number and coefficient of B is even number so if we arrange equation, 56-7A should be even number and should be divisible by both 2 & 3. Only A=2 satisfies required condition.

So statement 1st is sufficient.

Let's consider statement 2nd.

(2) The customer purchased conventional and organic bananas in the ratio of 7:2 respectively

=> .7A+.6B = 7x+2y it can be anything. so statement 2nd is not sufficient.

Answer : A
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Re: A certain customer at a health food store purchased organic bananas at  [#permalink]

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New post 16 Jul 2019, 04:26
2
Let the number of organic bananas bought be o and that of conventional bananas be c
We need to find the value of (o+c).
It is also given that \(o,c\geq{1}\)

Statement 1:
0.7o+0.6c = 5.6
At first, we may think that this info is not sufficient.
However, let us substitute different values of o and c to satisfy this eqn (given that \(o,c\geq{1}\))

As 0.6c results in a number with an even tenth digit for any integral value of c, 0.7o should have an even tenth digit as the RHS (5.6) has an even tenth digit. So, o should have only even values to satisfy the equation. Just think about it!

o=2, c=0.42/.6=7; We get an integral value for c. Hence, this is a possible case.
o=4, c=0.28/.6; \(c\neq{Integer}\). Hence, not possible.
o=6, c=0.14/.6; \(c\neq{Integer}\). Hence, not possible.
o=8, c=0/.6=0; But we know that \(c\geq{1}\). Hence, not possible.
Higher values of o give negative values of c which are not possible.

Hence, the only possible solution is o=2, c=7. Therefore, o+c=9.
Statement 1 is sufficient.

Statement 2: \(\frac{c}{o}=\frac{7}{2}\)
We can't find the value of o+c just by knowing the ratio o:c.
Statement 2 is insufficient.

Hence, option (A).
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Re: A certain customer at a health food store purchased organic bananas at  [#permalink]

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New post 16 Jul 2019, 05:49
1
The price of Organic bananas - $0.7 each
The price of Conventional bananas - $0.6 each

And The customer purchased at least one banana from both types of bananas

Statement1: in Total, The customer spent $5.6 on bananas
The equation will be 0.7x+0.6y=5.6

(x- the number of Organic bananas
y-the number of Conventional bananas)

In order to find the total bananas, we need the ratio of x and y:
if
x=1, 0.6y=5.6-0.7=4.9
y=4.9/0.6 (y cannot be integer, y should be integer)
x=2, 0.6y=5.6-1.4=4.2
y=7 (integer, we'll keep it).
We have to check the all possibility, because if two answers satisfies, statement will be insufficient)
x=3, 0.6y=5.6-2.1=3.5 (y cannot be a integer)
x=4, 0.6y=5.6-2.8=2.8 (y cannot be a integer)
x=5, 0.6y=5.6-3.5=2.1 (y cannot be a integer)
x=6, 0.6y=5.6-4.2=1.4 (y cannot be a integer)
Yes, we have just one ratio of x and y that will satisfies the equation. 2 organic bananas and 7 conventional bananas.
Sufficient.

Statement2:
Conventional / organic bananas = 7 : 2 respectively
Given ratio of two types bananas is good, but we don't have a total amount expenses on bananas.
That's why, The total number purchased bananas could be:
7+2 =9
14+4=18
21+6=27 and so on...
Insufficient

The Answer choice is A.
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Re: A certain customer at a health food store purchased organic bananas at  [#permalink]

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New post 16 Jul 2019, 06:28
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A certain customer at a health food store purchased organic bananas at a price of $.70 each, and conventional bananas at a price of $.60 each. How many total bananas did the customer purchase, if he purchased both organic and conventional bananas?

(1) In total, the customer spent $5.60 on bananas
Customer purchased both kinds of bananas a (organic) and b (conventional) and positive integers, we can consider:
0.7a + 0.6b = 5.6
By guessing possible numbers, we can get that 2 (a-organic) and 7 (b-conventional)
This condition makes A sufficient for us.


(2) The customer purchased conventional and organic bananas in the ratio of 7:2 respectively
Could be a lot of different situations, so the customer could buy:
7a+2b = 9 (a=1, b=1)
7a+2b=18 (a=2, b=2)
7a+2b=27 (a=3, b=3)
7a+2b=36 (a=4, b=4) etc...
So this condition is not sufficient to find exact quantity bought bananas.

A is the answer :heart
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Re: A certain customer at a health food store purchased organic bananas at   [#permalink] 16 Jul 2019, 06:28
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