At a certain fruit stand, oranges cost $0.35 each and grapefruits cost

Question

At a certain fruit stand, oranges cost $0.35 each and grapefruits cost $0.45 each. In a single transaction, Jean bought some oranges and some grapefruits. How many oranges did Jean buy in the transaction?

(1) Jean spent a total of $1.60 for the oranges and grapefruits at the stand.

This implies that

Since both "o" (oranges) and "g" (grapefruits) must be positive integers, this equation has only one set of values that satisfy it: o = 2 and g = 2. Sufficient.

(2) Jean bought an equal number of oranges and grapefruits.

This is clearly insufficient.

Answer: A.

Bunuel · Accepted Answer

At a certain fruit stand, oranges cost $0.35 each and grapefruits cost $0.45 each. In a single transaction, Jean bought some oranges and some grapefruits. How many oranges did Jean buy in the transaction?

(1) Jean spent a total of $1.60 for the oranges and grapefruits at the stand.

This implies that

35o + 45g = 160
7o + 9g = 32

Since both "o" (oranges) and "g" (grapefruits) must be positive integers, this equation has only one set of values that satisfy it: o = 2 and g = 2. Sufficient.

(2) Jean bought an equal number of oranges and grapefruits.

This is clearly insufficient.

Answer: A.

Kinshook · Answer

@najanapatGiven: At a certain fruit stand, oranges cost $0.35 each and grapefruits cost $0.45 each. In a single transaction, Jean bought some oranges and some grapefruits.
Asked: How many oranges did Jean buy in the transaction?
Let the number of oranges and grapefruits be o & g respectively

(1) Jean spent a total of $1.60 for the oranges and grapefruits at the stand.
.35o + .45g = 1.6
(o, g) = (2,2) since o & g are integers
SUFFICIENT

(2) Jean bought an equal number of oranges and grapefruits.
o=g
The information is not sufficient since absolute value of total purchase is not provided
NOT SUFFICIENT

IMO A

Vanshikakataruka · Answer

In such type of eqns how do we know that an eqn as in statement A will have only one answer? Is there a way to know cz trial and error might take up lots of time

Bunuel · Answer

Trial and error, combined with some number sense, is pretty much the only way. It's actually not that complex. You simply need to check if 32 minus a multiple of 9 results in a multiple of 7. We get 32 - 2*9 = 14 = 2*7.

Rokrrishi · Answer

Vanshikakataruka  you could just write the multiples of 35 and 45 and check if more than a pairs add to 160. If there are two or more pairs, the statement is insufficient, and if there is just one pair (as here, 70 + 90, which makes two oranges and two grapefruits), the statement is sufficient.

I have observed that many problems can very easily be solved just by listing out the possibilities (except for in probability and in combinations, in which this approach may be more cumbersome).

Posted from my mobile device

Pratidwandi · Answer

You're given an onscreen calculator for the data section, so using that reduces time for trial and error.

MarkGMATMentor · Answer

To avoid lengthy guessing and checking on questions like this, use the concept of exchange with the lowest common multiple.

Let's say we're working on Statement 1, and we've already seen that 2 oranges and 2 grapefruits total to $1.60, but we're unsure if there is another combination that works. Since both oranges and grapefruits must be integers, the only way for the total to remain at $1.60 is if a certain number of oranges are exchanged for a certain number of grapefruits, or vice versa. The total value of the oranges and the total value of the grapefruits would need to be equal.

What is that total value? It must be a multiple of $0.35 and a multiple of $0.45. The lowest common multiple of $0.35 and $0.45 will give us the smallest total value we can exchange.

Since 35 = 5 * 7 and 45 = 5 * 9, the LCM is 5 * 7 * 9 = $3.15. In other words, we can exchange 9 oranges, worth $3.15, for 7 grapefruits, worth $3.15. Because we only have 2 of each, there's no way to make the exchange, so 2 oranges and 2 grapefruits are the only combination that works.

The numbers here are small enough that guessing and checking works fine, but if the numbers were bigger, this becomes quite helpful. For example, let's say the total was $6.40 and you know that 8 oranges and 8 grapefruits will work (8*35 + 8*45). You could also exchange 7 grapefruits for 9 oranges to get 17 oranges and 1 grapefruit (17*35 + 1*45).

When two unknown values must be integers, see if you can exchange some of one for some of the other by using the lowest common multiple.

ohhno · Answer

Let:
 
 Oranges = xx 
 Grapefruits = yy  
Prices:
 
 Orange = $0.35 
 Grapefruit = $0.45  
We need to find  xx .
Statement (1)
Total spent = $1.60.
0.35x+0.45y=1.600.35x+0.45y=1.60
Multiply by 20:
7x+9y=327x+9y=32
Since x,yx,y are positive integers:
 
 y=1⇒7x=23y=1 \Rightarrow 7x=23, not integer 
 y=2⇒7x=14⇒x=2y=2 \Rightarrow 7x=14 \Rightarrow x=2  
So Jean bought  2 oranges and 2 grapefruits .
Statement 1  alone is sufficient .
Statement (2)
Equal number of oranges and grapefruits:
x=yx=y
This doesn't tell us the actual number. Could be 1 each, 2 each, 3 each, etc.
Statement 2  alone is not sufficient .
Answer:  (A)  ✅
 Statement 1 alone is sufficient, but statement 2 alone is not.

At a certain fruit stand, oranges cost $0.35 each and grapefruits cost

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At a certain fruit stand, oranges cost $0.35 each and grapefruits cost

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