A cafe tracked the number of online orders over three consecutive days

Question

A cafe tracked the number of online orders over three consecutive days. If on the second day, the number of orders was greater than it was on the first day, and on the third day, the number of orders was less than it was on the second day, was the number of online orders on the third day greater than the number on the first day?

(1) The percent increase in the number of online orders from the first day to the second day was greater than the percent decrease in the number of online orders from the second day to the third day.

(2) The percent increase in the number of online orders from the first day to the second day was three times the percent decrease in the number of online orders from the second day to the third day.

Bunuel · Accepted Answer

GMAT Club Official Solution:

A cafe tracked the number of online orders over three consecutive days. If on the second day, the number of orders was greater than it was on the first day, and on the third day, the number of orders was less than it was on the second day, was the number of online orders on the third day greater than the number on the first day?

Let the percent increase from day 1 to day 2 be x, and let the percent decrease from day 2 to day 3 be y. If the number of online orders on the first day was N, then the number on the third day was

N(1 + x/100)(1 - y/100)

We want to know whether this was greater than N:

N(1 + x/100)(1 - y/100) > N

Since N > 0, divide both sides by N:

(1 + x/100)(1 - y/100) > 1

Expand:

1 + x/100 - y/100 - xy/10000 > 1

x - y - xy/100 > 0

So the question is whether x - y - xy/100 is positive.

(1) The percent increase in the number of online orders from the first day to the second day was greater than the percent decrease in the number of online orders from the second day to the third day.

This tells us that x > y.

If y is very close to 0 and x is some large enough number, then x - y - xy/100 = x - (a number close to 0) - (a number close to 0) ≈ x, so positive.

However, if x = 60 and y = 50, then x - y - xy/100 = 60 - 50 - 3000/100 = -20, so negative.

Not sufficient.

(2) The percent increase in the number of online orders from the first day to the second day was three times the percent decrease in the number of online orders from the second day to the third day.

This gives x = 3y. Substitute into the question:

Is x - y - xy/100 > 0?

Is 3y - y - 3y^2/100 > 0?

Is y(2 - 3y/100) > 0?

Since y > 0, divide both sides by y:

Is 2 - 3y/100 > 0?

If y is close to 0, this expression is positive.

However, if y is large enough, the expression becomes negative.

Not sufficient.

(1) + (2): Statement (2) already includes statement (1), because if x = 3y (and y > 0), then x > y automatically. So statement (1) adds nothing to statement (2). Since statement (2) is insufficient, the two statements together are also insufficient.

Answer: E.

ankushsambare · Answer

Let Day 1 = 100 (easy base)

We know:
Day 2 > Day 1
Day 3 < Day 2

Simplified Question:
is Day 3 > 100?

Statement 1:
increase % > decrease %

a. Take increase = 50%, decrease = 40%
Day 2 = 100 → 150
Day 3 = 150 → 90
So Day 3 = 90 < 100

b. Now try another case
increase = 50%, decrease = 20%
Day 2 = 100 → 150
Day 3 = 150 → 120
So Day 3 = 120 > 100

Both answers possible → not sufficient

Statement 2:
increase = 3 × decrease

decrease = 40%, increase = 120%
Day 2 = 100 → 220
Day 3 = 220 → 132
So Day 3 > 100

Now try a big decrease
decrease = 80%, increase = 240%
Day 2 = 100 → 340
Day 3 = 340 → 68
Now Day 3 < 100

Different answers → not sufficient

Both together:
statement 2 already fixes the relation, statement 1 adds nothing new

Still different answers possible

Final answer: E

Adit_ · Answer

We are given x<y>z.

Here when we take an equal increase and decrease usually x>z as the the decrease percentage always brings the value down more than an equal increase.
Just like how if we increase by 1/4 or 5/4x we need to decrease by 1/5x or 4/5x to get to equality meaning decreasing value always has more weightage as compared to increasing.

Statement 1:
Here we are given that percentage increase IS GREATER than decrease.
Now we know when they are equal x>z for sure. but here we are given increase is greater. Now this is NOT the critical point.
Critical point would be if decrease was greater than increase then we know for sure that x>z.
Decrease holds higher weightage, thus increasing just being more is NOT enough to tell which is bigger. As increase percent may or may not offset the decrease.
INSUFFICIENT.

Statement 2:
This is interesting. We are now given increase is 3x the decrease.
Is it still not enough?
Take 2 with 150% increase = 5.
Take 50% decrease of 5 = 2.5.
We clearly see that z>x meaning it has not offset the decrease percent!

Also the fact that decrease percent cant go above 100% as values cant go negative!
Increase can go beyond 300% making decrease to go beyond 100% by which point its not possible thus, the value after increase may be well more than decrease value.
INSUFFICIENT.

Together also clearly insufficient.

Answer:  Option E

A cafe tracked the number of online orders over three consecutive days

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A cafe tracked the number of online orders over three consecutive days

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