Data Sufficiency

Julie bought 2 adult tickets, 1 child ticket, and 1 senior ticket to an amusement park. She received 30% off the adult tickets and 15% off the child and senior tickets. Was the total amount of the 4 discounts greater than 20% of the sum of the regular prices of the 4 tickets?

1. The original price of an adult ticket was $48, the original price of the senior ticket was $24, and the original price of the child ticket was less than the senior ticket.

2. The original price of the child ticket was $12.


I can translate the equation given in the problem but I'm having trouble determining that the 1st statement is sufficient without doing a whole bunch of time consuming calculations. Am I missing a shortcut on this question?

sgraves wrote:Julie bought 2 adult tickets, 1 child ticket, and 1 senior ticket to an amusement park. She received 30% off the adult tickets and 15% off the child and senior tickets. Was the total amount of the 4 discounts greater than 20% of the sum of the regular prices of the 4 tickets?

1. The original price of an adult ticket was $48, the original price of the senior ticket was $24, and the original price of the child ticket was less than the senior ticket.

2. The original price of the child ticket was $12.


I can translate the equation given in the problem but I'm having trouble determining that the 1st statement is sufficient without doing a whole bunch of time consuming calculations. Am I missing a shortcut on this question?

Consider one extreme scenario that violates the statement. Imagine Adult ticket = $48, Senior ticket = $24 and Children's ticket = $24.

So there are $48 spent on Adult tickets and a total of $48 spent on non-adult tickets (Senior + Children's tickets.) If you save 30% on the adult tickets and 15% on non-adult tickets, and you spent the same amount on both types of tickets, then your savings will be exactly halfway between 30% and 15%, or 22.5%.

However, we know that the children's tickets are less than $24, which means we'll have spent $48 on adult tickets and less than $48 on non-adult tickets. If we spent more $ on adult tickets, the average will be weighted more heavily towards 30%, so we know that the savings had to be greater than 22.5%.



(We know the children's ticket is actually less than $24, so we know that

(An alternative way to conceptualize the same idea: imagine that you have a business. Now imagine that your return on equity is 30% and that your return on debt is 15%. If you have equal amounts of equity and debt, your overall return will be 22.5%. If you have more equity than debt, your overall return will be pulled towards the higher end, and therefore will be greater than 22.5%.)

I think I kind of get the explanation but it still seems a little complex. There isn't a easier way?

sgraves wrote:I think I kind of get the explanation but it still seems a little complex. There isn't a easier way?

Imagine the discounts on a number line. (30% for the adults, 15% for the seniors and children. )

15----20----------30

The only way the overall savings would be under 20% would be if the total amount spent was much more heavily weighted towards the seniors + children, as 20% is much closer to 15% (the senior + children discount) than it is to 30% (adult discount.)

Rephrased question: Was the total amount spent heavily weighted towards seniors and children?

Statement 1: we know that more was spent on adults than seniors + children. Sufficient.

Julie bought 2 adult tickets, 1 child ticket, and 1 senior ticket to an amusement park. She received 30% off the adult tickets and 15% off the child and senior tickets. Was the total amount of the 4 discounts greater than 20% of the sum of the regular prices of the 4 tickets?

1. The original price of an adult ticket was $48, the original price of the senior ticket was $24, and the original price of the child ticket was less than the senior ticket.

2. The original price of the child ticket was $12.

Statement 1: The original price of an adult ticket was $48, the original price of the senior ticket was $24, and the original price of the child ticket was less than the senior ticket.
Test EXTREMES.

Case 1: Test the THRESHOLD price for the original price of the child ticket.
Since the original price of the child ticket must be less than the original price of the senior ticket, the threshold price for the child ticket = 24.
In this case:
(sum of the discounts)/(sum of the original prices) = (.3*2*48 + .15*24 + .15*24)/(2*48 + 24 + 24) = (28.8 + 3.6 + 3.6)/144 = 36/144 = 25%.

Case 2: Test a very small value for the original price of the child ticket.
Let the original price of a child ticket = 1 cent.
Here, the original price and the discounted price of the child ticket will both be so small that they can be disregarded in our calculations.
In this case:
(sum of the other discounts)/(sum of the other original prices) = (.3*2*48 + .15*24)/(2*48 + 24) = (28.8 + 3.6)/120 ≈ 32/120 = 26.66%.

Since the discount is greater than 20% in each case, SUFFICIENT.

Statement 2: The original price of the child ticket was $12.
No way to determine whether the sum of the discounts was greater than 20%.
INSUFFICIENT.

The correct answer is A.

sgraves wrote:Julie bought 2 adult tickets, 1 child ticket, and 1 senior ticket to an amusement park. She received 30% off the adult tickets and 15% off the child and senior tickets. Was the total amount of the 4 discounts greater than 20% of the sum of the regular prices of the 4 tickets?

1. The original price of an adult ticket was $48, the original price of the senior ticket was $24, and the original price of the child ticket was less than the senior ticket.

2. The original price of the child ticket was $12.

Alternate approach:

Let A = the regular price of an adult ticket, C = the regular price of a child ticket, and S = the regular price of a senior ticket.

Was the total amount of the 4 discounts greater than 20% of the sum of the regular prices of the 4 tickets?
More specifically:
Is (30% of 2A) + (15% of C) + (15% of S) > 20% of (2A + C + S)?

Or:
(30)(2A) + 15C + 15S > 20(2A + C + S) ?
60A + 15C + 15S > 40A + 20C + 20S ?
20A > 5C + 5S ?
4A > C + S ?

Questions stem, rephrased:
Is 4A > C + S?

Statement 1:
Subsituting A=48, S=24 and C<24 into 4A > C + S, we get:
4*48 > (less than 24) + 24 ?
192 > less than 48 ?
YES.
SUFFICIENT.

Statement 2:
No way to determine whether 4A > C + S.
INSUFFICIENT.

The correct answer is A.