Jack purchased two chairs at a discount of 50%

Are these like extremely difficult questions which guarantees you a 700+

Source : Manhattan Advanced Quant Question No. 9

OFFICIAL SOLUTION

Let x and y represent the undiscounted prices of Jack's two chairs (we do not know whether the chairs have the same price). We are told that z represents the undiscounted price of Sally's table.

With a 50% discount on his two chairs, Jack spent more than Sally, who got an 80% discount:

\(0.5 x + 0.5 y > 0.2 z\)
\(0.5( x + y) > 0.2 z\)
\(5( x + y) > 2 z\)
\(x + y > 0.4 z\)

With a 60% discount on his two chairs, Jack would have spent less than Sally, who again got an 80% discount:

\(0.4 x + 0.4 y < 0.2 z\)
\(0.4( x + y) < 0.2 z\)
\(4( x + y) < 2 z\)
\(x + y < 0.5 z\)

Therefore, \(0.4 z < x + y < 0.5 z\). The average price of Jack's chairs is \(\frac{( x + y)}{2}\), so divide the entire inequality by two: \(0.2 z < \frac{( x + y)}{2} < 0.25 z.\)

The only answer choice in this range is 0.22 z.

The correct answer is B.

Hi Aniket,

This is a great question where finding a range of values will help.

If Jack purchased two chairs at a discount of 50% then he ends up paying 50% the price of both chairs. Let us keep the price of both the chairs as 'c'. So the amount that Jack pays for both the chairs is '0.5c'

If Sally purchased a table at a discount of 80% then she ends up paying 20% the price of the table. Let us keep the price of the table to be 'z'. So the amount that Sally pays for the table is '0.2z'.

Now since Jack spent more on his two chairs than Sally did on her table ------> 0.5c > 0.2z ------> c > 0.4z.

Since since the cost of both chairs is c, the average undiscounted price of the two chairs will be c/2

If c > 0.4z then c/2 > 0.2z

If Jack had purchased his two chairs at a discount of 60% then he would have spent less than Sally. so 0.4c < 0.2z ------> c < 0.5z

Since since the cost of both chairs is c, the average undiscounted price of the two chairs will be c/2

If c < 0.5z then c/2 < 0.25z

So the range of c/2 is 0.2z < c/2 < 0.25z.

The only possible answer that lies in this range is 0.22z

Hope this helps!

CrackVerbal Academics Team

@Bunuel- where I'm lost with this explanation is x > 0.2Z

I keep thinking 2x > .02Z - if 2x is the price of 2 undiscounted chairs and jack spent more those 2 tables than sally's 1 table - where I am misreading this?

Undiscounted price of Sally's table = z
Discounted price of Sally's table = 0.2z

Say, undiscounted price of Jack's 2 chairs = c
At 50% discount, Jack's chairs cost = 0.5c
At 60% discount, Jack's chairs cost = 0.4c

0.5c > 0.2z
c > .4z

Also
0.4c < 0.2z
c < .5z

So .4z < c < .5z
Average price of each chair will c/2
.2z < c/2 < .25z

Answer (B)

It has been given that the undiscounted price of Sally's table is z.
As Sally bought the table at a 80% discount, the price she paid for it would be 20% of z, which is .2z

If x is the undiscounted price of the 2 chairs,
We are also told that the prices of Jack's chairs at 50% discount is more than this value(.2z) (50% of x < 20% of z)
Hence if z = 200, max value of x(cost of two chairs) > 80
Cost of 1 chair > 40(which is 20% of the cost)
The first value in the list which makes sense for the cost of one chair is 22% of the table(which is 44) as it is greater than 40(Option B)

Jack spent more on his two chairs than Sally did on her table.

Jack spent on his two chairs 0.5*2x = x (because Jack purchased two chairs at a discount of 50%)
Sally spent on her table 0.2z (because Sally purchased a table at a discount of 80%).

So, x > 0.2z.

Hope it's clear.

Let the average undiscounted price Jack paid for each chair be c. Then, the two chairs will cost 2c, and since Jack buys the chairs at a 50% discount, he pays (0.5)(2c) = c.

Since Sally buys the table at a discount of 80%, she pays 0.2z.

Since Jack paid more for the chairs than Sally did for the table, c > 0.2z.

If Jack had purchased the two chairs at 60% discount instead of a 50% discount, he would have paid (0.4)(2c) = 0.8c. Since we are given that this would cost less than what Sally spent, 0.8c < 0.2z. Multiplying this inequality by 5, we get 4c < z. Dividing each side by 4, we get c < z/4, which is equivalent to saying c < 0.25z.

Combining the two inequalities, we get 0.20z < c < 0.25z. We see that the only answer choice that would satisfy this inequality when substituted for c is B.

Answer: B

smart numbers can be helpful
as let the cp of both chairs be 100
after discount both cost 50
let the cp of table be 225 =z
50>0.2z (45)
40<0.2z (45)

average of chairs is 100/2 =50
put z=225 in options
answer is b

LOLOLOL @ variables. Why wouldn't we just use a combination of Plugging In and Plugging In The Answers (PITA)?!?

Let's make z = 100. The discounted price of the table is $20.

Okay, let's Plug In The Answers. I like trying B or D.

B: .22(100) = 22. That's the undiscounted price of a chair, so the undiscounted price of two chairs would be $44. At 50% off, they come to $22. At 60% off, they come to that minus another 10%, so $22-$4.4 = $17.60. Hey, that works!

Answer choice B.

ThatDudeKnowsPluggingIn
ThatDudeKnowsPluggingInTheAnswers
ThatDudeKnowsPITA