A certain movie star's salary for each film

[melguy]

A certain movie star's salary for each film she makes consists of a fixed amount, along with a percentage of the gross revenue the film generates. In her last two roles, the star made $32 million on a film that grossed $100 million, and $24 million on a film that grossed $60 million. If the star wants to make at least $40 million on her next film, what is the minimum amount of gross revenue the film must generate?

$110 million
$120 million
$130 million
$140 million
$150 million

The solution to this problem suggests that "One way to do this problem is to recognize that the star earned $8M more ($32M - $24M = $8M) when her film grossed $40M more ($100M - $60M = $40M). She wants to earn $40M on her next film, or $8M more than she earned on the more lucrative of her other two films. Thus, her next film would need to gross $40M more than $100M, or $140M. "

I am fine with the solution but I am curious to know when using the short cut how can we be sure about it when we do not know the fixed amount, along with a percentage of the gross revenue.

e.g.

60 -> 24 (24 is 40% of 60)
100 -> 32 (32%)
140 -> 40 (30% approx)

So we do not have a fixed percentage to work with. What am I missing? Thanks

[lunarpower]

One way to do this problem is with straight algebra:

A certain movie star's salary for each film she makes consists of a fixed amount, along with a percentage of the gross revenue the film generates.

let x = gross revenue, y = salary.
then y = mx + b, where m is the percentage (expressed as a decimal), and b is the "fixed" part of the salary.

then you can just plug in (x, y) = (100, 32) and (60, 24) to find the values of m and b. once you have those, you can plug in y = 40 and solve for x.

--

the shortcut is to realize that this is a linear relationship -- i.e., one quantity "steps up" by the same amount whenever another quantity is increased by some amount. (this is exactly the same concept as the slope of a line.)

from the given points, you know that the "step size" is an additional $8m in salary per each additional $40m in gross revenues.
conveniently, the problem just wants you to "step up" by another $8m, so you can just add another $40m to the gross revenue.

[lunarpower]

60 -> 24 (24 is 40% of 60)
100 -> 32 (32%)
140 -> 40 (30% approx)

So we do not have a fixed percentage to work with. What am I missing? Thanks

what you're missing is the "fixed amount" part.
i.e., the 24 is not calculated as 40% of 60; it's calculated as (fixed amount) + (smaller percentage of 60).
similarly, the 32 is calculated as (same fixed amount) + (same smaller percentage of 100).
etc.

if you solve the equations above -- which is illustrative when you're reviewing the problem, but is unnecessary in actually solving the problem (since you can just use the concept of step size as illustrated above) -- then you'll find out that it's a fixed amount of $12m, plus 20% of gross revenues.
using these values:
$12m plus 20% of $60m is ... $24m. check.
$12m plus 20% of $100m is ... $32m. check.
$12m plus 20% of $140m is ... $40m.

[melguy]

Thanks Ron. Makes perfect sense now although I have one (silly) question. I am unable to understand how to solve for the magic numbers 12m and 20% . Do we test by plugging in numbers ?? Thanks

[lunarpower]

Thanks Ron. Makes perfect sense now although I have one (silly) question. I am unable to understand how to solve for the magic numbers 12m and 20% . Do we test by plugging in numbers ?? Thanks

scroll up to my first post (3 posts above this one), where i outlined how to do that.
you'll get "b" = 12 and "m" = 0.2.

[melguy]

Thanks. Sorry my bad.

[GMATGuruNY]

A certain movie star's salary for each film she makes consists of a fixed amount, along with a percentage of the gross revenue the film generates. In her last two roles, the star made $32 million on a film that grossed $100 million, and $24 million on a film that grossed $60 million. If the star wants to make at least $40 million on her next film, what is the minimum amount of gross revenue the film must generate?

$110 million
$120 million
$130 million
$140 million
$150 million

An alternate approach is to GUESS AND CHECK the percentage that the star earns from the gross revenue.

$60 million dollar film:
A fraction of the $60 million gross contributes to the salary.
Since this fraction is described as a PERCENTAGE, the correct fraction is almost certainly one that is easily converted to a percentage.
Most likely options:
Case 1: 3/60 = 1/20 = 5%.
Case 2: 6/60 = 1/10 = 10%.
Case 3: 9/60 = 3/20 = 15%.
Case 4: 12/60 = 1/5 = 20%.
Case 5: 15/60 = 1/4 = 25%.

The numerators above -- which represent the amount ADDED to the fixed amount to yield the $24 million salary -- imply the following options for the fixed amount:
Case 1: 24-3 = 21.
Case 2: 24-6 = 18.
Case 3: 24-9 = 15.
Case 4: 24-12 = 12.
Case 5: 24-15 = 9.

$100 million film:
Subtracting the above options for the fixed amount from the $32 million salary, we get the following options for the portion earned from the $100 million gross:
Case 1: 32-3 = 29.
Case 2: 32-6 = 26.
Case 3: 32-9 = 23.
Case 4: 32-12 = 20.

We have a match!
In Case 4, for each film, the percentage of the gross = 20%.
Thus:
Fixed amount = $12 million.
Percentage of the gross = 20%.

Next film:
Subtracting the fixed amount of $12 million from the $40 million total to be earned on the next film, we get:
40-12 = 28.
This is the portion that must be earned from the gross revenue.
Thus, 20% of the correct answer choice must yield $28 million, implying that the correct answer is D:
.2(140) = 28.

The correct answer is D.