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This problem can be done mentally without any calculations

Readout indicated 24 min 18 sec when she had completed 10% of her exercise.

This means remaining 90% of the work will take 24 min 18 sec => 30% will take 8 min 6 sec

so 40% completed i.e. 60% remaining of the work will take (24min 18 sec- 8 min 6 sec)= 16min 12 sec.
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CoffeeCreamSugar
Dara ran on a treadmill that had a readout indicating the time remaining in her exercise session. When the readout indicated 24 min 18 sec, she had completed 10% of her exercise session. The readout indicated which of the following when she had completed 40% of her exercise session?

A. 10 min 48 sec
B. 14 min 52 sec
C. 14 min 58 sec
D. 16 min 6 sec
E. 16 min 12 sec

We are given that Dara completed 10% of her exercise session when the readout indicated 24 min and 18 seconds as her time remaining. In other words she had 90% left of her workout at that time. We need to determine the readout after she has completed 40% of her exercise, or when she has 60% left.

We can create a proportion to determine the time when Dara has 60% left. First, let’s convert 24 min 18 sec to minutes.

24 min 18 sec = (24 + 18/60) min = (24 + 3/10) min = 24.3 min

We can also express 90 percent as 90/100 = 0.9 and 60 percent as 60/100 = 0.6. We define x as the number of minutes remaining in her workout when she has 60%, or 0.6, of her workout left to do.

Our proportion is as follows:

0.9/24.3 = 0.6/x

9/243 = 0.6/x

9x = 145.8

x = 145.8/9 = 16.2 min

Thus, x = (16 + 2/10) min = (16 + 12/60) min = 16 min 12 sec

General Discussion
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CoffeeCreamSugar
Dara ran on a treadmill that had a readout indicating the time remaining in her exercise session. When the readout indicated 24 min 18 sec, she had completed 10% of her exercise session. The readout indicated which of the following when she had completed 40% of her exercise session?

A. 10 min 48 sec
B. 14 min 52 sec
C. 14 min 58 sec
D. 16 min 6 sec
E. 16 min 12 sec

Let's call 24min and 18sec Y.
24min and 18 sec are: 243/10.
Dara completed 10% of her session, so:
243/10 = 9x/10
x=27 min.
27-Y= 2min+42sec
If Dara completed 40% of her session, the she worked for 4*(2min+42sec) so that the rest of her session will be:
27min-10min&48sec = 16 min 12 sec
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24.3 minutes/.9=27 minutes=total time of session
(27)(.6)=16.2 minutes=time remaining when session is 40% completed
16.2 minutes=16 min 12 sec
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24 min 18 sec = 1458 sec --> this is 90% of the total workout (w)

.9w = 1458 sec
w = 1620 sec

The readout when she had done 40% of the workout will be 60% of the total

1620*.6 = 972 --> 972/60 = 16 min 12 sec

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CoffeeCreamSugar
Dara ran on a treadmill that had a readout indicating the time remaining in her exercise session. When the readout indicated 24 min 18 sec, she had completed 10% of her exercise session. The readout indicated which of the following when she had completed 40% of her exercise session?

A. 10 min 48 sec
B. 14 min 52 sec
C. 14 min 58 sec
D. 16 min 6 sec
E. 16 min 12 sec

Time remaining when 10% of session completed = 24 mins and 18 seconds.
Let us covert everything in seconds to calculate the total time.

24 mins 18 seconds = 24*60 + 18 = 1440 + 18 = 1458
90% of time = 1458 seconds.

100% of time = 1458*10/9 = 1620 seconds
40% completed = 60% remaining.
Time remaining = 0.6*1620 = 972seconds = 960 + 12 seconds = 16 minutes 12 seconds.

Correct Option: E
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I disregarded the minutes and only focused on the seconds since all answer options have a different seconds count.
18 seconds is 90% --> 20 seconds is 100% --> 60% is 12 seconds --> answer E
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Although I usually tend to rely on the algebraic approach described for this question by other posters, I want to provide an additional approach.

We are told that 90% of the remaining time equals 24 min and 18 seconds.

We can divide this one 90% portion into three 30% portions:

24 min divided by 3 = 3 portions of 8 min

18 sec divided by 3 = 3 portions of 6 sec

Since 30% + 30% = 60%, we then can make the sum of two of the portions to estimate the remaining time when Dara has completed 40% of the exercise session (or the remaining 60%):

(2 portions of 8 min = 16 min) + (2 portions of 6 sec = 12 sec)

Total = 16 min and 12 sec. Option E.

Hope it helps.
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I disregarded the minutes and only focused on the seconds since all answer options have a different seconds count.
18 seconds is 90% --> 20 seconds is 100% --> 60% is 12 seconds --> answer E
This approach is incorrect, it gives the correct answer out of luck only because the 60% of the total minutes is an integer.

If you want you can separate them, but should be like this:

(1) 24 mins in 90%. Means that in 60%: 60*24/90 = 16 mins (simplify to calculate that)
(2) 18 secs in 90%. Means that in 60%: 60*18/90 = 12 secs (also simplifying)

if (1) turned out to be a fraction, then the answer would not just rely on the 12 secs you calculated.
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CoffeeCreamSugar
Dara ran on a treadmill that had a readout indicating the time remaining in her exercise session. When the readout indicated 24 min 18 sec, she had completed 10% of her exercise session. The readout indicated which of the following when she had completed 40% of her exercise session?

A. 10 min 48 sec
B. 14 min 52 sec
C. 14 min 58 sec
D. 16 min 6 sec
E. 16 min 12 sec

$$Mental:$$

$$90 percent$$ = $$24 min 18 sec$$

$$60 percent$$ = $$\frac{2}{3} * 24 min 18 sec$$

= $$16 min 12 sec$$, i.e $$\frac{2}{3}$$ of each of $$24$$ and $$18$$ respectively

Hence, E.
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Dara passed 10%, so 90% remains ahead of her.
We are asked how much time will remain when she completes 40%, which actually means "how much time will remain when 60% of the run remains ahead of her"?

90% is 9 times 10%, 60% is 6 times 10%, call 10% X. The original time left is 9X, the time left after 40% of the workout is 6X. 6 = 2/3 * 9

If 9X = 24+(18/60):
6X = 2/3 *( 24 + 18/60 ) = 2/3 * 24 + 2/3 *18/60 = 16 + 12/60
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KarishmaB - I was able to understand your approach, but I wanted to solve it using inverse variation, and would really appreciate if you can tell me, what's wrong with my logic.

First understanding the relationships
1. Exercise completed has a "Direct relationship" with "time completed", as exercise % goes up, time completed comes up.
2. Exercise completed has an "Inverse relationship" with "time remaining", as exercise % goes up, time remaining comes down.

* Next, the stem gives us, two data points
1. "Time remaining" and
2. "% completed".
i.e. we have the data points for the "inverse relationship".

So technically, time rem. * % completed = k, hence I should in theory be able to set-up a relationship like the below:

24 (18/60)*10% = x*40%
But this produces the wrong answer. Wonder where I am getting this wrong?
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KarishmaB - I was able to understand your approach, but I wanted to solve it using inverse variation, and would really appreciate if you can tell me, what's wrong with my logic.

First understanding the relationships
1. Exercise completed has a "Direct relationship" with "time completed", as exercise % goes up, time completed comes up.
2. Exercise completed has an "Inverse relationship" with "time remaining", as exercise % goes up, time remaining comes down.

* Next, the stem gives us, two data points
1. "Time remaining" and
2. "% completed".
i.e. we have the data points for the "inverse relationship".

So technically, time rem. * % completed = k, hence I should in theory be able to set-up a relationship like the below:

24 (18/60)*10% = x*40%
But this produces the wrong answer. Wonder where I am getting this wrong?

Check out the video on Variation here: https://youtu.be/AT86tjxJ-f0

Exercise done (work done) and time taken vary directly - correct (assuming rate is constant)
If 10% work is done, time taken is say t. If 20% work is done (twice of before), time taken is 2t - Correct

Work remaining and time remaining also vary directly - correct (because rate is constant)
If 80% work is left, say time remaining is T.
If 40% work is left, time remaining will be T/2 (Half) - Correct

But that doesn't mean that work done and time remaining is inversely varying.

Say 1 work has to be done in 60 mins.
So when 10% work is done, 6 mins have passed. (10% work is done and 54 mins are left)
When 20% work is done, 12 mins have passed. (20% work is done and 48 mins are left - work done is now twice but time remaining is not half)

This is so because time passed and time remaining have an additive relation, not multiplicative. They add up to give a fixed value.
Intern
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­At a constant rate,

if 10% of her workout is complete, then 10% of her time is also complete => (timeleft @ start of workout * 0.9)

Amount of workout left........................Amount of time remaining
100%                                                      X
90%                                                 X(0.9)= 24m18s
= [ 24+(18/60) ] min
= 24.3 min
(X*0.9 = 24.3 => X = 27)