Two taps can fill a cistern in 20 minutes and 30 minutes. The first tap was opened initially for x minutes after which the second tap was opened. If it took a total of 15 minutes for the tank to be filled, what is the value of x?
A. 5.0
B. 7.5
C. 9.0
D. 10.0
E. 12.5
The OA is B.
I know the rate of the first tap is 1/20 minutes, and the rate of the second tap is 1/30.
Then, the first was working x/20 minutes, and after which the second tap was opened.
The total time was 15 minutes. I stuck here.
I'm confused by this PS question. Experts, any suggestion? Thanks in advance.
Two taps can fill a cistern in 20 minutes and 30 minutes...
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Hi LUANDATO.LUANDATO wrote:Two taps can fill a cistern in 20 minutes and 30 minutes. The first tap was opened initially for x minutes after which the second tap was opened. If it took a total of 15 minutes for the tank to be filled, what is the value of x?
A. 5.0
B. 7.5
C. 9.0
D. 10.0
E. 12.5
The OA is B.
I know the rate of the first tap is 1/20 minutes, and the rate of the second tap is 1/30.
Then, the first was working x/20 minutes, and after which the second tap was opened.
The total time was 15 minutes. I stuck here.
I'm confused by this PS question. Experts, any suggestion? Thanks in advance.
Let's continue your explanation.
The first tap rate is 1/20 per minute.
The second tap rate is 1/30 per minute.
Now, we have to set the following equation: $$rate\ tap\ 1\ during\ x\ \text{minutes}\ +\ \left(time\ both\ taps\ worked\ together\right)\left(rate\ tap\ 1\ +\ rate\ tap\ 2\right)$$ $$=\ whole\ \text{tank}$$ $$\frac{x}{20}+\left(15-x\right)\left(\frac{1}{20}+\frac{1}{30}\right)=1\ $$ $$\frac{x}{20}+\frac{15-x}{20}+\frac{15-x}{30}=1\ \Leftrightarrow\ \ \frac{15}{20}+\frac{15-x}{30}=1$$ $$450+20\left(15-x\right)=600\ \Leftrightarrow\ \ 300-20x=150\ \Leftrightarrow\ 20x=150\ \Leftrightarrow\ x=7.5$$ This is why the correct answer is the option B.
I hope this answer may help you.
If you have a doubt, let me know.
Regards.
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For the first x minutes, the first tap works alone at a rate of 1 cistern/20 minutes. We can express the total amount filled in this time as:
$$x\ \min\left(\frac{1\ cistern}{20\ \min}\right)$$
For the remainder of the 15 minutes (in other words, for 15 - x minutes), the two taps work together at their respective rates. We can express the total amount filled in this time as:
$$\left(15\ -x\right)\ \min\left(\frac{1\ cistern}{20\ \min}+\frac{1\ cistern}{30\ \min}\right)$$
If we add the total amount filled in the first x minutes to the total amount filled in the remained of the 15 minutes, we should get one full cistern:
$$x\ \min\left(\frac{1\ cistern}{20\ \min}\right)+\left(15\ -x\right)\ \min\left(\frac{1\ cistern}{20\ \min}+\frac{1\ cistern}{30\ \min}\right)=1\ cistern$$
Or with our units cancelled:
$$x\left(\frac{1}{20}\right)+\left(15\ -x\right)\left(\frac{1}{20}+\frac{1}{30}\right)=1$$
Then we simplify to solve for x:
$$\frac{x}{20}+\frac{15-x}{20}+\frac{15-x}{30}=1$$ $$\frac{3x}{60}+\frac{45-3x}{60}+\frac{30-2x}{60}=1$$ $$3x+45-3x+30-2x=60$$ $$75-2x=60$$ $$2x=15$$ $$x=7.5$$
So the value of x is 7.5.
$$x\ \min\left(\frac{1\ cistern}{20\ \min}\right)$$
For the remainder of the 15 minutes (in other words, for 15 - x minutes), the two taps work together at their respective rates. We can express the total amount filled in this time as:
$$\left(15\ -x\right)\ \min\left(\frac{1\ cistern}{20\ \min}+\frac{1\ cistern}{30\ \min}\right)$$
If we add the total amount filled in the first x minutes to the total amount filled in the remained of the 15 minutes, we should get one full cistern:
$$x\ \min\left(\frac{1\ cistern}{20\ \min}\right)+\left(15\ -x\right)\ \min\left(\frac{1\ cistern}{20\ \min}+\frac{1\ cistern}{30\ \min}\right)=1\ cistern$$
Or with our units cancelled:
$$x\left(\frac{1}{20}\right)+\left(15\ -x\right)\left(\frac{1}{20}+\frac{1}{30}\right)=1$$
Then we simplify to solve for x:
$$\frac{x}{20}+\frac{15-x}{20}+\frac{15-x}{30}=1$$ $$\frac{3x}{60}+\frac{45-3x}{60}+\frac{30-2x}{60}=1$$ $$3x+45-3x+30-2x=60$$ $$75-2x=60$$ $$2x=15$$ $$x=7.5$$
So the value of x is 7.5.
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We are given that the rate of the first cistern is 1/20 and the rate of the second cistern is 1/30. The first tap worked for 15 minutes, and the second tap worked for (15 - x) minutes. Thus:BTGmoderatorLU wrote:Two taps can fill a cistern in 20 minutes and 30 minutes. The first tap was opened initially for x minutes after which the second tap was opened. If it took a total of 15 minutes for the tank to be filled, what is the value of x?
A. 5.0
B. 7.5
C. 9.0
D. 10.0
E. 12.5
(1/20)(15) + (1/30)(15 - x) = 1
3/4 + (15 - x)/30 = 1
Multiplying the entire equation by 60, we have:
45 + 30 - 2x = 60
2x = 15
x = 7.5
Answer: B
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