Question: At exactly what time past 7:00 will the minute and hour hands of an accurate working clock be precisely perpendicular to each other for the first time?
- \(20\frac{13}{21}\) minutes past 7:00
- \(20\frac{13}{17}\) minutes past 7:00
- \(20\frac{3}{23}\) minutes past 7:00
- \(20\frac{9}{11}\) minutes past 7:00
- \(20\frac{4}{9}\) minutes past 7:00
“At exactly what time past 7:00 will the minute and hour hands of an accurate”- is the topic from the GMAT Quantitative problem set. To solve GMAT Problem Solving questions a student must have knowledge about a good number of qualitative skills. The GMAT Quant section consists of 31 questions in total. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.
Solution and Explanation:
Approach Solution 1:
The given condition states that a perfectly working clock needs to be precisely perpendicular to each other for the first time. Accordingly, it is important to find from the case the exact time past 7:00. Here the minute and the hour hands would be accurately in a position to be perpendicular to each other. In order to find the same, the fastest way to solve the problem is as follows:
Speed of hour hand/min = \(\frac{360}{20}*60=\frac{1}{2}\).
Speed of minute hand/min = \(\frac{360}{60}\) = 6.
Relative speed = \(\frac{11}{2}\) degrees/min
At 7 the two, including the hour hand and the minute hand are 210 degrees apart.
From 210 degrees to 90 degrees, the minute hand has to gain 120 over the hour hand to ensure that they are perpendicular to each other.
So time required will be the 120 degrees compared to the relative speed of the clock which can be implied as
= \(\frac{120}{Relative speed}\)
= \(120*\frac{2}{11}\)
= \(21\frac{9}{11}\)
Thus, option D is the correct answer with 21 9/11 minutes past 7:00.
Correct Answer: D
Approach Solution 2:
The stated condition indicates that the first pair of flawlessly running clocks must be precisely perpendicular to one another. Therefore, it is crucial to determine the precise time past 7:00 from the instance. Here, the minute and hour hands would be precisely positioned such that they were parallel to one another. To discover the same, the following is the problem's quickest solution:
Minutes per hour hand speed.
Minute hand movement/minute = 6.
degrees/min relative speed
The minute hand and the two are 210 degrees apart at seven o'clock.
To guarantee that they are perpendicular to one another, the minute hand must gain 120 degrees over the hour hand from 210 degrees to 90 degrees.
Therefore, the amount of time needed will be 120 degrees in relation to the relative speed of the clock, which may be expressed as
=\(\frac{120}{Relative speed}\)
=\(120*\frac{2}{11}\)
= \(21\frac{9}{11}\)
Thus, with 21 minutes after 7:00 on September 11, option D is the right response.
Correct Answer: D
Approach Solution 3:
According to the given requirement, the first pair of perfectly functioning clocks must be perfectly perpendicular to one another. Determining the exact time past 7:00 from the instance is therefore vital. The minute and hour hands would be placed precisely such that they were parallel to one another in this arrangement. The easiest way to find the same is to do what is described below:
Time in minutes per hour.
Minute hand motion per minute equals 6.
relative speed in degrees/min
At seven o'clock, the two are separated by 210 degrees from the minute hand.
The minute hand must advance by 120 degrees over the hour hand, from 210 degrees to 90 degrees, to ensure that they are perpendicular to one another.
Given that the relative speed of the clock is 120 degrees, the amount of time required may be represented as
=\(\frac{120}{Relative speed}\)
=\(120*\frac{2}{11}\)
= \(21\frac{9}{11}\)
So, option D is the appropriate reaction on September 11, 21 minutes after 7:00.
Correct Answer: D
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