Question 1

What is the ratio of 18 minutes to one hour, expressed in simplest fractional form?

Accepted Answer

Introduction / Context: Ratio questions are very common in aptitude tests because they check your understanding of proportional comparison between quantities with the same units. Here, you are given 18 minutes and one hour, and you need to express the ratio of these times in simplest fractional form. This is a foundational concept that appears in time and work, speed, and percentage questions as well. Given Data / Assumptions: First quantity: 18 minutes. Second quantity: 1 hour. We know that 1 hour equals 60 minutes. We must express the ratio 18 minutes : 60 minutes and simplify it. Concept / Approach: When forming a ratio, both quantities must be in the same unit. After converting, the ratio a : b can often be simplified by dividing both terms by their greatest common divisor. Once converted and simplified, the ratio can be written either as a : b or as a fraction a / b. For time, using minutes for both values makes the comparison straightforward. Step-by-Step Solution: Step 1: Convert one hour to minutes. One hour equals 60 minutes. Step 2: Now write the ratio in terms of minutes: 18 minutes : 60 minutes. Step 3: To simplify, divide both numbers by their greatest common divisor. The greatest common divisor of 18 and 60 is 6. Step 4: Divide both terms by 6: 18 / 6 = 3 and 60 / 6 = 10. Step 5: So the simplified ratio is 3 : 10. Step 6: Expressing this as a fraction gives 3/10. Verification / Alternative check: You can also convert 18 minutes to a fraction of an hour directly. Since 60 minutes is one hour, 18 minutes is 18 / 60 hours. Simplifying this fraction, divide numerator and denominator by 6 to get 3 / 10 hours. This confirms again that the ratio of 18 minutes to one hour is 3/10, because you are essentially comparing 0.3 hours to 1 hour. Why Other Options Are Wrong: 1/5: This equals 0.2, which is less than 18 / 60 = 0.3, so it understates the fraction of an hour. 3/4: This equals 0.75 hours, much larger than 18 minutes and therefore incorrect. 1/7: This is about 0.1428, which is too small compared with 0.3. 2/5: This is 0.4, which is greater than the true value of 0.3 for 18 minutes out of an hour. Common Pitfalls: A frequent mistake is to forget to convert hours into minutes or vice versa before forming the ratio. Some learners keep the units mixed, which can lead to meaningless comparisons. Others simplify 18 : 60 incorrectly by dividing by the wrong common factor or by skipping the simplification step altogether. Always ensure that units are consistent and that the numbers are reduced to simplest form. Final Answer: The ratio of 18 minutes to one hour is 3/10.

Question 2

An accurate clock shows exactly 7 o clock in the morning. Through how many degrees will the hour hand rotate by the time the clock shows 3 o clock in the afternoon?

Accepted Answer

Introduction / Context: This question tests your understanding of how far the hour hand of a clock rotates over a period of time. Instead of asking for the angle between the hour and minute hands, it asks for the total angular rotation of the hour hand from one time to another. Problems like this appear frequently in aptitude exams under the topic of clocks and require only a simple relationship between time and angle. Given Data / Assumptions: Initial time: 7 o clock in the morning. Final time: 3 o clock in the afternoon. The clock is accurate and moves uniformly. We must find the total angle through which the hour hand rotates between these two times. Concept / Approach: On a standard clock, the hour hand completes one full revolution of 360 degrees in 12 hours. Therefore, in one hour the hour hand rotates 360 / 12 = 30 degrees. To find the total rotation over a given number of hours, we simply multiply the number of hours by 30 degrees per hour. The key step is to count the number of hours between the two times correctly, noting the change from morning to afternoon. Step-by-Step Solution: Step 1: Calculate the time interval from 7 o clock in the morning to 3 o clock in the afternoon. Step 2: From 7 a.m. to 12 noon is 5 hours. From 12 noon to 3 p.m. is another 3 hours. Total time = 5 + 3 = 8 hours. Step 3: The hour hand rotates 30 degrees in one hour. Step 4: Therefore, in 8 hours, the rotation of the hour hand is 8 * 30 degrees = 240 degrees. Step 5: Hence, the hour hand sweeps through an angle of 240 degrees between 7 a.m. and 3 p.m. Verification / Alternative check: You can also think in terms of hour marks on the dial. At 7 o clock, the hour hand points at 7. At 3 o clock, it points at 3. Moving forward from 7 to 3 goes through the positions 8, 9, 10, 11, 12, 1, 2, and finally 3. That is 8 hour steps. Each step represents 30 degrees, so 8 * 30 = 240 degrees. This count of hour intervals provides the same result and confirms the calculation. Why Other Options Are Wrong: 144 degrees: This would correspond to 144 / 30 = 4.8 hours, not 8 hours, so it does not match the actual interval. 168 degrees: This equals 5.6 hours of rotation, again not compatible with the 8 hour time difference. 180 degrees: This would correspond to 6 hours, the rotation from 7 to 1, not all the way to 3 p.m. 150 degrees: This equals 5 hours of rotation, matching only the interval from 7 a.m. to 12 noon. Common Pitfalls: A common mistake is miscounting the number of hours between the two times, especially when crossing from morning to afternoon. Another error is confusing the hour hand with the minute hand and using 6 degrees per minute instead of 30 degrees per hour. Always identify which hand is being discussed and check whether the time span crosses noon or midnight. Final Answer: The hour hand rotates through 240 degrees between 7 a.m. and 3 p.m.

Question 3

What is the smaller angle between the hour hand and the minute hand of a clock when the time shown is 9:15?

Accepted Answer

Introduction / Context: Angle between hands questions are classic clock problems in aptitude exams. They test your ability to convert a given time into angular positions for the hour and minute hands, and then find the angle between those positions. Here, the time is 9:15, and you are asked to find the smaller angle between the hour hand and the minute hand at that instant. Given Data / Assumptions: Time shown on the clock is 9:15. The clock is accurate and circular with 360 degrees total. The minute hand moves 360 degrees in 60 minutes, that is, 6 degrees per minute. The hour hand moves 360 degrees in 12 hours, that is, 30 degrees per hour, or 0.5 degrees per minute. We must report the smaller angle between the two hands. Concept / Approach: The standard method is to compute the separate angular positions of the minute and hour hands measured from the 12 o clock position. The minute hand position is straightforward: minutes multiplied by 6 degrees. The hour hand position depends on both the hour and the extra movement caused by the minutes past the hour. Once both angles are known, we find their absolute difference. If this difference is greater than 180 degrees, we subtract it from 360 degrees to obtain the smaller angle between the two hands. Step-by-Step Solution: Step 1: Find the angle of the minute hand at 9:15. The minute hand is at 15 minutes, so its angle from 12 is 15 * 6 = 90 degrees. Step 2: Find the angle of the hour hand at 9:15. At exactly 9:00, the hour hand is at 9 * 30 = 270 degrees. In the next 15 minutes, it moves further by 0.5 degrees per minute, that is, 15 * 0.5 = 7.5 degrees. Step 3: Total angle of the hour hand from 12 o clock at 9:15 is 270 + 7.5 = 277.5 degrees. Step 4: Compute the difference between the two angles: difference = |277.5 - 90| = 187.5 degrees. Step 5: The smaller angle between the hands is 360 - 187.5 = 172.5 degrees. Step 6: Therefore, the required smaller angle is 172.5 degrees. Verification / Alternative check: A quick consistency check is to note that at exactly 9:00 the angle between the hands is 90 degrees (hour at 270 degrees and minute at 0 degrees). As time passes to 9:15, the minute hand moves forward and the hour hand also moves forward slightly, increasing the separation. The result being slightly less than 180 degrees is reasonable. Using the general angle formula |30H - 5.5M| with H = 9 and M = 15 gives |30 * 9 - 5.5 * 15| = |270 - 82.5| = 187.5; then the smaller angle is 360 - 187.5 = 172.5 degrees, confirming the answer. Why Other Options Are Wrong: 165 degrees: This is close but does not match the precise calculation from the standard formula. 112.5 degrees: This would be the difference if the separation were much smaller; it is not supported by the positions of the hands. 125.5 degrees: This is not derived from any correct application of the 30H minus 5.5M formula. 187.5 degrees: This is the larger reflex angle between the hands, not the smaller angle requested in the question. Common Pitfalls: A common mistake is to forget that the hour hand moves continuously with the minutes, so taking the hour hand only at 270 degrees for 9 o clock leads to an incorrect difference. Another error is to stop at the raw difference of 187.5 degrees and forget to find the smaller angle by subtracting from 360 degrees. Always apply the formula carefully and remember to choose the smaller of the two possible angles between the hands. Final Answer: The smaller angle between the hour hand and minute hand at 9:15 is 172.5 degrees.

Question 4

Between 3 o clock and 4 o clock, after how many minutes past 3 will the hour hand and the minute hand of a clock exactly coincide?

Accepted Answer

Introduction / Context: This is a standard clock question that asks for the exact time between two hours when the hour hand and the minute hand coincide. Problems of this type are very common in competitive exams because they combine angle speed, relative speed, and simple algebra. Here, we want to know how many minutes after 3 o clock the two hands will be exactly together between 3 and 4. Given Data / Assumptions: We are interested in the interval between 3 o clock and 4 o clock. At 3 o clock, the hour hand is exactly at the 3 mark. The minute hand is at the 12 mark at 3 o clock. The hour hand moves at 0.5 degrees per minute, and the minute hand at 6 degrees per minute. We must find the number of minutes t after 3 o clock when both hands coincide. Concept / Approach: The idea is to treat the motion of both hands as uniform circular motion and then use relative speed. At 3 o clock, the hour hand is ahead of the minute hand by a certain angle. As time passes, the minute hand moves faster and gradually catches up. When they coincide, both hands are at the same angular position measured from the 12 o clock reference. We set up an equation equating the positions of the two hands after t minutes and solve for t. Step-by-Step Solution: Step 1: At 3 o clock, the hour hand is at 3 * 30 = 90 degrees from the 12 o clock position. Step 2: The minute hand at exactly 3 o clock is at 0 degrees because it points to 12. Step 3: After t minutes, the minute hand angle becomes 6t degrees. Step 4: After t minutes, the hour hand angle becomes 90 + 0.5t degrees. Step 5: For coincidence, these two angles must be equal: 6t = 90 + 0.5t. Step 6: Rearranging, 6t - 0.5t = 90 gives 5.5t = 90. Step 7: So t = 90 / 5.5 = 90 * 2 / 11 = 180 / 11 minutes. Step 8: Convert 180 / 11 into a mixed fraction: 11 goes into 180 sixteen times (16 * 11 = 176) with a remainder of 4. So t = 16 4/11 minutes. Step 9: Therefore, the hands coincide 16 4/11 minutes past 3 o clock. Verification / Alternative check: There is a well known shortcut formula for the time in minutes after H o clock when the hands coincide: t = (60 * H) / 11. For H = 3, t = (60 * 3) / 11 = 180 / 11 = 16 4/11 minutes. This confirms the result from the detailed angular method. Both methods agree, so the value is reliable. Why Other Options Are Wrong: 11 4/11 minutes: This is too early for the minute hand to fully catch up to the hour hand, as the angular gap would still be positive. 13 4/11 minutes: Also smaller than 16 4/11 and does not satisfy the equation 6t = 90 + 0.5t. 15 4/11 minutes: Close, but still not enough time for the minute hand to close the initial 90 degree gap at a relative speed of 5.5 degrees per minute. 18 4/11 minutes: This overshoots; by this time the minute hand has already moved ahead of the hour hand, so they no longer coincide. Common Pitfalls: A common error is to forget that the hour hand also moves during these minutes and to equate 6t directly to 90, ignoring the extra 0.5t term. Another pitfall is to miscalculate the mixed fraction from 180 / 11 or to round the decimal instead of keeping the exact fractional expression, which is usually required in exam answers. Remembering and correctly using the shortcut formula 60H / 11 can save time once the concept is understood. Final Answer: The hands of the clock coincide 16 4/11 minutes after 3 o clock.

Question 5

At what exact time between 9 o clock and 10 o clock will the hour hand and the minute hand of a watch be together?

Accepted Answer

Introduction / Context: Finding the exact time when the hour and minute hands coincide between two specified hours is a central type of clock problem. This question focuses on the interval between 9 o clock and 10 o clock and asks when, during this interval, the two hands are exactly together. It requires understanding of how fast each hand moves and how to set up the catch up equation. Given Data / Assumptions: We consider the period between 9 o clock and 10 o clock. At 9 o clock, the hour hand is at the 9 mark, and the minute hand is at the 12 mark. The minute hand moves at 6 degrees per minute. The hour hand moves at 0.5 degrees per minute. We must find the number of minutes t after 9 o clock when both hands coincide. Concept / Approach: At 9 o clock, the hour hand is ahead of the minute hand by a certain fixed angle. Because the minute hand rotates faster, it eventually catches up to the hour hand somewhere between 9 and 10. We can calculate the angular position of each hand after t minutes, set them equal for coincidence, and solve for t. A shortcut formula t = (60 * H) / 11, where H is the hour, can also be used after the underlying reasoning is understood. Step-by-Step Solution: Step 1: At 9 o clock, the hour hand is at 9 * 30 = 270 degrees from the 12 o clock position. Step 2: At the same moment, the minute hand is at 0 degrees (pointing to 12). Step 3: After t minutes, the angle of the minute hand is 6t degrees. Step 4: After t minutes, the angle of the hour hand is 270 + 0.5t degrees. Step 5: For coincidence, set the angles equal: 6t = 270 + 0.5t. Step 6: Rearranging gives 6t - 0.5t = 270, so 5.5t = 270. Step 7: Therefore, t = 270 / 5.5 = 270 * 2 / 11 = 540 / 11 minutes. Step 8: Convert 540 / 11 to a mixed fraction. 11 goes into 540 forty nine times (49 * 11 = 539) with a remainder of 1. So t = 49 1/11 minutes. Step 9: Hence, the hands are together at 9:49 1/11. Verification / Alternative check: Use the shortcut formula for coincidence after H o clock: t = (60 * H) / 11 minutes past H. Here H = 9, so t = (60 * 9) / 11 = 540 / 11 = 49 1/11 minutes, which matches the algebraic result. This confirms that the time of coincidence between 9 and 10 is 9:49 1/11. Why Other Options Are Wrong: 49 3/11 minutes past 9: This is slightly later than the correct time and would cause the minute hand to have already passed the hour hand. 45 1/11 minutes past 9: This is too early; the minute hand has not yet fully closed the initial angular gap at this time. 45 7/11 minutes past 9: Also too early for coincidence, because the minute hand still lags behind the hour hand. 47 1/11 minutes past 9: This is between the earlier incorrect values and still does not satisfy the equality of angles. Common Pitfalls: Learners often forget to include the 0.5t term for the hour hand or mistakenly assume that the hour hand stays fixed at 270 degrees. Another frequent error is to use an incorrect version of the shortcut formula or to round the fraction too early. To avoid these mistakes, always write the full angular expressions for both hands and solve the equation carefully before converting the fraction into mixed form. Final Answer: The hands of the watch are together at 49 1/11 minutes past 9.

Question 6

In 16 minutes, by how many degrees does the minute hand of a clock gain over the hour hand?

Accepted Answer

Introduction / Context: This question concerns the relative speed of the minute hand and the hour hand on a clock. Instead of asking for a specific time of coincidence, it asks directly by how many degrees the minute hand advances ahead of the hour hand during a period of 16 minutes. Such questions emphasise the concept of relative angular speed, which is fundamental to many clock problems. Given Data / Assumptions: Time interval considered: 16 minutes. The minute hand moves 360 degrees in 60 minutes. The hour hand moves 360 degrees in 12 hours, that is, 720 minutes. We must find the angle through which the minute hand gains over the hour hand in this period. Concept / Approach: First, we determine the angular speed of each hand. The minute hand has a higher angular speed than the hour hand. The difference between these speeds is the relative angular speed, that is, the rate at which the minute hand gains on the hour hand. Once we know this relative speed, we multiply it by the time interval to find the net angular gain of the minute hand over the hour hand in that period. Step-by-Step Solution: Step 1: Compute the speed of the minute hand. It completes 360 degrees in 60 minutes, so its speed is 360 / 60 = 6 degrees per minute. Step 2: Compute the speed of the hour hand. It completes 360 degrees in 12 hours, or 12 * 60 = 720 minutes, so its speed is 360 / 720 = 0.5 degrees per minute. Step 3: Determine the relative speed of the minute hand compared with the hour hand: relative speed = 6 - 0.5 = 5.5 degrees per minute. Step 4: In 16 minutes, the gain in angle is: gain = 5.5 * 16 degrees. Step 5: Multiply 5.5 by 16: 5.5 * 16 = (5 * 16) + (0.5 * 16) = 80 + 8 = 88 degrees. Step 6: Therefore, in 16 minutes the minute hand gains 88 degrees over the hour hand. Verification / Alternative check: To verify, separately compute the actual rotation of each hand in 16 minutes. The minute hand rotates 6 * 16 = 96 degrees. The hour hand rotates 0.5 * 16 = 8 degrees. The gain is 96 - 8 = 88 degrees, which matches the result from the relative speed method. This cross check confirms that the calculation is correct. Why Other Options Are Wrong: 16 degrees: This would be the rotation of a hand moving at 1 degree per minute over 16 minutes, which is not the relative speed here. 80 degrees: This is the approximate rotation of the minute hand minus almost no movement of the hour hand, but the hour hand does move and the precise difference is 88 degrees. 94 degrees: Larger than the correct gain and not supported by the exact speeds of the hands. 72 degrees: This might come from incorrectly using 4.5 degrees per minute as the relative speed, which is not correct for clock hands. Common Pitfalls: Students sometimes forget that the hour hand moves during the interval and incorrectly take the gain as equal to the full movement of the minute hand. Others miscalculate the speed of the hour hand, using 30 degrees per hour but forgetting to convert this correctly into degrees per minute. Always calculate both individual speeds and then subtract to get the relative speed for these types of questions. Final Answer: In 16 minutes, the minute hand gains 88 degrees over the hour hand.

Question 7

How many degrees will the minute hand of a clock move in the same time that the second hand moves through 4800 degrees?

Accepted Answer

Introduction / Context: This question links the motions of the second hand and the minute hand on a clock. Instead of directly using time, it describes the second hand moving through a specific angular distance and asks how far the minute hand moves during exactly the same interval. The problem therefore tests understanding of angular speeds of different hands and proportional reasoning. Given Data / Assumptions: The second hand moves through 4800 degrees. The second hand completes 360 degrees in 60 seconds. The minute hand completes 360 degrees in 60 minutes, that is, 3600 seconds. We must find the angle moved by the minute hand in the same time the second hand uses to move 4800 degrees. Concept / Approach: First, we use the angular speed of the second hand to find the time taken for it to move 4800 degrees. Once this time is known, we use the angular speed of the minute hand to compute how far it moves in that same interval. The key is to keep units consistent, typically using seconds for both hands, and then apply simple direct proportion. Step-by-Step Solution: Step 1: Calculate the speed of the second hand. It completes 360 degrees in 60 seconds, so its speed is 360 / 60 = 6 degrees per second. Step 2: Time taken by the second hand to move 4800 degrees is: time = 4800 / 6 seconds. Step 3: Compute this value: 4800 / 6 = 800 seconds. Step 4: Now find the angular speed of the minute hand. It completes 360 degrees in 3600 seconds, so its speed is 360 / 3600 = 0.1 degrees per second. Step 5: In the same 800 seconds, the minute hand moves: angle = 0.1 * 800 degrees. Step 6: Multiply to get angle = 80 degrees. Step 7: Therefore, the minute hand moves 80 degrees while the second hand moves 4800 degrees. Verification / Alternative check: Another way to see this is as a ratio problem. The speeds of the second and minute hands are 6 degrees per second and 0.1 degrees per second respectively. The ratio of their speeds is 6 : 0.1 = 60 : 1. So, when the second hand moves 4800 degrees, the minute hand moves 4800 / 60 = 80 degrees. This shortcut matches the detailed calculation, confirming the result. Why Other Options Are Wrong: 160 degrees: This is double the correct value and would result from dividing by 30 instead of 60. 140 degrees: Not supported by any correct ratio of speeds for the two hands. 135 degrees: This is a common approximate value in some angle problems but does not fit the exact proportion here. 60 degrees: This would correspond to a 1:80 ratio instead of 1:60, which is not correct for these hand speeds. Common Pitfalls: Typical errors include mixing up minutes and seconds or assigning incorrect speeds to the hands. Some students incorrectly assume that the minute hand moves 6 degrees per minute and then forget to convert time units consistently. Always convert both motions into the same time unit, preferably seconds, and then apply direct proportionality using the correct angular speeds. Final Answer: In the given interval, the minute hand moves 80 degrees.

Question 8

In one full day of 24 hours, how many times are the hands of a clock exactly at right angles to each other?

Accepted Answer

Introduction / Context: This question asks for the number of times in a day that the hour hand and the minute hand of a clock are at a right angle, that is, separated by 90 degrees. Such counting questions are common in clock problems and require understanding how often certain angular relationships occur in a 12 hour cycle and then extending that to 24 hours. Given Data / Assumptions: A day has 24 hours. We consider an ordinary analog clock with hour and minute hands. We define a right angle as a 90 degree separation between the two hands. The motions of the hands are uniform. Concept / Approach: In 12 hours, the hour and minute hands are at right angles a certain number of times. Because 24 hours is simply two cycles of 12 hours, we can count the number of right angle occurrences in one 12 hour period and then double that count. The main task is to know or derive how many right angle positions exist in 12 hours. For each hour interval, there are usually two moments when the hands form a right angle, though in some cases one of these can fall outside the interval. Step-by-Step Solution: Step 1: Over 12 hours, the minute hand makes 12 full revolutions, while the hour hand makes one. Step 2: It is a known result from clock theory that in 12 hours, the hands are at right angles 22 times. Step 3: One way to see this is that, in each hour, the relative motion of the hands leads to approximately two right angle positions, so across 11 closer intervals there are 22 such occurrences. Step 4: Because 24 hours equals two blocks of 12 hours, the pattern of right angle occurrences repeats in the second half of the day. Step 5: Therefore, in 24 hours, the total number of times the hands are at right angles is 22 * 2 = 44. Step 6: Hence, the answer is 44 times in one day. Verification / Alternative check: For a more detailed reasoning, note that the relative angular speed of the hands is 5.5 degrees per minute. The angle between them increases or decreases at this rate. The condition for a right angle is that the difference in their positions is 90 degrees or 270 degrees. Solving these equations for all times in a 12 hour range leads to 22 distinct solutions. Doubling this count for 24 hours again gives 44, confirming the standard result. Why Other Options Are Wrong: 54: This would imply 27 right angles in 12 hours, which is more than the known maximum for the relative motion. 64: This would mean more than 2.5 right angle occurrences per hour on average, which is impossible given the uniform movement. 22: This is the number for 12 hours only, not for the entire day. 36: This count does not correspond to any standard cycle result and is neither 1.5 nor 2 times the known 12 hour value. Common Pitfalls: Many learners confuse the counts for coincidences and right angles. The hands coincide 22 times in 24 hours, but they are at right angles twice as often, that is, 44 times. Another mistake is assuming exactly two right angle positions in every single hour without considering the transitions near 11 and 12, which can slightly alter the straightforward count. Remembering the benchmark values 22 coincidences and 44 right angle positions per day is very helpful. Final Answer: The hands of a clock are at right angles 44 times in one day.

Question 9

How many degrees will the minute hand of a clock move in the same time that the second hand moves through 5400 degrees?

Accepted Answer

Introduction / Context: This problem, like an earlier one, compares the movement of the second hand and the minute hand. The second hand is given to move through 5400 degrees, and we are asked how far the minute hand moves in that same period. This is a straightforward proportional reasoning question based on angular speeds of different clock hands. Given Data / Assumptions: The second hand moves 5400 degrees. The second hand completes 360 degrees in 60 seconds. The minute hand completes 360 degrees in 3600 seconds. We must find the angular movement of the minute hand in the same time interval. Concept / Approach: First, determine the time taken for the second hand to move 5400 degrees. To do that, we use its known angular speed in degrees per second. Once we know the duration, we use the angular speed of the minute hand to calculate how far it moves in that same number of seconds. This is a classic speed multiplied by time approach, using consistent units. Step-by-Step Solution: Step 1: The second hand completes 360 degrees in 60 seconds, so its speed is 360 / 60 = 6 degrees per second. Step 2: Time taken to move 5400 degrees is: time = 5400 / 6 seconds. Step 3: Compute this: 5400 / 6 = 900 seconds. Step 4: The minute hand completes 360 degrees in 3600 seconds, so its speed is 360 / 3600 = 0.1 degrees per second. Step 5: In 900 seconds, the minute hand moves: angle = 0.1 * 900 degrees. Step 6: Calculate the angle: 0.1 * 900 = 90 degrees. Step 7: Therefore, the minute hand moves 90 degrees during the time the second hand moves 5400 degrees. Verification / Alternative check: Again, this can be verified using the ratio of speeds. The second hand moves at 6 degrees per second, the minute hand at 0.1 degrees per second. The ratio of their speeds is 6 : 0.1 = 60 : 1. Hence, for every 60 degrees moved by the second hand, the minute hand moves 1 degree. For 5400 degrees of movement by the second hand, the minute hand movement is 5400 / 60 = 90 degrees. This confirms the previous calculation. Why Other Options Are Wrong: 85 degrees: Close to the correct value but not equal to the exact ratio, so it cannot be correct. 60 degrees: Would correspond to a 90 : 1 speed ratio, which is not the case for these hands. 45 degrees: This is half the correct answer and might result from accidentally dividing by 120 instead of 60. 75 degrees: Does not match the precise 60 : 1 proportion between second hand and minute hand speeds. Common Pitfalls: Learners may mistakenly treat the minute hand speed as 6 degrees per minute rather than converting everything into degrees per second. Others may divide by the wrong factor when using the ratio method. Always check the basic angular speeds and ensure that time units match before applying any proportion or calculation. Final Answer: The minute hand moves 90 degrees in the time that the second hand moves 5400 degrees.

Question 10

Between 3 o clock and 4 o clock, at what exact time (in clock form) will the hour hand and the minute hand coincide?

Accepted Answer

Introduction / Context: This question is another version of the classic coincidence problem, but this time the answer is requested in full clock time format instead of just minutes past the hour. We must determine the precise time between 3 o clock and 4 o clock when the hour hand and minute hand are exactly together. Understanding how to convert the computed minutes into standard time notation is an essential skill for such problems. Given Data / Assumptions: Time interval of interest: between 3 o clock and 4 o clock. At 3 o clock, the hour hand points at 3 and the minute hand points at 12. Minute hand speed: 6 degrees per minute. Hour hand speed: 0.5 degrees per minute. We must express the final answer as a clock time, such as 3:16 4/11. Concept / Approach: As in an earlier question, the approach is to apply relative speed. At the start, the hour hand is ahead in angle. Because the minute hand moves faster, it eventually catches up and coincides with the hour hand. We equate the angular positions of both hands after t minutes and solve for t. Once t is obtained, we add it to 3 o clock and express the result in the required format H:MM fraction. Step-by-Step Solution: Step 1: At 3 o clock, the hour hand is at 3 * 30 = 90 degrees. Step 2: The minute hand at 3 o clock is at 0 degrees. Step 3: After t minutes, the minute hand is at 6t degrees. Step 4: After t minutes, the hour hand is at 90 + 0.5t degrees. Step 5: For coincidence, set the angles equal: 6t = 90 + 0.5t. Step 6: Rearranging gives 6t - 0.5t = 90, so 5.5t = 90. Step 7: Solving, t = 90 / 5.5 = 180 / 11 minutes. Step 8: Express 180 / 11 as a mixed number. 11 goes into 180 sixteen times (16 * 11 = 176) with a remainder of 4, so t = 16 4/11 minutes past 3. Step 9: Therefore, the exact clock time is 3:16 4/11. Verification / Alternative check: The shortcut formula for the time in minutes after H o clock when the hands coincide is t = (60 * H) / 11. With H = 3, t = (60 * 3) / 11 = 180 / 11 = 16 4/11 minutes. This matches our algebraic solution. Converting this value into clock notation gives 3:16 4/11, confirming the result. Why Other Options Are Wrong: 3:16 7/11: This corresponds to a slightly larger value of t than the correct 16 4/11 and would result in the minute hand overshooting the hour hand. 3:16 11/4: This expression is not even a valid minute fraction less than one minute and clearly does not correspond to the required small fractional part. 3:30: At 3:30, the minute hand is at 180 degrees, while the hour hand is halfway between 3 and 4 at 105 degrees, so they are not coincident. 3:18 4/11: This is another time slightly later than 3:16 4/11 and does not satisfy the exact coincidence condition. Common Pitfalls: A major source of error is mismanaging fractional minutes and incorrectly converting an improper fraction into a mixed number. Some learners also forget that the hour hand moves during the time t and mistakenly set 6t = 90. Using the shortcut formula as a check can prevent such mistakes, but only if the formula is correctly remembered and applied. Final Answer: The hands coincide at 3:16 4/11.

Question 11

When the time shown by a clock is 4:15, what is the angle between the minute hand and the hour hand?

Accepted Answer

Introduction / Context: Angle at a given time questions require converting time into the angular positions of each hand on the clock. Here, the clock shows 4:15, and you need to find the angle between the minute hand and the hour hand. This involves calculating both positions from the 12 o clock mark and then finding the difference between them. It is a standard and important concept in aptitude questions on clocks. Given Data / Assumptions: Time is 4:15. The minute hand moves 360 degrees in 60 minutes, so its speed is 6 degrees per minute. The hour hand moves 360 degrees in 12 hours, that is, in 720 minutes, so its speed is 0.5 degrees per minute. We measure angles from the 12 o clock position in a clockwise direction. Concept / Approach: The minute hand angle is straightforward: multiply the number of minutes by 6 degrees. The hour hand angle depends on both the hour and the minutes past the hour. Once the separate angles are known, take the absolute difference between them to find the angle between the hands. If needed, we may also choose the smaller of the two possible angles, but at 4:15 the direct difference is already less than 180 degrees. Step-by-Step Solution: Step 1: Compute the minute hand angle. At 4:15, the minute hand points at 15 minutes, so its angle is 15 * 6 = 90 degrees from 12 o clock. Step 2: Compute the hour hand angle. At 4:00, the hour hand is at 4 * 30 = 120 degrees. Step 3: From 4:00 to 4:15, the hour hand moves further because it moves continuously at 0.5 degrees per minute. In 15 minutes, it moves 15 * 0.5 = 7.5 degrees. Step 4: Therefore, at 4:15, the hour hand angle from 12 is 120 + 7.5 = 127.5 degrees. Step 5: Find the angle between the hands by taking the absolute difference: difference = |127.5 - 90| = 37.5 degrees. Step 6: Since 37.5 degrees is less than 180 degrees, it is already the smaller angle between the hands, and no further adjustment is needed. Verification / Alternative check: We can use the general formula for the angle between hands: angle = |30H - 5.5M|, where H is the hour and M is minutes. Here H = 4 and M = 15. Compute: |30 * 4 - 5.5 * 15| = |120 - 82.5| = 37.5 degrees. This matches the value obtained from direct calculations, verifying that the answer is correct. Why Other Options Are Wrong: 0 degrees: This would mean the hands coincide, which clearly does not happen at 4:15. 27 degrees: Not obtained from any correct use of the formula or from the actual angles 90 and 127.5 degrees. 15 degrees: This is too small and might result from incorrectly ignoring the continuous motion of the hour hand. 52.5 degrees: Slightly larger than the correct value and likely arises from miscalculating either the hour hand or minute hand angle. Common Pitfalls: One frequent mistake is to assume that at 4:15 the hour hand is still exactly at 120 degrees, which ignores its movement during the 15 minutes. Another pitfall is misusing the formula and substituting the wrong values for H and M. Careful conversion of time into angles and simple absolute differences are enough to avoid these errors. Final Answer: The angle between the hands at 4:15 is 37.5 degrees.

Question 12

At 6 o clock a clock strikes 6 times, and the time between the first and the last strike is 30 seconds. Assuming the interval between any two consecutive strikes is constant, how long (in seconds) will it take to strike 12 times at 12 o clock?

Accepted Answer

Introduction / Context: This question is about the striking pattern of a clock rather than the rotation of its hands. Many aptitude exams include such problems where you are given the time between the first and last strike for a certain hour and asked to predict the time taken to strike a different number of times. The key idea is that the time refers to the intervals between strikes, not to the number of strikes themselves. Given Data / Assumptions: At 6 o clock, the clock strikes 6 times. The time between the first strike and the last strike at 6 o clock is 30 seconds. The interval between any two consecutive strikes is constant. We must find the time between the first and last strike when the clock strikes 12 times at 12 o clock. Concept / Approach: If a clock strikes n times, then there are not n intervals but rather n minus 1 equal intervals between the first and last strike. The given time refers to the sum of these intervals. Once we know the length of one interval, we can easily compute the time required for any other number of strikes by multiplying the interval by the number of gaps, that is, the new number of strikes minus one. Step-by-Step Solution: Step 1: At 6 o clock the clock strikes 6 times, creating 6 - 1 = 5 equal intervals between the first and last strike. Step 2: The total time for these 5 intervals is given as 30 seconds. Step 3: Therefore, the time per interval is: interval time = 30 / 5 = 6 seconds. Step 4: At 12 o clock, the clock must strike 12 times. Step 5: The number of intervals between the first and last strike at 12 o clock is 12 - 1 = 11 intervals. Step 6: Each interval is 6 seconds, so the total time for all 11 intervals is: total time = 11 * 6 = 66 seconds. Step 7: Thus, it will take 66 seconds for the clock to complete 12 strikes at 12 o clock. Verification / Alternative check: An easy way to verify the logic is to test with a smaller example. If a clock struck 2 times, there would be 1 interval between them. If that interval was 6 seconds, then the time between the first and second strike would be 6 seconds. Extending this pattern, for 3 strikes we would have 2 intervals, taking 2 * 6 = 12 seconds, and so on. This pattern is linear in the number of intervals and confirms that using 11 intervals for 12 strikes is correct. Why Other Options Are Wrong: 55 seconds: This would correspond to a per interval time of 5 seconds for 11 intervals, which contradicts the 30 second figure at 6 o clock. 36 seconds: This equals 6 intervals of 6 seconds each, but 12 strikes require 11 intervals, not 6. 24 seconds: Much too small and would require an unrealistically short interval between strikes compared with the given data. 60 seconds: This would correspond to 10 intervals of 6 seconds each, which is still short of the required 11 intervals. Common Pitfalls: The most common mistake is to equate the number of strikes directly with the number of intervals, forgetting that n strikes create only n minus 1 gaps. Students may also mistakenly compute interval time as 30 / 6 instead of 30 / 5. Keeping track of intervals instead of just counts helps ensure accurate calculations. Final Answer: At 12 o clock, the clock will take 66 seconds to strike 12 times.

Question 13

Between 3 o clock and 4 o clock, at what exact time will the hour hand and the minute hand be on the same straight line but facing in opposite directions (forming a 180 degree angle)?

Accepted Answer

Introduction / Context: This problem is about the hour and minute hands being in a straight line but pointing in opposite directions, so that the angle between them is 180 degrees. Unlike coincidence problems where the angle is 0, here we want the hands to form a straight line between 3 o clock and 4 o clock. This involves setting the angular difference between the hands equal to 180 degrees and solving for the time. Given Data / Assumptions: We are considering the interval between 3 o clock and 4 o clock. At 3 o clock, the hour hand is on the 3 and the minute hand is on the 12. The minute hand moves 6 degrees per minute. The hour hand moves 0.5 degrees per minute. We require the angle between the two hands to be 180 degrees. Concept / Approach: At any time t minutes after 3 o clock, the angle of the minute hand and the angle of the hour hand can be expressed from the 12 o clock position. The angle between the hands is the absolute difference of these two angles. For the hands to be in a straight line but opposite each other, this difference must be 180 degrees. We set up an equation based on this condition and solve for t to find the required time. Step-by-Step Solution: Step 1: At 3 o clock, the hour hand angle from 12 is 3 * 30 = 90 degrees. Step 2: The minute hand angle from 12 at exactly 3 o clock is 0 degrees. Step 3: After t minutes, the minute hand angle is 6t degrees. Step 4: After t minutes, the hour hand angle is 90 + 0.5t degrees. Step 5: The angle between the hands is |6t - (90 + 0.5t)| degrees. Step 6: For the hands to be opposite each other, set this equal to 180 degrees: 6t - (90 + 0.5t) = 180. Step 7: Simplify: 6t - 90 - 0.5t = 180, so 5.5t - 90 = 180. Step 8: Solve for t: 5.5t = 270, so t = 270 / 5.5 = 540 / 11 minutes. Step 9: Convert 540 / 11 to a mixed number. 11 goes into 540 forty nine times (49 * 11 = 539) with a remainder of 1, so t = 49 1/11 minutes. Step 10: The required time is therefore 3:49 1/11. Verification / Alternative check: We can check quickly by computing the angles at t = 49 1/11 minutes. The minute hand angle is 6 * (540 / 11) = 3240 / 11 degrees. The hour hand angle is 90 + 0.5 * (540 / 11) = 90 + 270 / 11 = (990 + 270) / 11 = 1260 / 11 degrees. The difference is (3240 - 1260) / 11 = 1980 / 11 = 180 degrees exactly. This confirms that at 3:49 1/11 the hands are 180 degrees apart and therefore lie on the same straight line in opposite directions. Why Other Options Are Wrong: 3:15 2/8: This is much earlier than the correct time and the hands are far from being opposite. 3:49: This is close but slightly earlier; the angle between the hands will be slightly less than 180 degrees. 3:51: Later than the correct time, by which point the minute hand angle has gone past the exact 180 degree separation. 3:45: A commonly guessed time, but at 3:45 the angle between the hands is not exactly 180 degrees. Common Pitfalls: Common mistakes include using 90 degrees instead of 180 degrees in the equation, which corresponds to a right angle rather than a straight line, or neglecting to take the absolute value of the angle difference. Some students also stop as soon as they find a value that looks close, such as 3:49, without converting the exact fraction or verifying the result by substituting it back into the angle expressions. Final Answer: The hands are on the same straight line but opposite each other at 3:49 1/11.

Question 14

In one full day of 24 hours, how many times do the hour hand and the minute hand of a clock exactly coincide?

Accepted Answer

Introduction / Context: This question asks for the number of times in a full day that the hour and minute hands of a clock exactly coincide, that is, occupy the same position. Counting such events is a standard result in clock problems and reinforces understanding of relative motion between the hands over a 12 hour cycle and a 24 hour day. Given Data / Assumptions: A day has 24 hours. We consider a normal analog clock with hour and minute hands. Two hands coincide when the angle between them is zero degrees. The motion of the hands is uniform and continuous. Concept / Approach: In 12 hours, the minute hand laps or overtakes the hour hand a certain number of times. Each overtaking corresponds to one coincidence. Because 24 hours is simply two such 12 hour cycles back to back, we can determine the number of coincidences in 12 hours and then double this value to cover the whole day. The key is to remember that the hands do not coincide 12 times in 12 hours but slightly fewer times due to their relative speeds. Step-by-Step Solution: Step 1: Consider the relative speed of the hands. The minute hand moves at 6 degrees per minute, and the hour hand at 0.5 degrees per minute, so the relative speed is 5.5 degrees per minute. Step 2: To coincide, the minute hand must gain a full 360 degrees over the hour hand. Step 3: The time between two consecutive coincidences is: interval = 360 / 5.5 minutes = 65 5/11 minutes. Step 4: Now look at a 12 hour period. In 12 hours, the minute hand laps the hour hand exactly 11 times, resulting in 11 coincidences. Step 5: Because 24 hours contains two such 12 hour periods, the total number of coincidences in a full day is 11 * 2 = 22. Step 6: Therefore, the hands coincide 22 times in one day. Verification / Alternative check: We can convince ourselves by considering the positions of the hour hand at each coincidence. In 12 hours, there are 11 coincidences, each roughly one hour and five minutes apart. If we start at 12 o clock at noon and track through the next 12 hours, we see that the last coincidence before 12 midnight occurs slightly before the hour, so we do not get a twelfth coincidence in that range. The pattern repeats in the next 12 hours, giving another 11 coincidences, consistent with the total of 22 per day. Why Other Options Are Wrong: 20: This would imply only 10 coincidences in 12 hours, which is fewer than the known result. 21: Also inconsistent with the 11 coincidences in a single 12 hour period. 23: This would mean one of the 12 hour periods has 12 coincidences, which is not possible. 24: A common incorrect guess based on one coincidence per hour, but the hands do not coincide every hour exactly. Common Pitfalls: Students often assume that since there are 12 hours, the hands coincide 12 times in 12 hours, and hence 24 times in a day. This ignores the exact timing and relative speed of the hands. Another mistake is confusing the number of right angle positions (44 times per day) with the number of coincidences (22 times per day). Keeping these two counts separate helps avoid confusion. Final Answer: The hour and minute hands coincide 22 times in one day.

Question 15

In every 60 minutes, by how many minutes on the dial does the minute hand gain over the hour hand?

Accepted Answer

Introduction / Context: This question is phrased in terms of minutes on the dial rather than degrees. It asks how many dial minutes the minute hand gains over the hour hand in one hour, that is, in 60 minutes. This is another way of talking about relative speed, but instead of using degrees per minute, we use positions on the clock face measured in minute divisions. Given Data / Assumptions: We consider a time interval of 60 minutes. On the clock dial, there are 60 minute divisions around the circle. The minute hand moves 60 minute divisions in 60 minutes, one full revolution. The hour hand moves from one hour mark to the next in 60 minutes. We must find how many minute divisions separate the two hands after 60 minutes, that is, the gain of the minute hand over the hour hand. Concept / Approach: On a 12 hour clock face, the hour hand moves from one number to the next in 60 minutes. Each hour mark corresponds to 5 minute divisions (because 12 * 5 = 60). Therefore, in 60 minutes, the hour hand moves 5 minute divisions, while the minute hand moves 60 minute divisions. The gain of the minute hand is the difference between these two movements. Step-by-Step Solution: Step 1: In 60 minutes, the minute hand completes one full revolution, moving through all 60 minute divisions on the clock face. Step 2: The hour hand moves from one hour mark to the next in 60 minutes. Each hour mark is separated by 5 minute divisions, so the hour hand moves 5 minute divisions in the same 60 minutes. Step 3: The gain of the minute hand over the hour hand is: gain = movement of minute hand minus movement of hour hand. Step 4: Substitute the values: gain = 60 - 5 = 55 minute divisions. Step 5: Therefore, in every 60 minutes, the minute hand gains 55 minutes on the dial over the hour hand. Verification / Alternative check: We can also express this in degrees to cross check. One minute division corresponds to 6 degrees (since 360 degrees / 60 divisions = 6 degrees per division). A gain of 55 minute divisions corresponds to 55 * 6 = 330 degrees. If we compute the relative angular speed earlier as 5.5 degrees per minute, then in 60 minutes the gain in degrees is 5.5 * 60 = 330 degrees, which again matches 55 minute divisions. This confirms that the gain is 55 minutes on the dial. Why Other Options Are Wrong: 53: This would correspond to 318 degrees of gain, not matching the 330 degree result from relative angular speed. 54: This is 324 degrees and still less than the true gain. 56: This would represent 336 degrees, which is larger than the possible relative gain in 60 minutes. 50: This is a round number but does not align with the actual calculations in either minutes or degrees. Common Pitfalls: Some learners mistakenly assume that the hour hand does not move during the hour and conclude that the gain is all 60 minutes. Others misinterpret the phrase gain in minutes and confuse it with minutes of time instead of dial divisions. Carefully distinguishing between dial divisions and minutes of real time is essential for solving such questions correctly. Final Answer: In every 60 minutes, the minute hand gains 55 minutes on the hour hand.

Question 16

When the time shown by a clock is 4:20, what is the angle between the minute hand and the hour hand?

Accepted Answer

Introduction / Context: This question again deals with finding the angle between the hour hand and the minute hand at a specific time, here 4:20. These problems test your ability to convert time into angular positions and then calculate their difference. By practising such questions at different times, you strengthen your understanding of how both hands move continuously. Given Data / Assumptions: Time is 4:20. The minute hand moves 360 degrees in 60 minutes, that is, 6 degrees per minute. The hour hand moves 360 degrees in 12 hours, that is, 0.5 degrees per minute. Angles are measured from the 12 o clock position. Concept / Approach: We calculate the position of the minute hand by multiplying the number of minutes by 6 degrees. To find the position of the hour hand, we start with the base angle for the hour and add the extra motion caused by the minutes past the hour. Then we take the absolute difference between the two angles to obtain the angle between the hands. If needed, we can use the general formula |30H - 5.5M| for quick computation. Step-by-Step Solution: Step 1: Find the angle of the minute hand at 4:20. It is at 20 minutes, so its angle from 12 o clock is 20 * 6 = 120 degrees. Step 2: Find the angle of the hour hand. At 4:00, the hour hand is at 4 * 30 = 120 degrees. Step 3: From 4:00 to 4:20, the hour hand moves further at 0.5 degrees per minute. In 20 minutes it moves 20 * 0.5 = 10 degrees. Step 4: Thus, at 4:20 the hour hand angle is 120 + 10 = 130 degrees. Step 5: The angle between the hands is the absolute difference: difference = |130 - 120| = 10 degrees. Step 6: Since 10 degrees is less than 180 degrees, this is the required angle. Verification / Alternative check: Use the general formula angle = |30H - 5.5M| with H = 4 and M = 20. Compute: |30 * 4 - 5.5 * 20| = |120 - 110| = 10 degrees. This matches the detailed angle calculations, confirming the result. Why Other Options Are Wrong: 0 degrees: This would correspond to the hands coinciding, which does not happen at 4:20. 30 degrees: This is a common angle at some times but not at 4:20 based on the actual positions. 60 degrees: Much larger than the actual separation; might come from a rough guess rather than calculation. 20 degrees: Close but still double the correct value, often arising from mixing up angles or misusing the formula. Common Pitfalls: One common error is to forget the additional movement of the hour hand during the 20 minutes and assume it is still at 120 degrees. Another pitfall is incorrectly substituting into the formula, for example using minutes only and ignoring the hour. Always compute each hand position carefully and then take the absolute difference. Final Answer: At 4:20, the angle between the hands is 10 degrees.

Question 17

If the current time in the USA is 4:15 a.m. and Indian Standard Time (IST) is always 9 hours 30 minutes ahead of USA time, what will be the time in India exactly 18 hours from now?

Accepted Answer

Introduction / Context: This is a time zone problem that combines a fixed time difference between two locations with an additional time interval into the future. You are told the present time in the USA, the offset between USA time and Indian Standard Time, and asked to find what the time in India will be 18 hours from now. Such questions test your ability to handle time zone conversions and to track day changes correctly. Given Data / Assumptions: Current time in the USA is 4:15 a.m. IST is ahead of USA time by 9 hours 30 minutes (IST = USA time + 9 hours 30 minutes). We must find the time in India exactly 18 hours from the current moment. We assume the date itself is not required, only the time and whether it is the same day or the next day. Concept / Approach: First, determine the current time in India using the given offset between USA time and IST. Once the current IST is known, we add 18 hours to it to find the future IST. Because adding 18 hours can extend into the next calendar day, we must handle the wrap around of the 24 hour clock and state clearly whether the result is on the same day or the next morning. Step-by-Step Solution: Step 1: Start with the current USA time: 4:15 a.m. Step 2: Add the time zone difference of 9 hours 30 minutes to convert to IST: 4:15 a.m. + 9 hours 30 minutes = 13:45, which is 1:45 p.m. in India. Step 3: Thus, the current time in India is 1:45 in the afternoon. Step 4: Now we must find the time in India 18 hours from this instant. Step 5: Add 18 hours to 1:45 p.m. First, 12 hours after 1:45 p.m. is 1:45 a.m. the next day. Step 6: There are still 6 more hours to add (since 18 = 12 + 6). Add 6 hours to 1:45 a.m. This gives 7:45 a.m. on the next day in India. Step 7: Therefore, the required time in India is 7:45 in the next morning. Verification / Alternative check: In 24 hour format, 1:45 p.m. is 13:45. Add 18 hours to 13:45: 13:45 + 18:00 = 31:45. Since 24:00 corresponds to midnight, subtract 24 hours: 31:45 - 24:00 = 7:45. Because we crossed midnight, the time belongs to the next day. This matches the earlier step by step reasoning and confirms that the time is 7:45 a.m. the next morning. Why Other Options Are Wrong: 1:45 in the afternoon: This is the current IST, not the time 18 hours later. 7:45 in the evening: That would be only 6 hours after 1:45 p.m., not 18 hours. 1:45 in the next morning: This corresponds to 12 hours after the starting IST, not 18 hours. 10:45 in the evening: This time does not match any natural decomposition of the 18 hour addition from 1:45 p.m. Common Pitfalls: A common mistake is to add 18 hours directly to the original USA time without first converting to IST, which mixes time zones incorrectly. Others miscount the day change when the sum exceeds 24 hours and may forget to mark that the result occurs the next day. Using 24 hour format often makes the addition clearer and helps avoid these off by one day errors. Final Answer: Eighteen hours from now, the time in India will be 7:45 in the next morning.

Question 18

Between 3 o clock and 4 o clock, after how many minutes past 3 will the hour hand and minute hand of a clock exactly coincide?

Accepted Answer

Introduction / Context: This is another version of the standard clock problem asking when the hour and minute hands coincide between 3 o clock and 4 o clock. The focus is again on computing the precise number of minutes after 3 o clock at which the hands are exactly together. Such repeated practice reinforces the method of forming and solving the coincidence equation. Given Data / Assumptions: We consider the time interval between 3 o clock and 4 o clock. At 3 o clock, the hour hand is at the 3 mark and the minute hand at the 12 mark. The minute hand moves at 6 degrees per minute. The hour hand moves at 0.5 degrees per minute. We must find a time t minutes after 3 o clock when the two hands are coincident. Concept / Approach: As before, we model the movement of each hand as uniform circular motion. At 3 o clock the hour hand leads the minute hand by a fixed angle. Over time, the faster minute hand gains on the hour hand. When the hands coincide, their angular positions measured from 12 o clock are equal. Setting these angular expressions equal leads to a simple linear equation in t, which we solve and then convert to a mixed fraction in minutes. Step-by-Step Solution: Step 1: At 3 o clock, the hour hand angle from 12 is 3 * 30 = 90 degrees. Step 2: The minute hand angle at 3 o clock is 0 degrees because it points at 12. Step 3: After t minutes, the minute hand angle is 6t degrees. Step 4: After t minutes, the hour hand angle is 90 + 0.5t degrees. Step 5: For coincidence, set these angles equal: 6t = 90 + 0.5t. Step 6: Rearranging, 6t - 0.5t = 90 gives 5.5t = 90. Step 7: Solve for t: t = 90 / 5.5 = 90 * 2 / 11 = 180 / 11 minutes. Step 8: Convert 180 / 11 to mixed form. Since 11 * 16 = 176, t = 16 4/11 minutes. Step 9: Hence, the hands coincide exactly 16 4/11 minutes past 3. Verification / Alternative check: Using the standard formula for the time after H o clock when the hands are together, t = (60 * H) / 11. Substituting H = 3, we get t = (60 * 3) / 11 = 180 / 11 = 16 4/11 minutes. This matches our full derivation and confirms that the computed time is correct. Substituting t back into the angles 6t and 90 + 0.5t also shows that they are numerically equal. Why Other Options Are Wrong: 11 1/11 minutes: This is too early; the minute hand has not yet closed the full 90 degree gap by that time. 15 4/11 minutes: Slightly less than the correct value and does not satisfy the equation 6t = 90 + 0.5t exactly. 14 4/11 minutes: Even earlier and farther from the true coincidence moment. 12 4/11 minutes: Clearly too early for the faster minute hand to have caught up with the hour hand. Common Pitfalls: As with similar questions, errors often arise from forgetting the continuous motion of the hour hand and incorrectly setting 6t equal to 90. Another issue is mishandling the fraction 180 / 11 when converting to mixed form, leading to an incorrect expression of the final answer. Remembering the shortcut formula and verifying that it matches the detailed calculation is a good way to avoid mistakes. Final Answer: The hands coincide 16 4/11 minutes after 3 o clock.

Question 19

In a standard analog clock, when the hour hand and the minute hand are in exactly opposite directions (that is, they form a straight line of 180 degrees), how many minute spaces apart are the two hands on the dial?

Accepted Answer

Introduction / Context: Questions on clocks often test your understanding of how the circular dial is divided and how angles between the hour and minute hands are related to these divisions. In this problem, you are asked to find how many 'minute spaces' separate the hands when they are in exactly opposite directions, forming a straight line (180 degrees). Knowing the relationship between minute spaces and degrees makes this type of question very quick to solve once the concept is clear. Given Data / Assumptions: • The clock dial is a full circle of 360 degrees. • The dial is divided into 60 equal minute spaces around the circumference. • When the hands are in opposite directions, the angle between them is 180 degrees. • Each minute space corresponds to a fixed angle in degrees. Concept / Approach: The key idea is that the clock face is divided into 60 equal minute spaces. Since a full circle is 360 degrees, each minute space represents 360 / 60 degrees. Once we know the angle represented by one minute space, we can determine how many such spaces add up to 180 degrees, which corresponds to the hands being opposite each other in a straight line. Step-by-Step Solution: Step 1: A clock has 60 minute spaces over a full angle of 360 degrees. Step 2: Angle per minute space = 360 / 60 = 6 degrees. Step 3: When the hands are in opposite directions, they make an angle of 180 degrees. Step 4: Number of minute spaces corresponding to 180 degrees = 180 / 6. Step 5: 180 / 6 = 30 minute spaces. Step 6: Therefore, when the hands are opposite each other, they are 30 minute spaces apart on the dial. Verification / Alternative check: Another way to think about it is to note that 30 minute spaces is exactly half of the 60 spaces on the dial. Being opposite each other means the hands are separated by half the circle, which is consistent with a straight angle of 180 degrees. Since half of 60 is 30, this again confirms that the hands must be 30 minute spaces apart when they are opposite. Why Other Options Are Wrong: • 15 minute spaces: 15 * 6 = 90 degrees, which is a right angle, not opposite directions. • 20 minute spaces: 20 * 6 = 120 degrees, which is an obtuse angle but not a straight line. • 25 minute spaces: 25 * 6 = 150 degrees, still less than 180 degrees, so the hands are not exactly opposite. Common Pitfalls: Students sometimes confuse right angles (90 degrees) with opposite directions (180 degrees). Another common mistake is to forget that each minute space represents 6 degrees, and instead assume some other value or mix up hour divisions with minute divisions. Carefully remembering that 360 degrees divided by 60 minute marks gives 6 degrees per minute space helps avoid these errors. Also, mixing up the idea of overlap (0 degrees) and opposite directions (180 degrees) can lead to incorrect answers. Final Answer: When the two hands of a clock are in opposite directions, they are separated by 30 minute spaces on the dial.

Question 20

At 3:40 on an analog clock, what is the angle in degrees between the hour hand and the minute hand (taking the smaller angle between them)?

Accepted Answer

Introduction / Context: This question checks your understanding of how to calculate the angle between the hour hand and the minute hand of a clock at a given time. At 3:40, both hands have moved from their original positions, and you must carefully account for each hand's movement to find the smaller angle between them. Such problems are very common in aptitude tests and help build precision in handling relative motion on a circular dial. Given Data / Assumptions: • Time considered: 3:40 (3 hours and 40 minutes). • The minute hand moves 6 degrees per minute (360 degrees / 60 minutes). • The hour hand moves 30 degrees per hour (360 degrees / 12 hours). • The hour hand also moves 0.5 degrees per minute (30 degrees / 60 minutes). • We need the smaller of the two possible angles between the hands. Concept / Approach: To compute the angle between the two hands at a given time, we find their absolute positions in degrees from 12 o'clock and then take the absolute difference. The standard formulas are: Minute hand angle = 6 * minutes. Hour hand angle = 30 * hours + 0.5 * minutes. The required angle is the smaller of the two possible angles between the hands, usually obtained by taking the absolute difference between these angles and, if necessary, subtracting from 360 degrees to ensure it is the acute or smaller angle. Step-by-Step Solution: Step 1: At 3:40, the minute hand is at 40 minutes. Step 2: Minute hand angle = 6 * 40 = 240 degrees from 12 o'clock. Step 3: The hour hand at 3:00 is at 30 * 3 = 90 degrees. Step 4: In 40 minutes, the hour hand moves further by 0.5 * 40 = 20 degrees. Step 5: Total hour hand angle at 3:40 = 90 + 20 = 110 degrees. Step 6: Difference between the hands = |240 - 110| = 130 degrees. Step 7: 130 degrees is already less than 180 degrees, so it is the smaller angle between the two hands. Verification / Alternative check: We can quickly check the reasonableness: at 3:30, the angle between the hands is 75 degrees. By 3:40, the minute hand has moved 10 more minutes (60 degrees), and the hour hand has moved 5 more degrees. So the angle should increase by about 55 degrees. Starting from 75 degrees, 75 + 55 = 130 degrees, which matches our exact calculation. This confirms that 130 degrees is consistent and correct. Why Other Options Are Wrong: • 110 degrees: This would be the hour hand angle alone, not the difference between the two hands. • 120 degrees: This value does not match the calculated difference and ignores precise movement of both hands. • 140 degrees: This overestimates the separation and does not fit the accurate calculations using standard formulas. Common Pitfalls: A frequent mistake is to ignore the movement of the hour hand within the hour and assume it stays fixed at the hour mark, which would give an incorrect angle. Some students also forget to take the absolute difference or mistakenly compute 360 minus the difference when the direct difference is already the smaller angle. Carefully using the formulas for each hand and then taking the absolute difference helps avoid these errors. Final Answer: The smaller angle between the hour and minute hands of the clock at 3:40 is 130 degrees.

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