Number System Geometry Integers Fractions and Decimals Data Handling Mensuration Algebra Ratio and Proportion Symmetry and Practical Geometry Example 1: What is the smaller angle between the two hands of the clock at 5:00 O’clock? Sol : At 5 O’clock, the hour hand will be at 5 and the minute hand will be at 12. Now, they have a gap of 5 hour spaces and each hour space is 30 degrees. That means the angle between the two hands will be 30 × 5 = 150 degrees.
 Example 2: What is the smaller angle between the two hands of the clock at 2:20 pm? Sol: At 2 O’clock the hour hand will be at 2 and the minute hand will be at 12. Now, they have a gap of 2 hour spaces and each hour space is 30 degrees. That means at 2:00 clock the angle will be = 30 × 2 = 60 degrees. After that in 20 min, the relative coverage of the minute hand will be 20 × 11/2 = 110 degrees. Now take the difference between the two angles, which will become our answer i.e. 110 - 60 = 50 degrees. Required angle is the difference between the two = 120 – 70 = 50 degrees. Example 3: At what time between 3 and 4 are the minute and hour hand together? Sol: At 3 O’clock, the relative distance between the hour and the minute hand is 15 minutes. To catch up with the hour hand, the minute hand has to cover a relative distance of 15 minutes at the relative speed of 11/12 minutes per minute. Thus, time required = 15/(11/12) = 15 × 12/11 = 180/11 = 16 4/11 min. Example 4: The minute hand of a clock overtakes the hour hand at intervals of 63 minutes of correct time. How much does the clock lose or gain in a day? Sol: In a correct clock, the minute hand and hour hand should meet after every 65 5/11 min. But we know that they are meeting after every 63 minutes. So gain in 63 minutes is 27/11 minutes. Gain in 24 hours =(24 ×60/63) × (27/11) = 56 8/77 min. Example 5: At what time between 4 and 5 are the minute and hour hand at right angles? Sol: At 4 O’clock, the relative distance between the hour and the minute hand is 20 minutes. To make a 90–degree angle with the hour hand, the minute hand has to cover a relative distance of 5 minutes at the relative speed of (11/12) minutes per minute. Thus, time required = 5/(11/12) = 60/11 or 5 (5/11) minutes. As explained above in important points, there is still one more case. When a relative distance of 35 minutes has been covered, even then the angles would be at right angles. Time required = 35/(11/12) = 420/11 or 38(2/11) minutes You can also think that the difference between the two right angles themselves be equal to 30 min. In one of the right angles, the minute hand will be 15 min before the hour hand and in the other; it will be 15 min ahead of the hour hand. Example 6: A watch gains uniformly. It was observed that it was 6 min slow at 12 o’clock in the night on Monday. On Friday at 6 pm it was 4 min 48 second fast. When was it correct? Sol:The time between 12 O’clock on Monday night and 6 pm on Friday = 90 hours. Now, the watch gains 6 + 4 4/5 min 6 + 24/5 = 54/5 minutes in 90 hours. So, the watch gains 6 minutes in (90×5×6)/54 = 50 hours Add 50 hours in Monday 12’O clock night, thus the watch is correct at 2:00 am on Thursday. Example 7: A clock is set right at 10 am. The clock gains 5 minutes in 12 hours. What will be the true time when the clock indicates 3 pm on the next day? Sol: Time from 10 am to 3 pm on the following day is 29 hrs. Now, 12 hrs 5 min i.e. 145/12 hrs of this clock = 12 hours of correct clock. So, 29 hours of this clock is (29×12×12)/145 = 144/5 = 28 hours 48 minutes. So, the time is 12 minutes before 3 pm. Example 8: The minute hand of a clock overtakes the hour hand at intervals of 65 minutes of correct time. How much does the clock lose or gain in 12 hours? Sol: In a correct clock, the minute hand and hour hand should meet after every 65 5/11 min. But we know that they are meeting after every 65 minutes. So, the gain in 65 minutes is 5/11 minutes. Gain in 12 hours = (120×60/65) × (5/11) = 720/143 = 5 5/143 min.
Example 9: At what time between 4 and 5, the minute hand will be 2 minutes spaces ahead of hour hand? Sol: At 4 O’ clock, the two hands are 20 min spaces apart. In this case, the min hand will have to gain (20 + 2) i.e. 22 – minute spaces. So, 22 – minute spaces will be gained in (60/55) × 22 = 24 min.
What is the angle formed by the minute hand and the hour hand at 4:45? Possible Answers:
Correct answer: Explanation: The angle measure between any two consecutive numbers on a clock is Call the "12" point on the clock the zero-degree point. At 4:45, the minute hand is at the "9" - that is, at the Therefore, the angle between the hands is
What is the angle between the hour hand and the minute hand at 4:40? Possible Answers:
Correct answer: Explanation: At 4:40, the minute hand is on the 8, and the hour hand is two-thirds of the way from the 4 to the 5. That is, the hands are three and one-third number positions apart. Each number position is thirty degrees around the clock, so the hands form an angle of
It is 4 o’clock. What is the measure of the angle formed between the hour hand and the minute hand? Possible Answers:
Correct answer: Explanation: At four o’clock the minute hand is on the 12 and the hour hand is on the 4. The angle formed is 4/12 of the total number of degrees in a circle, 360. 4/12 * 360 = 120 degrees
The hour hand on a clockface points to the Possible Answers:
Correct answer: Explanation: There are Create a fraction out of these two quantities to use later as a conversion rate: Between the
What is the measure of the smaller angle formed by the hands of an analog watch if the hour hand is on the 10 and the minute hand is on the 2? Possible Answers:
Explanation: A analog clock is divided up into 12 sectors, based on the numbers 1–12. One sector represents 30 degrees (360/12 = 30). If the hour hand is directly on the 10, and the minute hand is on the 2, that means there are 4 sectors of 30 degrees between then, thus they are 120 degrees apart (30 * 4 = 120).
Thomas is trying to determine the angle between the hands of his clock. Right now it reads Possible Answers:
Correct answer: Explanation: You can think of a clock in two ways: 1. Out of 12 hours, or 2. In terms of a circle with If you try to solve it in terms of #1: Goal: Find the angle measurement between the hour and the minute hands. We only want to find the degrees between the hours of 9 and 12 So we are looking at 3 hours out of the 12 total hours on a clock. As a fraction: So that means that the clock hands are making an angle that is 1/4 of the clock (which is a circle). So knowing that a circle has 1/4 of a circle is ____________________________________________________________ 2. If you think of the clock as a circle first you can determine the angle that the clock hands create very quickly. Since there are
What is the measure of the larger angle formed by the hands of a clock at Possible Answers:
Correct answer: Explanation: Like any circle, a clock contains a total of
What is the measure, in degrees, of the acute angle formed by the hands of a 12-hour clock that reads exactly 3:10? Possible Answers:
Explanation: The entire clock measures 360°. As the clock is divided into 12 sections, the distance between each number is equivalent to 30° (360/12). The distance between the 2 and the 3 on the clock is 30°. One has to account, however, for the 10 minutes that have passed. 10 minutes is 1/6 of an hour so the hour hand has also moved 1/6 of the distance between the 3 and the 4, which adds 5° (1/6 of 30°). The total measure of the angle, therefore, is 35°.
Find the angle in degrees between clock hands at 3:30. Possible Answers:
Correct answer: Explanation: On first glance, this problem may seem simple in that the angle between the 3 and 6 on a clock is one quarter of a circle or 90 degrees. However, you must take into account that in the half hour that has passed from 3:00 to 3:30, the hour hand has moved half the distance between the 3 and 4. To find the degrees, simply divide the total number of degrees in a circle by 12 to find the degrees between each consecutive number, and then multiply that number by 2.5 because you have half the distance to the 4, and the the full distance to the 5 and the full distance to the 6. Thus,
A clock shows that the time is 9:00am. What is the angle between the minute and the hour hands? Possible Answers:
Correct answer: Explanation: By dividing the clock into pieces, we can determine that the angle between the two hands is Within a clock, just like any circle, there are 360 total degrees. Within an clock, there are 60 total minutes. Each minute that passes, the minute hand advances 6 degrees. The hour hand advances .5 degrees. But since it is 9am on the dot, we will just be using the minute hand to count the degrees. Since the minute hand is covering a total of 15 minutes in between it and the hour hand, we can do the math:
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At what time between 3 and 4 oclock is the angle between the minute hand and the hour hand of a clock 75?
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