Clock Aptitude Reasoning Tricks & Problems - Finding Angle Between The Hands of a Clock Given Time
Summary
TLDRThis video script offers a detailed explanation on calculating the angle between the hour and minute hands of an analog clock at different times. It covers various examples, such as 12:30, 1:20, 11:15, and 10:25, illustrating the step-by-step process to determine the angles. The script also explains how to find the shortest angle and the longer angle by subtracting from 360 degrees, providing a comprehensive guide for understanding clock angles.
Takeaways
- 🕒 The angle between the hour and minute hands of a clock can be calculated using the positions of the hands relative to the clock's numbers.
- 🔢 Each hour on the clock represents an angle of 30 degrees, as the full circle of 360 degrees is divided by the 12 hours.
- 📏 At 12:30, the minute hand is at the 6, and the hour hand is halfway between 12 and 1, creating an angle of 15 degrees from the 12 o'clock position.
- 📉 To find the angle at 12:30, add the angle from the 12 to the 6 (150 degrees) and the angle from the 12 to the halfway point between 12 and 1 (15 degrees), totaling 165 degrees.
- 📍 For the time 1:20, the hour hand is between 1 and 2, and the minute hand is at the 4, creating an angle that can be calculated by fractions of the hour.
- 📈 The position of the hour hand relative to the hour marks is found by dividing the minute value by 60, which gives the fraction of the hour passed.
- 🔄 At 1:20, the hour hand is two-thirds of the way between 1 and 2, and one-third of the way from 1 to 2, which translates to angles of 20 degrees and 40 degrees respectively.
- 📐 The total angle at 1:20 is found by adding the angles between the hour marks and the fractions of those angles, resulting in 80 degrees.
- 🕘 For 11:15, the minute hand is at the 3, and the hour hand is one-fourth of the way from 11 to 12, leading to an angle calculation involving fractions of 30 degrees.
- 🔢 Calculating the angle at 11:15 involves determining the fraction of the hour passed (one-fourth) and using it to find a partial angle from the 11 o'clock position (22.5 degrees).
- 📉 The total angle at 11:15 is the sum of the angle from the 12 to the 11 (90 degrees) and the partial angle (22.5 degrees), equaling 112.5 degrees.
- 🕒 For 10:25, the shortest angle between the hour and minute hands is found by considering the positions and calculating the smaller of the two possible angles, resulting in 162.5 degrees.
Q & A
What is the angle between the minute hand and the hour hand at 12:30?
-At 12:30, the minute hand is at the 6, and the hour hand is halfway between 12 and 1. Since each hour represents 30 degrees, the angle between 12 and 1 is 30 degrees. Being halfway, the angle is 15 degrees. Adding the 150 degrees from 12 to 6 gives a total angle of 165 degrees.
How do you calculate the angle for the time 1:20 on an analog clock?
-At 1:20, the minute hand is at the 4, and the hour hand is between 1 and 2. The hour hand is one-third of the way from 1 to 2. Each hour represents 30 degrees, so the angle between 1 and 2 is 30 degrees. Two-thirds of 30 degrees is 20 degrees. Adding the 30 degrees between 1 and 3 gives a total angle of 80 degrees.
What is the method to find the angle between the hour hand and the minute hand at 11:15?
-At 11:15, the minute hand is at the 3, and the hour hand is between 11 and 12. The hour hand is one-fourth of the way from 11 to 12. Each hour represents 30 degrees, so the angle between 11 and 12 is 30 degrees. Three-fourths of 30 degrees is 22.5 degrees. Adding the 90 degrees between 12 and 3 gives a total angle of 112.5 degrees.
How do you determine the shortest angle between the hour hand and the minute hand at 10:25?
-At 10:25, the minute hand is at the 5, and the hour hand is between 10 and 11. The hour hand is five-twelfths of the way from 10 to 11. Each hour represents 30 degrees, so the angle between 10 and 11 is 30 degrees. Five-twelfths of 30 degrees is 12.5 degrees. The angle between the hour hand and the minute hand is less than 180 degrees, which is 162.5 degrees.
What is the longest angle between the hour hand and the minute hand at 10:25?
-The longest angle at 10:25 is the supplementary angle to the shortest angle of 162.5 degrees, which can be found by subtracting the shortest angle from 360 degrees. So, the longest angle is 360 - 162.5 = 197.5 degrees.
How can you find the angle between the hour hand and the minute hand if the clock says 2:40?
-At 2:40, the minute hand is at the 8, and the hour hand is between 2 and 3. The hour hand is two-thirds of the way from 2 to 3. Each hour represents 30 degrees, so the angle between 2 and 3 is 30 degrees. Two-thirds of 30 degrees is approximately 20 degrees. Adding the 60 degrees between 12 and 2 gives a total angle of 80 degrees.
What is the significance of dividing the minutes by 60 when calculating the angle between the hour and minute hands?
-Dividing the minutes by 60 gives you the fraction of the hour that has passed. This fraction is used to determine how far the hour hand has moved from the last hour mark towards the next one, which is essential for calculating the angle between the hour and minute hands.
Why is it important to consider the shortest angle between the hour and minute hands?
-Considering the shortest angle is important because it represents the actual visual angle between the two hands on the clock face. It is often the most relevant measurement for practical purposes, such as determining the time until the next hour.
How does the position of the hour hand change as time passes?
-The hour hand moves continuously as time passes, covering 30 degrees for each hour. It moves at a slower pace than the minute hand, and its position relative to the hour marks changes as the minutes increase.
Can you provide a formula to calculate the angle between the hour and minute hands at any given time?
-Yes, the formula to calculate the angle between the hour and minute hands is: (hour * 30) + (minute / 2) for the hour hand, and (minute * 6) for the minute hand. The absolute difference between these two values gives the angle in degrees.
Outlines
🕒 Understanding Clock Angles at 12:30
The paragraph explains how to calculate the angle between the hour and minute hands of an analog clock at 12:30. It starts by noting that each hour represents 30 degrees (360 degrees / 12 hours). At 12:30, the minute hand is at the 6, while the hour hand is halfway between 12 and 1. The calculation involves determining the angle from the 12 to the 1 (30 degrees) and then finding half of that (15 degrees) because the hour hand is at the midpoint. Adding these gives a total angle of 165 degrees between the two hands.
🕗 Calculating Angles for 1:20 Using Fractions
This section teaches how to find the angle between the clock hands at 1:20. It emphasizes using fractions to determine the hour hand's position relative to the hour marks. The minute hand at the 4 indicates 20 minutes past the hour. The calculation involves finding two-thirds of an hour's 30-degree angle (20 degrees) and adding it to the angle between the 1 and 4 (60 degrees), resulting in an 80-degree angle between the hands.
🕘 Finding Shortest Angle at 10:25
The final paragraph focuses on calculating the shortest angle between the clock hands at 10:25. It explains that the minute hand at the 5 corresponds to 25 minutes, and the hour hand is between 10 and 11, slightly closer to 10. The calculation involves determining the fraction of the hour passed (5/12) and using it to find the angle between the hour hand and both the 10th and 11th hours. The shortest angle, which is less than 180 degrees, is found by subtracting the smaller angle (5/12 of 30 degrees) from 150 degrees (the angle between the 10th and 5th hours), resulting in a 162.5-degree angle.
Mindmap
Keywords
💡Analog Clock
💡Angle
💡Hour Hand
💡Minute Hand
💡Degrees
💡Complete Revolution
💡Fraction
💡Shortest Angle
💡Clock Face
💡Time Calculation
Highlights
At 12:30, the minute hand is at the 6, and the hour hand is halfway between 12 and 1.
Each hour on a clock represents an angle of 30 degrees, as 360 degrees is divided by 12 hours.
The angle between the hour hand and the minute hand at 12:30 is calculated by adding 150 degrees and 15 degrees, resulting in 165 degrees.
At 1:20, the hour hand is between 1 and 2, and the minute hand is at the 4.
The hour hand's position at 1:20 is one-third of the way from 1 o'clock towards 2 o'clock.
The angle between the hour hand and the minute hand at 1:20 is found by adding 20 degrees, 30 degrees, and 30 degrees, totaling 80 degrees.
At 11:15, the minute hand is at the 3, and the hour hand is between 11 and 12.
The hour hand at 11:15 is one-fourth of the way from 11 o'clock towards 12 o'clock.
The angle between the hour hand and the minute hand at 11:15 is calculated by adding 90 degrees and 22.5 degrees, resulting in 112.5 degrees.
At 10:25, the minute hand is at the 5, and the hour hand is between 10 and 11.
The hour hand at 10:25 is five-twelfths of the way from 10 o'clock towards 11 o'clock.
The shortest angle between the hour hand and the minute hand at 10:25 is found by adding 12.5 degrees and 150 degrees, totaling 162.5 degrees.
The longer angle at 10:25 can be found by subtracting the shorter angle from 360 degrees, resulting in 197.5 degrees.
The sum of all three angles in a clock face must equal 360 degrees.
The method for calculating the angle between clock hands involves understanding the position of each hand relative to the hour markers.
The angle calculation changes depending on whether the minute hand is ahead or behind the hour hand.
For times when the minute hand is ahead of the hour hand, the shorter angle is calculated by adding the angles between the hour markers.
For times when the minute hand is behind the hour hand, the shorter angle is calculated by subtracting the smaller angle from 360 degrees.
Transcripts
the time on an analog clock
reads 12 30.
what is the angle between the minute
hand and the hour hand of the clock
so on a clock this is going to be 12
we have 3 on the right
six below
and nine
so at 12 30
the minute hand that's the long hand is
going to be at the six
and the short hand the hour hand
is not exactly at 12
because if it was exactly at 12 it would
be 12 o'clock
if it wasn't one o'clock the hour hand
would be at one
but if it's 12 30
the hour hand
which is the shorthand has to be between
12 and 1.
exactly right in the middle i'm gonna
represent the hour hand in red
so that's twelve thirty our goal is to
find the angle
between
those two hands
so how can we do so
now first it's helpful to know
the measure of one hour
so let's focus on the three
what's the angle between the two and the
three
notice that it takes 12 hours for the
hour hand to make a complete revolution
so 12 hours
correlates to 360 degrees because that's
the entire circle
so one hour is going to be 360 divided
by 12 so every hour represents an angle
of 30.
so therefore
from one o'clock
to six o'clock
that's a time period of five hours
so if one hour represents 30 degrees
then 5 hours is 5 times 30 which is 150
degrees
so therefore
from 1
all the way to six
that angle is 150 degrees
so now we need to find it from
this point
to this point this angle here
now we know from 12 to 1 it represents
30 degrees
so the red line is right between 12 and
1 because we're at 12 30 we're halfway
between 12 and 1.
so if we're halfway
we need to multiply 30 by half
so therefore the angle on the inside
is 15 degrees
so we got to add
150 and 15.
therefore the angle between the hour
hand and the minute hand is 165
which is less than 180
because if the hour hand was at 12 and
the minute hand was at 6 which is
impossible
that would be a straight line that would
be 180 but it has to be less than 180
and this answer is reasonable
here's another example
convert the time 120
to degrees
that is
find the angle between the hour hand and
the minute hand if the clock says 120
so first let's draw a picture
here's 12 this is going to be 3
6
and 9
and then here's one
two
four
and five
so if it's 120 that means the hour hand
is between
one and two
the minute hand has to be at four so i'm
going to use blue to represent the
minute hand which is the long one and
red to represent the hour hand
now if the out hand is one is between
one and two
where exactly is it
so focus on the minute hand which is at
20.
20 divided by 60
is a third
so what this means is that
the hour hand is one third away from
the one o'clock hour or the value of one
and it's two thirds away
from the second hour or from two
so that's why you want to take whatever
your minute value is and divided by 60
so you can get the fraction of how far
it is from one of the hours
and to find the distance from the other
one it's going to be one minus the
original fraction so one minus one third
will give you two thirds
so now with this information we could
find everything we need
so keep in mind
the angle between one hour is always 30
degrees
it's 360 divided by 12. so between 3 and
4 is 30 degrees and between 2 and 3 is
30.
now here is where it gets interesting
so this is when you want to use this
fraction
this angle
is two-thirds of an hour
that's the missing 40 minutes
so the angle for one hour is always
going to be 30 that's not going to
change we got to find two-thirds of 30.
30 divided by 3 is 10 times 2 is 20. so
therefore this angle is 20. now all you
need to do
is add up these three angles 20 plus 30
plus 30 is equal to 80.
so that is the angle between the hour
hand and the minute hand and that's how
you do it
so now it's your turn
try this example
let's say the time on an analog clock is
11 15.
go ahead and find the angle between the
hour hand and the minute hand
so feel free to pause the video and work
out this problem
now let's always begin with a picture so
this is 12 3
6
and 9.
the minute ham
is at 3
because it represents 15. 3 is for 15 6
is 30
9 is 45 12 is well
zero
so every increment of five
is for each number
so for example
let's put down the other numbers
so let's say if it was 1105
the minute hand will be at five
eleven ten
it will be at two
fifteen is for three twenty is four
twenty five is for five
thirty is for six 35 is for seven and so
forth 55 is for 11. this is 50
45 40 and then back to zero
so just in case you're wondering those
are the numbers that
you need to know
so the 15 points to three
now where is the hour hand
we know that the hour hand is between
eleven and twelve
but it's closer to eleven
what we need is the fraction
so take fifteen
the value of the minutes and divided by
sixty
fifteen over 60 reduces to one-fourth
so therefore
the hour hand is one-fourth its way from
the 11th hour which means that if we
subtract one by a fourth
that's four over four minus one over
four four minus one is three so
three-fourths
is between the hour hand and twelfth so
it's three-fourths it's one-fourth of
the weight going from eleven to twelve
it has three fourths left to get to
eleven and twelve
so basically
it's twenty five percent of the way
between eleven and twelfth
now that we have that let's go ahead and
calculate the angle
so every hour represents 30 degrees
now the angle between the hour hand and
12 is going to be 3 4 of 30.
so what is three-fourths of 30
30 divided by 4 is 7.5 7.5 times 3 is
22.5
so that's the missing angle
so if we add 30 three times that's 90.
so we got to add 90
plus
22.5
and so that's going to be
112.5
so that is the angle between the hour
hand and the minute hand
it's 112.5
so here's the last example
find the shortest angle
for
the time of 10 25 so find the shortest
angle between the hour hand and the
minute hand when the clock says 10 25
this is going to be 12 3
six and nine
so 25 corresponds to five
that's where the minute hand is going to
be located
now the hour hand
is between 10
and 11. so it's very close to the middle
but it's slightly closer to 10.
so there's the hour hand
now let's find the fraction of 25
divided by 60.
so twenty five over sixty
twenty five is five times five sixty is
five times twelve
so it's five over twelve
therefore between the hour hand and 10
is 5 12 of 30 degrees
and between
the hour hand and 11
that's going to be
1 minus 5 over twelve which is twelve
over twelve minus five over twelve
so that's seven over twelve
so seven twelfths of thirty degrees is
the angle between the hour hand and
the eleventh hour
so now that we have that we can find the
angle so we're looking for the angle of
the shortest
we're looking for the shortest angle
between the hour hand and the minute
hand
this side is clearly the longer angle
that's going to be more than 180
so we need to find the angle
of that side between the hour hand and
the minute hand
that's less than 180.
now we know that one hour
represents 30 degrees
so if we find the angle from the fifth
hour
to the tenth hour
that's five hours ten minus five is five
and five hours represent an angle
of 150 degrees
so now we just gotta find this angle
here
which is five twelfths of thirty
so what's five twelfths of thirty
so thirty times five is one fifty and
150 divided by 12
is 12.5 degrees
so we need to add
12.5 and 150 which will give us a final
answer of
162.5 degrees
so that is the angle well that is the
shorter angle
between
the the hour hand and the minute hand
when the clock says 10 25 by the way if
you want to find the longer angle
that is this angle
it's simply
360 minus this answer so 360 minus 162.5
that will give you the other angle of
197.5
and keep in mind this is 12.5 on the
inside
so all three of these angles have to add
up to 360.
浏览更多相关视频
CAT exam preparation videos 2024 |Time & Distance clocks 1 | | Quantitative Aptitude
Find the time between 2 and 3 when angle is 50 between hour and minute hands
Interior and Exterior Angles of a Polygon
Angle Addition Postulate explained with examples
Example: Identify 4 Possible Polar Coordinates for a Point Using Radians
Dot Product and Force Vectors | Mechanics Statics | (Learn to solve any question)
5.0 / 5 (0 votes)