Jeff took 8 hours to travel from Town A to Town B.
Nick took 12 hours to travel from Town B to Town A
Both Jeff and Nick started at 0930.
At what time would they pass each other on the way?
Actually, this is a relatively simple question... If you know how to do it, it is very easy.
The solution is below.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
METHOD 1:
Let the distance between Town A and Town B be 24km.
Speed of Jeff : Dist/Time = 24km/8h = 3km/h
Speed of Nick : Dist/Time = 24km/12h = 2km/h
Time taken to meet = Dist/Total speed
= 24/ 3+2
= 24/5
= 4.8h = 4h 48 min
0930 --4h 48 min ------ 1418--> ANSWER!
Simple???
METHOD 2 :
Jeff : 8h --> 1 journey
1h --> 1/8 journey
Nick : 12h --> 1 journey
1h --> 1/12 journey
1/8 + 1/12 = 3/24 + 2/24 = 5/24
5/24 journey = 1h
24/24 journey = 24/5 h = 4.8h = 4h 48min
Friday, April 3, 2009
Wednesday, April 1, 2009
Applying the concept of simple and compound interest on practical situations
You may ask, Why do I even need to learn simple/compound interest? All I have to do is not borrow and not lend money! Well, that thinking is very childish! GROW UP! In our life, I bet that we will borrow or lend money, and simple interest is for, well, simple amounts. For example, in primary school you may come across people saying"You lend me $1 I return you $2 tomorrow" This is a perfect and simple example of simple interest. You borrow money at a 100% interest.(Which is very stupid)
Now, compound interest. Its pretty much the same as simple interest, and you can think of compound interest as a series of back-to-back simple interest contracts. The interest earned in each period is added to the principal of the previous period to become the principal for the next period. For example, you borrow $10,000 for three years at 5% annual interest compounded annually:
p = principal (original amount borrowed or loaned)
i = interest rate for one period
n = number of periods
interest year 1 = p x i x n = 10,000 x .05 x 1 = 500(i1)
interest year 2 = (p2 = p1 + i1) x i x n = (10,000 + 500) x .05 x 1 = 525(i2)
interest year 3 = (p3 = p2 + i2) x i x n = (10,500 + 525) x.05 x 1 = 551.25
Total interest earned over the three years = 500 + 525 + 551.25 = 1,576.25. Compare this to 1,500 earned over the same number of years using simple interest. You will find that you earned A LOT of money as compared to charging simple interest.
So there you have it for those aspiring entrepreneurs, charge compound interest for any amount you lend, and you will be rolling in the green stuff if the lender is a procrastinator and takes ages to pay back. Ditto loan sharks
=)
Source: http://www.getobjects.com/Components/Finance/TVM/iy.html
Now, compound interest. Its pretty much the same as simple interest, and you can think of compound interest as a series of back-to-back simple interest contracts. The interest earned in each period is added to the principal of the previous period to become the principal for the next period. For example, you borrow $10,000 for three years at 5% annual interest compounded annually:
p = principal (original amount borrowed or loaned)
i = interest rate for one period
n = number of periods
interest year 1 = p x i x n = 10,000 x .05 x 1 = 500(i1)
interest year 2 = (p2 = p1 + i1) x i x n = (10,000 + 500) x .05 x 1 = 525(i2)
interest year 3 = (p3 = p2 + i2) x i x n = (10,500 + 525) x.05 x 1 = 551.25
Total interest earned over the three years = 500 + 525 + 551.25 = 1,576.25. Compare this to 1,500 earned over the same number of years using simple interest. You will find that you earned A LOT of money as compared to charging simple interest.
So there you have it for those aspiring entrepreneurs, charge compound interest for any amount you lend, and you will be rolling in the green stuff if the lender is a procrastinator and takes ages to pay back. Ditto loan sharks
=)
Source: http://www.getobjects.com/Components/Finance/TVM/iy.html
Non-routine Mathematics Questions
Here are some non-routine maths questions for you to have some fun with.
Q1. 3 men go on a business trip and pay $30 altogether at a hotel. The manager of the hotel discovers that he has overcharged them and refunds them $5. The 3 men decide to accept $1 each and give the $2 to a bellboy. So, each man has spent $9 so they have an expenditure of $27. Including the tip, that's $29. So where is the missing $1?( Remember, they paid $30)
Answer: Actually, there is no missing $1. The men do pay a total of $9.00 each (total of $27.00) for the room, but only after you add in the amount they tipped the bellboy($2). They paid $25 for the room and they gave the bellboy $2 as a tip. ($25 + 2 = $27.00). Including the $3 the men received as a refund, you will get the full $30( $27+$3=$30).
Q2. A road is 300km long. Car A is at the very start of the road and Car B is at the very end of the road. They start driving at exactly the same time towards each other. A bird is on the roof of Car A. As soon as both cars start moving, the bird travels at a speed of 150km/h towards Car B. When it reaches Car B, it turns around and returns to Car A, then Car B, so on and so forth, until both cars reach the mid-point of the road. How far has the bird flown if Car A and Car B travel at 60km/h and 90km/h respectively?
<-------------------------------------------300km------------------------------------------->
Car A>------------------------------------>ROAD<--------------------------------------Car B
Answer: This question is really quite simple. It can be solved in just 2 steps. However, you must know the formula for calculating Distance, Time and Speed first. For your info, it is Distance= Time * Speed. Thus, before we find out the distance the bird has flown(" How far has the bird flown") , we must find out its speed and time. We already know its speed( 150km/h). So, how do we know its time?
We know the road is 300km long and Car A and Car B travel at a total speed of 150km/h( 60km/h + 90km/h). Hence, we can work out easily that they will take 2 hours to meet. That is the time given to the bird to travel.
So, since Distance= Speed * Time, Distance flown= Bird's Speed* Bird's time to travel=150km/h * 2h= 300km.( Answer)
Q1. 3 men go on a business trip and pay $30 altogether at a hotel. The manager of the hotel discovers that he has overcharged them and refunds them $5. The 3 men decide to accept $1 each and give the $2 to a bellboy. So, each man has spent $9 so they have an expenditure of $27. Including the tip, that's $29. So where is the missing $1?( Remember, they paid $30)
Answer: Actually, there is no missing $1. The men do pay a total of $9.00 each (total of $27.00) for the room, but only after you add in the amount they tipped the bellboy($2). They paid $25 for the room and they gave the bellboy $2 as a tip. ($25 + 2 = $27.00). Including the $3 the men received as a refund, you will get the full $30( $27+$3=$30).
Q2. A road is 300km long. Car A is at the very start of the road and Car B is at the very end of the road. They start driving at exactly the same time towards each other. A bird is on the roof of Car A. As soon as both cars start moving, the bird travels at a speed of 150km/h towards Car B. When it reaches Car B, it turns around and returns to Car A, then Car B, so on and so forth, until both cars reach the mid-point of the road. How far has the bird flown if Car A and Car B travel at 60km/h and 90km/h respectively?
<-------------------------------------------300km------------------------------------------->
Car A>------------------------------------>ROAD<--------------------------------------
Answer: This question is really quite simple. It can be solved in just 2 steps. However, you must know the formula for calculating Distance, Time and Speed first. For your info, it is Distance= Time * Speed. Thus, before we find out the distance the bird has flown("
We know the road is 300km long and Car A and Car B travel at a total speed of 150km/h( 60km/h + 90km/h). Hence, we can work out easily that they will take 2 hours to meet. That is the time given to the bird to travel.
So, since Distance= Speed * Time, Distance flown= Bird's Speed* Bird's time to travel=150km/h * 2h= 300km.( Answer)
Sunday, March 29, 2009
Compound Interest
Compound Interest is the act of adding accumulated interest back to the principal sum, so that interest is earned on the earlier gained interest from that moment onwards. It is different from Simple Interest in the way that Simple Interest does not involve any compounding.
Which means to say, Compound Interest includes the interest paid on the initial investment AND on the interest accrued while Simple Interest does not include the additional interest.
Let me demonstrate; say, a house mortgage has a 1% interest and compounds once every month.
(In 1 month)
Compound Interest: Number of times of Compounding= 1, Principal=$100,000, Interest= 1%.
Simple Interest: Number of times of Compounding=0, Principal= $100,000, Interest=1%.
Thus, in 1 month, the amount owned is $100,000 X 101%= $101,000 for both Simple and Compound Interest.
In 2 months, the amount owned will be 101% X (principal plus interest)( $101,000), which will equal $102,010 for Compound Interest. However, for Simple Interest, the outstanding mortgage loan will instead be 102%( 100%+ 1% X Number of Months) X principal, which will equal $102,000. In a 6 months, the amount owned will be approximately $106,152.02 and $106,000 for Compound and Simple Interest respectively.
Thus, we can tell that Simple Interest is mostly used for short term loans, usually within the span of half a year. Compound Interest is used for long durations or periods, such as a year or more as it catches up with Simple Interest in 6 months. So, if we need a loan for 5 months or less, we know that we should take a Compound Interest loan and if we need a loan for 6 months or more, we know that we should take a Simple Interest loan instead.
On the other hand, if we are investing in bonds, Simple Interest is the way to go for up to 5 months and Compound Interest from then on.
Math Joke:
Q: What do you get if you divide the circumference of a jack-o-lantern by its diameter?
A: Pumpkin Pi!
Singapore (Channel News Asia). At Changi International Airport today, a Chinese male who claimed to be a math teacher was arrested by airport officials on duty while attempting to board a flight while in possession of a compass, a protractor and a calculator.
According to the officials, he is believed to have ties with several terrorist networks, including the infamous Al-Coolda network. He will be charged in court on Monday for carrying weapons of math instruction.
Which means to say, Compound Interest includes the interest paid on the initial investment AND on the interest accrued while Simple Interest does not include the additional interest.
Let me demonstrate; say, a house mortgage has a 1% interest and compounds once every month.
(In 1 month)
Compound Interest: Number of times of Compounding= 1, Principal=$100,000, Interest= 1%.
Simple Interest: Number of times of Compounding=0, Principal= $100,000, Interest=1%.
Thus, in 1 month, the amount owned is $100,000 X 101%= $101,000 for both Simple and Compound Interest.
In 2 months, the amount owned will be 101% X (principal plus interest)( $101,000), which will equal $102,010 for Compound Interest. However, for Simple Interest, the outstanding mortgage loan will instead be 102%( 100%+ 1% X Number of Months) X principal, which will equal $102,000. In a 6 months, the amount owned will be approximately $106,152.02 and $106,000 for Compound and Simple Interest respectively.
Thus, we can tell that Simple Interest is mostly used for short term loans, usually within the span of half a year. Compound Interest is used for long durations or periods, such as a year or more as it catches up with Simple Interest in 6 months. So, if we need a loan for 5 months or less, we know that we should take a Compound Interest loan and if we need a loan for 6 months or more, we know that we should take a Simple Interest loan instead.
On the other hand, if we are investing in bonds, Simple Interest is the way to go for up to 5 months and Compound Interest from then on.
Math Joke:
Q: What do you get if you divide the circumference of a jack-o-lantern by its diameter?
A: Pumpkin Pi!
Singapore (Channel News Asia). At Changi International Airport today, a Chinese male who claimed to be a math teacher was arrested by airport officials on duty while attempting to board a flight while in possession of a compass, a protractor and a calculator.
According to the officials, he is believed to have ties with several terrorist networks, including the infamous Al-Coolda network. He will be charged in court on Monday for carrying weapons of math instruction.
Saturday, March 21, 2009
Finding Perfect Numbers
Okay, I found this very interesting way to find perfect numbers.By the way, this question came from Ms Wun's enrichment worksheets(how many of you actually DO them?)
Finding Perfect Numbers
1) 1+2=3(3 is prime)
3x2=6=perfect number =)
2)1+2+4=7 (7 is prime)
4x7=28=perfect no. =)
3)1+2+4+8=15 (15 is not prime)
8x15=120≠perfect no. =(
4)1+2+4+8+16=31 (31 is prime)
16x31=496=perfect no. =)
.
.
.
.
.
.
and so on...
In "1)" we get 1+2 because 1x2=2. In "2)" we get 1+2+4 because 2x2=4.
And so on
In "1)" we get 3x2 because the two numbers are between the equal sign. Ditto others.
HOWEVER this method will not work for "3)" because 15 is not prime.
Ta-Da! =)
GET IT?
Finding Perfect Numbers
1) 1+2=3(3 is prime)
3x2=6=perfect number =)
2)1+2+4=7 (7 is prime)
4x7=28=perfect no. =)
3)1+2+4+8=15 (15 is not prime)
8x15=120≠perfect no. =(
4)1+2+4+8+16=31 (31 is prime)
16x31=496=perfect no. =)
.
.
.
.
.
.
and so on...
In "1)" we get 1+2 because 1x2=2. In "2)" we get 1+2+4 because 2x2=4.
And so on
In "1)" we get 3x2 because the two numbers are between the equal sign. Ditto others.
HOWEVER this method will not work for "3)" because 15 is not prime.
Ta-Da! =)
GET IT?
Monday, March 16, 2009
Speed, Uniform Speed and Average Speed
Speed, uniform speed and average speed. Three topics which might confuse people. Actually, they are quite simple and are easy to understand. I will now explain them in simple terms.
What is speed? To put it in a typical Singaporean's terms, "speed is how fast something travels lor." However the real meaning according to Wikipedia is "the rate of motion, or equivalently the rate of change of distance"
Uniform speed is actually just constant speed if you are confused by the word. Just remember that your school uniform never changes so it is always the same. An example of uniform speed would be if a motorcycle was travelling at the same speed (50km/h) for 1 hour, the distance that he covered would be the speed he was travelling (50km). Of course this distance the motorcycle travelled starts only when it has reached 50km/h as it needs to accelerate.
The average speed of an object tells you the average rate at which it covers a certain distance. For example, if a bus is travelling at an average speed of 50km/h for two hours, it would have travelled 100km in that two hours.
To sum this up, speed is simply distance divided by time, distance is speed times time and time taken is distance divided by speed. Speed= Distance/Time, Distance=Speed X Time, Time= Distance/Speed.
Maths Joke:
Q: Why was 6 afraid of 7?
A: Because 7 8 9!
Explanation if you have no imagination: seven ate (Eight and ate? Geddit?) 9
sources:http://en.wikipedia.org/wiki/Speed,http://www.batesville.k12.in.us/Physics/PhyNet/Mechanics/Kinematics/AveSpeed.html
What is speed? To put it in a typical Singaporean's terms, "speed is how fast something travels lor." However the real meaning according to Wikipedia is "the rate of motion, or equivalently the rate of change of distance"
Uniform speed is actually just constant speed if you are confused by the word. Just remember that your school uniform never changes so it is always the same. An example of uniform speed would be if a motorcycle was travelling at the same speed (50km/h) for 1 hour, the distance that he covered would be the speed he was travelling (50km). Of course this distance the motorcycle travelled starts only when it has reached 50km/h as it needs to accelerate.
The average speed of an object tells you the average rate at which it covers a certain distance. For example, if a bus is travelling at an average speed of 50km/h for two hours, it would have travelled 100km in that two hours.
To sum this up, speed is simply distance divided by time, distance is speed times time and time taken is distance divided by speed. Speed= Distance/Time, Distance=Speed X Time, Time= Distance/Speed.
Maths Joke:
Q: Why was 6 afraid of 7?
A: Because 7 8 9!
Explanation if you have no imagination: seven ate (Eight and ate? Geddit?) 9
sources:http://en.wikipedia.org/wiki/Speed,http://www.batesville.k12.in.us/Physics/PhyNet/Mechanics/Kinematics/AveSpeed.html
Knowing the concept of simple interest and its formula
Knowing the concept of simple interest and its formula
Well, simple interest is indeed simple !
When money is borrowed, interest is charged for the use of that borrowed money for a certain period of time. When the money is paid back, the person would then have to pay the original amount of money he had borrowed together with the interest.
You may wonder -- Then what is interest?
Interest -- The amount of interest mainly depends on 3 main factors.
1) The amount of money borrowed
2) The length of time that the money is borrowed for
3) The interest rate
Therefore, the formula for finding simple interest is :
Interest = Principal* Rate * Time
Example : If $50 was borrowed for 2 years at a 10% interest rate, the interest would be
$50x10/ 50x2 = $10
OR
$50x 10% x2 = $5x2 = $10
Therefore, as a conclusion, simple interest is generally charged for borrowing money for short periods of time only. It is used for uncompounded stocks, shares and bonds as it is the amount of interest paid only on the original principal, not on the interest accrued or accumulated.
Maths Joke:
A group of mathematicians and a group of engineers are traveling together by train to attend a conference on mathematical methods in engineering. Each mathematician has a ticket whereas only one of the engineers has one. The mathematicians laugh at the unworldly engineers and look forward to the moment the conductor shows up.
Suddenly, one of the engineers shouts: "Conductor coming!"
All the engineers disappear into one washroom.
The conductor checks the ticket of each mathematician and then knocks at the washroom door: "Your ticket, please."
The engineers stick the one ticket they have under the door, the conductor checks it and leaves. A few minutes later, when it is safe, the engineers come out of the washroom. The mathematicians are impressed.
When the conference has come to an end, the mathematicians decide that they are at least as smart as the engineers and also buy just one ticket for the whole group. This time the engineers do not buy a ticket at all.
Again, one of the engineers shouts: "Conductor coming!".
All the mathematicians rush off to one washroom. One of the engineers goes to that washroom, knocks at the door, and says: "Your ticket, please..."
sources: http://www.aaamath.com/g84-simple-interest.html
Well, simple interest is indeed simple !
When money is borrowed, interest is charged for the use of that borrowed money for a certain period of time. When the money is paid back, the person would then have to pay the original amount of money he had borrowed together with the interest.
You may wonder -- Then what is interest?
Interest -- The amount of interest mainly depends on 3 main factors.
1) The amount of money borrowed
2) The length of time that the money is borrowed for
3) The interest rate
Therefore, the formula for finding simple interest is :
Interest = Principal* Rate * Time
Example : If $50 was borrowed for 2 years at a 10% interest rate, the interest would be
$50x10/ 50x2 = $10
OR
$50x 10% x2 = $5x2 = $10
Therefore, as a conclusion, simple interest is generally charged for borrowing money for short periods of time only. It is used for uncompounded stocks, shares and bonds as it is the amount of interest paid only on the original principal, not on the interest accrued or accumulated.
Maths Joke:
A group of mathematicians and a group of engineers are traveling together by train to attend a conference on mathematical methods in engineering. Each mathematician has a ticket whereas only one of the engineers has one. The mathematicians laugh at the unworldly engineers and look forward to the moment the conductor shows up.
Suddenly, one of the engineers shouts: "Conductor coming!"
All the engineers disappear into one washroom.
The conductor checks the ticket of each mathematician and then knocks at the washroom door: "Your ticket, please."
The engineers stick the one ticket they have under the door, the conductor checks it and leaves. A few minutes later, when it is safe, the engineers come out of the washroom. The mathematicians are impressed.
When the conference has come to an end, the mathematicians decide that they are at least as smart as the engineers and also buy just one ticket for the whole group. This time the engineers do not buy a ticket at all.
Again, one of the engineers shouts: "Conductor coming!".
All the mathematicians rush off to one washroom. One of the engineers goes to that washroom, knocks at the door, and says: "Your ticket, please..."
sources: http://www.aaamath.com/g84-simple-interest.html
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