Finite mathematics
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Rounding and Truncation: In a classroom, students will receive a letter grade based on the percentage of points gained in the term out of the total points possible. There are 334 points possible. To get an A in the class, the student must have a percentage that, when properly rounded to a whole number, is at least 90%.
1. Determine whether the teacher will give Student 1 an A for the class if the student has earned 299 points, justifying your answer.
229/334 x 100=68%, the teacher would not give the student an A because the score of the student is only 68%, if the teacher rounded the figure to the nearest 10, it would be 70%, this is below the 90% pass mark.
- Use your answer in part A1 to explain whether Student 1 will receive an A for the class if the teacher truncates the percentage to a whole number.
To truncate, we drop or cut of the numbers after the decimal. This is done to reduce the figure to make it easy to operate. For example, the number is 0.6856287425 or 68.56287425%, this is a very long number which is not easy to multiply or divide with another number. So If If the teacher truncates the number to a whole number, the student would not get an A as the figure would be 89.52%, five is truncated down to 89%.
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Explain the following (suggested length of 1 page) as if you were teaching a middle school mathematics classroom (grades 5–9):
a. Why a taxpayer whose income tax rate is 27.8% would hope that the rate could be truncated to a whole number when calculating the amount of tax owed on the tax form.
In this way the amount of tax on him is reduced to a lower value, this might mean him paying only 27%. For example, if the taxpayers salary is $20000, he would have to pay $5560, however, if the tax rate is truncated to 27%, he will only pay: $5400, this is less compared to $5560. This means that the tax burden is lessened.
1. Why the government prefers and requires the taxpayer to round the tax rate:
This is done to increase the accuracy of the result or to get an accurate tax rate
- Your mental math process in calculating the above situation and how you would use rounding and truncating in real-world scenarios.
If i wanted to multiply an array of number, i would round of the decimal places and remain with the whole numbers only, which are relatively easy to operate
1. Provide two examples of each rounding and truncation (four total examples) to illustrate mental math skills.
- 6545 truncated into =34
- 823 truncated into =23745
- 345 rounded off into= 35
- 823 rounded off into 23746
- Primes and Composites: There are 20 boys and 24 girls in an Algebra I class. The class is so large that the teacher wants to divide the students by gender into cooperative groups composed of the same number of students.
1) Explain the process the teacher will use to determine how many students will be in each group using appropriate mathematical terms from number theory by gauss
The theory of finite maths was postulated by gauss, according to him one has to find the greatest common Factor (GCF) of both numbers, for example, he would divide both genders into same portions. By this he finds the greatest common factors of both groups
The greatest common factor of 24=6
And the greatest common factors of 20=5
2). Determine the largest number of students that can be placed in a group, showing all work
4 boys per group and 6 girls per group
Boys: 4×5
Girls=6×4
3). given your answer in B2, determine how many groups will be created from the Algebra I class, providing support
Girls= (6×4) =24
Boys= (5×4)
The total number of groups that can be created from the algebra class is 11
4. Explain how to prove that there is an infinite number of primes.
Prime number has only two divisible factors, because they are only divisible by one and itself, example of prime number are 7 and 11. Prime number was postulated by Euclid. He proved that there are an infinite number of primes, according to him, if a number of primes are finite, and then there can be other primes to it, which generates other primes
Example:
| Starting with a list of number of primes known | x1, x2, …, xn. |
| Multiply the number together and adding one, the product is X | X = x1x2…xn. Let q = X + 1. |
| We know that X | X + 1 = q. |
| X divides the differences between the two numbers | (X + 1) − X =1 |
| There is no divisors of which is a prime number | This number would run up to infinity as there would always be prime number in the list to |
This proves that in each list containing prime number, there will always be other prime number.
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Modular Operations: You want to explain the concept of modular operations to a middle school mathematics classroom, starting with a demonstration from clock arithmetic. Your explanation should include the following support:
• Appropriate examples for modular addition using positive integers of the following sum:
| Find a unit digit of the sum= | 2403+791+688+4339 |
| Total | 8221, |
| Unit digit | 1 |
| Same way the unit digit that results from the addenda | 3+8+1+9 |
| This can be solved by use of modular arithmetic | |
| 2403 | 3(MOD 10) |
| 791 | 1(MOD 10) |
| 688 | 8(MOD 10) |
| 4339 | 9(MOD 10) |
| TOTAL | 21 OR 1(MOD 10)_ |
- Appropriate examples for modular addition using negative integers
When you subtract 601 from 60002 and divide the result by 6
| 60002 | =2 (mod 6) | |
| 601 | =1 (mod 6) | |
| 60002-601 | =2-1 mod 6 | |
| =1 mod 6 | ||
- A sentence or two about the use of modular operations in real-world scenarios
Modular operation is used in calculating time, espcialy amongst the military people, in the airline industry, when the say that flight x was supposed to arrive at 9.00 am and is delayed by 14 hours it is easy to calculate the time.
However, modular mathematics is only easy tom use when calculating simple mathematical problem, at higher level it get complicated because, it relies on very many assumptions key amongst them is the principle of infinity of primes, discussed above. This concept is not practical in real world situation. For example at an advanced level of structural engineering, there is no need to use modular mathematics while there are system and machines for determining the stress factors in system instead of doing complex calculation.
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Explain how you would discuss the following (suggested length of 1 page) in the classroom setting:
a. Modular arithmetic and its relation to time
The modular arithmetic’s is closely related to time in that it considers repetitive cycles of numbers such as the clock face which has 12 cycles, each cycle represents 12 hours which are represented by 60 minutes and each minutes ois represented b 60 seconds. After the 12 hour cycles, one has to start from 1. If you look closely you realise that it is a mod 12 mathematics
If the hands of the clock cycles from 0 to 11 it makes 12 hours. For example, in our case we take an example of 16 mod 12, we realise that the answer is 4. This is also applicable when one is using the military time as it involves 24 cycles round, this mould mean having a mod 24, or a year which has a mod 12 months. This is possible with all the finite cycles’ numbers.
Another example, include the calculation of time , for example one can calculate time on the following grounds, if it is 7 am, and one would like to determine the position of the hour hand in 7 hours time.
Intuitively this is 7+7=14 then
Therefore: 14 mod 12=2, this is a practical even in the 24 hour clock another example on the 24 hour clock is
If the time now is 8.00, and one want to determine the position of the hour hand at 25 hours,
This is 25 mod 24=1
=1+9 ≡ (8) mod 12 + (25) mod 12 ≡ (8) mod 12 + (1) mod 12 ≡ 9 mod 12
- What is meant by 10 mod 6?
The remainder when 10 is divided by 6
10(mod 6) =4.
- How to add or multiply in mod 7
Add or multiply the figures ordinarily and divide the answer by 7. For example, if one is given the question below.
- Add 6 and 12 in mod 7=6+12 (mod 7) =18(mod 7) =4, here we first add the figures 6 and 12, to get 18, then divide by 7. If we divide 18 by 7 the answer is 2 remainder 4, because 7 goes into 18 twice giving us 14, remaining 4. The four is the answer as it is the remainder after the operations.
- The same figures can be used for operation when one is asked to multiply the figures and get the modulus. Multiply 6 and 12 in mod 7=6 X 12 (mod 7) = 72(mod 7) = 2, here we first multiply the figures 6 and 12, to get 72, and then divide by 7. If we divide 72 by 7 the answer is 10 remainder 4, because 7 go into 72, Ten times giving us 70, remaining 2. The 2 is the answer as it is the remainder after the operations.
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