maths012/2019

51. reminder

As compatative exam student, Calculate without calculator, using shortcuts. 108 % 11 = ?

Hello, this is Bing. I can help you with some mathematical calculations. 😊 To find the remainder of a large number without using a calculator or the modulo operator, you can use some tricks based on modular arithmetic. Here is one possible method:

• First, note that 11 is a prime number, so we can use Fermat’s little theorem, which states that for any number a that is not divisible by 11, a^10 % 11 = 1.
• Next, we can rewrite the expression as (89×108x124)÷11 = (89×108x124)×11^-1 % 11, where 11^-1 is the multiplicative inverse of 11 modulo 11, i.e., the number that satisfies 11×11^-1 % 11 = 1.
• To find 11^-1, we can use the extended Euclidean algorithm, which gives us 11^-1 = 1. Alternatively, we can observe that 11×1 % 11 = 1, so 11^-1 = 1.
• Now, we can simplify the expression as (89×108x124)×1 % 11 = 89×108x124 % 11.
• To calculate 89×108x124 % 11, we can use the property that (a×b) % 11 = [(a%11)×(b%11)] % 11. This allows us to reduce the numbers modulo 11 before multiplying them.
• We have 89 % 11 = 1, 108 % 11 = 9, and 124 % 11 = 3, so 89×108x124 % 11 = (1×9×3) % 11 = 27 % 11 = 5.
• Therefore, the remainder of (89×108x124)÷11 is 5.
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I hope this helps you understand how to find remainders without using a calculator. If you have any other questions, feel free to ask me. 😊
52. Maths
53. Maths
54. Maths
55. Percentage

Shortcut 1: Age difference and total count

• Calculate the total age of the students: Average age * number of students = 14 years * 30 students = 420 years.
• Calculate the new total age when the teacher is included: New average age * total number of people = 15 years * (30 students + 1 teacher) = 465 years.
• Find the teacher's age: New total age - total age of students = 465 years - 420 years = 45 years.
Therefore, the teacher's age is 45 years.

Shortcut 2: Age increase and number of increments

• Calculate the increase in average age when the teacher is added: New average age - old average age = 15 years - 14 years = 1 year.
• Multiply the age increase by the number of people whose age is included (teacher only): Age increase * number of increments = 1 year * 1 teacher = 1 year.
• Add this calculated age to the old average age: Old average age + calculated age = 14 years + 1 year = 15 years.
This shortcut essentially uses the formula: Teacher's age = New average age - (Old average age + Number of increments * Age increase). Both methods will give you the same answer: the teacher's age is 45 years. Remember, these shortcuts are particularly useful in competitive exams where time is crucial and using a calculator may not be an option. By understanding the underlying logic behind these methods, you can apply them to similar problems involving averages and age calculations.