Question

Помогите 35 баллов 3. Представьте в виде обыкновенной дроби бесконечную переодическую дробь 1) 0,(5) 3) 5,(12) 5) 18,11(124) 2) 0,8(6) 4) 4,3(16) 6) 4,(173) 4. Представьте в виде обыкновенной дроби каждую бесконечную переодическую дробь и найдите значения выражения 9,(4)+3,(3)+2,(2)

Answer 1

Ответ:

3.

1) 0,(5) = 5/9;3) 5,(12)= 5 12/99; 5) 18,11(124)= 18 11124/99999;

2) 0,8(6)= 86/99; 4) 4,3(16)= 4 316/999 6) 4,(173)= 4 173/999;

4.

9 4/9 + 3 3/9 + 2 2/9 = 14 9/9

Пошаговое объяснение:

Answer 2

Representing Infinite Repeating Decimals as Fractions:

To represent infinite repeating decimals as fractions, we can use the concept of geometric series. A repeating decimal can be written as the sum of a finite decimal and an infinite geometric series.

Let's go through each of the given repeating decimals and represent them as fractions:

1) 0,(5): To represent 0,(5) as a fraction, we can consider it as the sum of 0.5 and an infinite geometric series with a common ratio of 0.01. Using the formula for the sum of an infinite geometric series, we can calculate the fraction as follows:

0,(5) = 0.5 + 0.05 + 0.0005 + ...

Using the formula for the sum of an infinite geometric series:

S = a / (1 - r)

where S is the sum of the series, a is the first term, and r is the common ratio, we can substitute the values:

S = 0.5 / (1 - 0.01)

Simplifying the expression:

S = 0.5 / 0.99

Therefore, the fraction representation of 0,(5) is 5/9.

2) 5,(12): To represent 5,(12) as a fraction, we can consider it as the sum of 5.12 and an infinite geometric series with a common ratio of 0.01. Using the formula for the sum of an infinite geometric series, we can calculate the fraction as follows:

5,(12) = 5.12 + 0.0512 + 0.000512 + ...

Using the formula for the sum of an infinite geometric series:

S = a / (1 - r)

where S is the sum of the series, a is the first term, and r is the common ratio, we can substitute the values:

S = 5.12 / (1 - 0.01)

Simplifying the expression:

S = 5.12 / 0.99

Therefore, the fraction representation of 5,(12) is 512/99.

3) 18,11(124): To represent 18,11(124) as a fraction, we can consider it as the sum of 18.11124 and an infinite geometric series with a common ratio of 0.00001. Using the formula for the sum of an infinite geometric series, we can calculate the fraction as follows:

18,11(124) = 18.11124 + 0.00011124 + 0.0000011124 + ...

Using the formula for the sum of an infinite geometric series:

S = a / (1 - r)

where S is the sum of the series, a is the first term, and r is the common ratio, we can substitute the values:

S = 18.11124 / (1 - 0.00001)

Simplifying the expression:

S = 18.11124 / 0.99999

Therefore, the fraction representation of 18,11(124) is 1811124/99999.

4) 0,8(6): To represent 0,8(6) as a fraction, we can consider it as the sum of 0.86 and an infinite geometric series with a common ratio of 0.01. Using the formula for the sum of an infinite geometric series, we can calculate the fraction as follows:

0,8(6) = 0.86 + 0.0086 + 0.000086 + ...

Using the formula for the sum of an infinite geometric series:

S = a / (1 - r)

where S is the sum of the series, a is the first term, and r is the common ratio, we can substitute the values:

S = 0.86 / (1 - 0.01)

Simplifying the expression:

S = 0.86 / 0.99

Therefore, the fraction representation of 0,8(6) is 86/99.

5) 4,3(16): To represent 4,3(16) as a fraction, we can consider it as the sum of 4.316 and an infinite geometric series with a common ratio of 0.0001. Using the formula for the sum of an infinite geometric series, we can calculate the fraction as follows:

4,3(16) = 4.316 + 0.0004316 + 0.000004316 + ...

Using the formula for the sum of an infinite geometric series:

S = a / (1 - r)

where S is the sum of the series, a is the first term, and r is the common ratio, we can substitute the values:

S = 4.316 / (1 - 0.0001)

Simplifying the expression:

S = 4.316 / 0.9999

Therefore, the fraction representation of 4,3(16) is 4316/9999.

Evaluating Expressions:

Now, let's evaluate the given expressions using the fraction representations we found:

1) 9,(4): To evaluate 9,(4), we can substitute the fraction representation of 9,(4) which is 9/9 into the expression:

9,(4) = 9/9

Simplifying the expression:

9,(4) = 1

Therefore, the value of 9,(4) is 1.

2) 3,(3): To evaluate 3,(3), we can substitute the fraction representation of 3,(3) which is 3/3 into the expression:

3,(3) = 3/3

Simplifying the expression:

3,(3) = 1

Therefore, the value of 3,(3) is 1.

3) 2,(2): To evaluate 2,(2), we can substitute the fraction representation of 2,(2) which is 2/2 into the expression:

2,(2) = 2/2

Simplifying the expression:

2,(2) = 1

Therefore, the value of 2,(2) is 1.

In summary, the fraction representations of the given repeating decimals are: - 0,(5) = 5/9 - 5,(12) = 512/99 - 18,11(124) = 1811124/99999 - 0,8(6) = 86/99 - 4,3(16) = 4316/9999

And the values of the given expressions are: - 9,(4) = 1 - 3,(3) = 1 - 2,(2) = 1

I hope this helps! Let me know if you have any further questions.