78.45679757: Round off the mixed repeating (recurring) decimal number to the nearest ten (2 whole places)

Rounding a number means replacing it with a simpler, shorter approximation, while keeping its value close to the initial one. The rounded number will be less accurate, but easier to work with.

A repeating decimal has a number of decimals that repeat over and over again, infinitely: 78.45679757 ≈ 78.456797577975779757...

Counting by tens (2 whole places at a time), our number is sitting on the axis of numbers between two consecutive neighboring numbers:
70 < 78.45679757 < 80

Our number is to be rounded off to one of these neighbors, the closer one.

The middle of this interval, the number that is equally close to the either neighbor, is: (70 + 80) ÷ 2 = 75

Our number, 78.45679757 ≈ 78.456797577975779757..., is larger than 75, so it is closer to the larger neighbor: 80

Except for 'To Ceiling, To Floor', the rounded off number (both the positive and the negative) will be equal only to this larger neighbor.

Rounding digit. Let's call the digit of the position place that is intended to round off to as the 'rounding digit'. The digit is 7: 78.456797577975779757...

In a positive number, if the digit to the right of the 'rounding digit' is larger than 5, then the number is rounded to the larger neighbor.

The digit is 8: 78.456797577975779757... ≈ 80 rounded to the nearest ten (2 whole places).

Or, even simpler.
In any number rounded to whole places, if the digit to the right of the 'rounding digit' is 5 (but it is not the last non-zero digit to the right) or more than 5, ie. 6, 7, 8 or 9, then increase the 'rounding digit' by 1 and replace all the digits to the right of it with zeros:

78.456797577975779757... ≈ 80 rounded to the nearest ten (2 whole places)

Half Up, Half Down, Half Away From Zero, Half Towards Zero, Half to Even, Half to Odd. To Ceiling, To Floor.

Number Round Half Up:

Numbers that are halfway between two neighbors are rounded off to the larger neighbor. 78.45679757 is not halfway between its neighbors.

78.45679757 ≈ 80

- 78.45679757 ≈ - 80

rounded to the nearest ten (2 whole places)

Number Round Half Down:

Numbers that are halfway between two neighbors are rounded off to the smaller neighbor. 78.45679757 is not halfway between its neighbors.

78.45679757 ≈ 80

- 78.45679757 ≈ - 80

rounded to the nearest ten (2 whole places)

Number Round Half Away From Zero:

Numbers that are halfway between two neighbors are rounded off to the neighbor which is farther away from zero. 78.45679757 is not halfway between its neighbors.

78.45679757 ≈ 80

- 78.45679757 ≈ - 80

rounded to the nearest ten (2 whole places)

Number Round Half Towards Zero:

Numbers that are halfway between two neighbors are rounded off to the neighbor which is closer towards zero. 78.45679757 is not halfway between its neighbors.

78.45679757 ≈ 80

- 78.45679757 ≈ - 80

rounded to the nearest ten (2 whole places)

Number Round Half to Even:
(Gaussian Rounding or Banker's Rounding)

Numbers that are halfway between two neighbors are rounded off to the neighbor with an even rounding digit. 78.45679757 is not halfway between its neighbors.

78.45679757 ≈ 80

- 78.45679757 ≈ - 80

rounded to the nearest ten (2 whole places)

Number Round Half to Odd:
(Gaussian Rounding or Banker's Rounding)

Numbers that are halfway between two neighbors are rounded off to the neighbor with an odd rounding digit. 78.45679757 is not halfway between its neighbors.

78.45679757 ≈ 80

- 78.45679757 ≈ - 80

rounded to the nearest ten (2 whole places)

Number Round Ceiling:

All the numbers that are between two neighbors are always rounded off to the larger neighbor.

78.45679757 ≈ 80

- 78.45679757 ≈ - 70

rounded to the nearest ten (2 whole places)

Number Round Floor:

All the numbers that are between two neighbors are always rounded off to the smaller neighbor.

78.45679757 ≈ 70

- 78.45679757 ≈ - 80

rounded to the nearest ten (2 whole places)