If there is an error in a value that is used in a calculation, that error can become more significant when the calculation is made. For example, if the radius of a circle is rounded from 2.57 cm to 2.6 cm, an error of 0.49 cm² would be made when calculating the area of the circle.

In this section we consider how errors introduced by rounding can be increased (or *propagated*) in subsequent calculations.

The radius of a circle is given as 31 cm, correct to the nearest cm. What are the possible errors when calculating its area?

As the radius, *r* cm, is given as 31 cm to the nearest cm, we have

30.5 ≤ *r* < 31.5

If r = 30.5, | A | = π × 30.5² |

= 2922.466566 cm² | ||

= 2920 cm² to 3 s.f. |

If r = 31, | A | = π × 31² |

= 3019.07054 cm² | ||

= 3020 cm² to 3 s.f. |

If r = 31.5, | A | = π × 31.5² |

= 3117.245311 cm² | ||

= 3120 cm² to 3 s.f. |

If *r* = 30.5, then the error is

3019.07054 – 2922.466566 = 96.603974 cm²

whilst, if *r* = 31.5, the error is

3117.245311 – 3019.07054 = 98.17477043 cm²

Hence the maximum possible error occurs when *r* = 31.5, and is approximately 98.2 cm². In other words, there is a potential error here of almost 100 cm² in the area if we calculate it from the rounded radius.

A rectangular plot of land has sides with lengths of 38 m and 52 m correct to the

Calculate the maximum and minimum possible values of:

(a)

the *perimeter* of the rectangle,

The sides have been given to the nearest metre, so

51.5 m ≤ length < 52.5 m

37.5 m ≤ width < 38.5 m

Minimum perimeter | = 2(37.5 + 51.5) |

= 178 m | |

Maximum perimeter | = 2(38.5 + 52.5) |

= 182 m |

(b)

the *area* of the rectangle.

Minimum area | = 37.5 × 51.5 |

= 1931.25 m² | |

Maximum area | = 38.5 × 52.5 |

= 2021.25 m² |

The values of *x* and *y* are given to 1 decimal place as *x* = 4.2 and *y* = 7.3

Determine the minimum and maximum values of:

(a)

*x* + *y*

First note that 4.15 ≤ *x* < 4.25 and 7.25 ≤ *y* < 7.35.

Minimum value of x + y | = 4.15 + 7.25 | (minimum value of x + minimum value of y) |

= 11.4 | ||

Maximum value of x + y | = 4.25 + 7.35 | (maximum value of x + maximum value of y) |

= 11.6 |

(b)

*y* – *x*

Minimum value of y – x | = 7.25 – 4.25 | (minimum value of y – maximum value of x) |

= 3 | ||

Maximum value of y – x | = 7.35 + 4.15 | (maximum value of y – minimum value of x) |

= 3.2 |

(c)

Minimum value of | = = 0.56462585 | (minimum value of x ÷ maximum value of y) |

= 0.565 to 3 s.f. | ||

Maximum value of | = = 0.586206896 | (maximum value of x ÷ minimum value of y) |

= 0.586 to 3 s.f. |

Note that, for *x* and *y* both positive,

To find *maximum* value of *x* + *y* or *xy*,

use the largest value of *x* and largest value of *y*.

To find the *minimum* value of *x* + *y* or *xy*,

use the smallest value of *x* and smallest value of *y*.

To find *maximum* value of *x* – *y* or ,

use the largest value of *x* and smallest value of *y*.

To find the *minimum* value of *x* – *y* or ,

use the smallest value of *x* and largest value of *y*.