Measures and accuracy: including upper and lower bounds (Higher tier only)

The idea

At Higher tier, upper and lower bounds are used in calculations with rounded values. For a quantity rounded to a given unit, the lower bound is the stated value minus half the unit, and the upper bound is the stated value plus half the unit. For example, a length of 8 cm rounded to the nearest cm has a lower bound of 7.5 cm and an upper bound of 8.5 cm, forming the error interval 7.5 ≤ length < 8.5. When adding or multiplying positive quantities, combine bounds to find the maximum or minimum possible result: for a sum, add the upper bounds for the maximum and lower bounds for the minimum. For a difference, choose bounds for the required extreme. For a positive quotient, the maximum is upper numerator ÷ lower denominator and the minimum is lower numerator ÷ upper denominator. Keep units consistent and use correct inequality signs.

What earns the marks

• Calculate the lower bound and upper bound of each rounded quantity by halving the unit of rounding and adding or subtracting it from the stated value.
• For a sum, find the maximum by adding the upper bounds and the minimum by adding the lower bounds.
• For a difference, find the maximum by subtracting the lower bound from the upper bound, and the minimum by subtracting the upper bound from the lower bound.
• For a product, find the maximum by multiplying the upper bounds and the minimum by multiplying the lower bounds, provided all quantities are positive.
• For a quotient, find the maximum by dividing the upper bound of the numerator by the lower bound of the denominator.
• For a quotient of positive quantities, find the minimum by dividing the lower bound of the numerator by the upper bound of the denominator.

Common mistakes and how to put them right

• Using the stated value instead of the bound in a calculation.

  Put it right: For example, using 8 cm rather than 8.5 cm when finding a maximum.

• Adding the same half-unit to both quantities when finding a minimum.

  Put it right: For a difference, the minimum uses the lower bound minus the upper bound.

• Dividing by the upper bound to find a maximum quotient.

  Put it right: For positive quantities, the maximum quotient uses the upper bound of the numerator and the lower bound of the denominator.

Exam tips

• Write down the lower and upper bound of every rounded quantity before starting the calculation, so you can see which combination to use.
• For a maximum quotient, ask whether you need the largest possible numerator and the smallest possible denominator.
• Check the direction of each inequality: lower bound ≤ value < upper bound for a value rounded to a given unit.