Approximations and rounding to hundredths — Mathematics, 11–13
Approximating a decimal to hundredths: from below, from above and by rounding. Mathematics, 11–13 years.
Two neighbours and one choice
Approximating a number to hundredths means replacing it with a decimal that has two decimal places and is very close to it. 2.834 lies between 2.83 and 2.84. The approximation by default (from below) is the neighbour that does not exceed the number (2.83); the approximation by excess (from above) is the neighbour above it (2.84). Rounding picks the closer neighbour: look at the thousandths digit and, if it is 5 or more, go up; if it is less than 5, stay.
Why we do not always want the exact value
You do not always need the exact value, and sometimes you cannot even write it: 10 ÷ 3 gives 3.3333… and never ends. An approximate value is easier to read, say and store. But it matters which side you err on: the value from below never exceeds the true one, the value from above is never short of it. If you need ribbon for a length of 2.834 m, buying 2.83 m falls short; buying 2.84 m is enough.
Approximating 2.834 and 5.678
Approximate 2.834 and 5.678 to hundredths. 2.834 lies between 2.83 and 2.84: by default, 2.83; by excess, 2.84. The thousandths digit is 4 (less than 5), so the rounding is 2.83. 5.678 lies between 5.67 and 5.68: by default, 5.67; by excess, 5.68. The thousandths digit is 8 (5 or more), so the rounding is 5.68.
Rounding is not always going up
Rounding is not the same as approximating by excess. For 2.834, the value by excess is 2.84, but the rounding is 2.83, because the thousandths digit is 4. There is also the chain-rounding mistake: for 2.3449, someone first rounds to thousandths (2.345) and then to hundredths (2.35). Wrong: look only at the thousandths digit (4) and get 2.34, because 2.3449 is less than 2.345.
Sharing costs and reading signs
An exact value like 10 ÷ 3 = 3.3333… cannot be paid in money: each of 3 friends pays 3.33 € (rounding to hundredths), and 3 × 3.33 = 9.99, so 1 cent is missing. A road sign may say 2.8 km when the route is 2.83 km: to drive, that is enough. But when cutting cloth or wood, the value by excess is the safer one, so you do not come up short.
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