### 數值估算的技巧

 ${\left({a}\right)}$ Reformulation strategy ${\left({i}\right)}$ Rounding off ${\left({i}{i}\right)}$ Using clustered value If the numbers involved in an estimation are close to each other, we can select a clustered value to represent them. ${\left({i}{i}{i}\right)}$ Using compatible numbers Compatible numbers are approximate value that make calculation easier. ${\left({b}\right)}$ Compensation strategy This method makes adjustment to the result of a rough estimation so that it is closer to the exact value. ${\left({c}\right)}$ Translation strategy This method rearranges the order of operations in a complicated expression so that the estimation can be simplified. ${\left({d}\right)}$ Taking larger or smaller approximations When isong estimation to solve real-life problems, we should determine whether to take larger or smaller approximations in order to obtain more useful estimates. Note:  1. To round up a number, we take an approximation which is slightly larger to replace the original number. 2. To round down a number, we take an approximation which is slightly smaller to replace the original number.

 Example of (a)  (a)(i) Estimate the average of the following numbers by rounding off them correct to the nearest integer. ${2.19}$ , ${3.27}$ , ${4.81}$ , ${4.56}$ , ${6.88}$ , ${8.01}$ (a)(ii) Use a clustered value to estimate the value of the expression. ${3800}+{3980}+{4075}+{4112}+{4208}$ (a)(iii) Use compatible numbers to estimate the value of the expression. ${0.67}\times{89.8}$ Solution  (a)(i) Average $=\dfrac{{{2.19}+{3.27}+{4.81}+{4.56}+{6.88}+{8.01}}}{{6}}$ $=\dfrac{{{2}+{3}+{5}+{5}+{7}+{8}}}{{6}}$ round off to the nearest integer $={5}$ (a)(ii) The value of each of the numbers is about ${4000}$ . clustered value ∴  ${3800}+{3980}+{4075}+{4112}+{4208}$ $\approx{4000}\times{5}$ $={20000}$ (a)(iii) ${0.67}\times{89.8}$ $\approx\dfrac{{2}}{{3}}\times{90}$ $={60}$ Example of (b)  Use compensation strategy to estimate the value of the following expression: ${4.25}+{3.76}-{0.19}-{1.92}+{2.95}$ Solution  ${4.25}+{3.76}-{0.19}-{1.92}+{2.95}$ $={\left({4}+{0.25}\right)}+{\left({3}+{0.76}\right)}+{\left(-{0.19}\right)}+{\left(-{1}-{0.92}\right)}+{\left({2}+{0.95}\right)}$ $={\left({4}+{3}+{0}-{1}+{2}\right)}+{\left({0.25}+{0.76}\right)}+{\left(-{0.19}-{0.92}+{0.95}\right)}$ Estimate the decimal parts as compensation $\approx{8}+{1}+{0}$ $={9}$ Example of (c)  Use translation strategy to estimate the value of ${3017}\times{19}\div{15}$.  Solution  ${3017}\times{19}\div{15}$ $\approx{3000}\times{20}\div{15}$ Replace ${3017}$ by ${3000}$ and ${19}$ by ${20}$ $={3000}\div{15}\times{20}$ Rearrange the order of operations $={200}\times{20}$ $={4000}$ Example of (d)  (d)(i) Peter wants to buy ${2}$ books for $${54.4}$ each and ${3}$ magazines for$${19.8}$ each. Suppose Peter has only $${180}$. Estimate whether Peter has enough money to buy them. (d)(ii) Mary wants to record ${8}$ files in a CD. The file sizes (in MB) of the files are listed below: ${128.3}$ , ${92.2}$ , ${110.8}$ , ${56.7}$ , ${82.1}$ , ${124.6}$ , ${76.3}$ , ${71.5}$ Suppose the CD can store ${700}$ MB of data. Determine whether the total file size exceeds the capacity of the CD. Solution  (d)(i) By taking larger approximations, the total price $={\left({55}\times{2}+{20}\times{3}\right)}$ $=$$${170}$ $\lt$\$${180}$ ∴   Peter has enough money to buy them. (d)(ii) By taking smaller approximations, the total file size $={\left({120}+{90}+{110}+{50}+{80}+{120}+{70}+{70}\right)}$ $={710}$MB $\gt{700}$MB ∴   The total file size exceeds the capacity of the CD.

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