Approximation, rounding and estimationIB MYP Maths Extended: Revision notes

To round to a given number of decimal places (d.p.), look at the digit after the last one you keep. If it is $5$ or more, round up; if it is less than $5$, leave the last digit as it is. $0.0063584$ to $3$ d.p.: the third decimal place is $6$ and the next digit is $3$, so the answer is $0.006$. Decimal places count digits after the decimal point, including zeros.

Significant figures (s.f.) count from the first non-zero digit. Zeros before the first non-zero digit are not significant. $0.0063584$ to $3$ s.f.: the significant figures start at $6$, so keep $6,3,5$; the next digit is $8$, so round up to $0.00636$. To $2$ s.f. it is $0.0064$. For large numbers use zeros as place holders: $48\,637$ to $2$ s.f. is $49\,000$, not $49$. To the nearest hundred, $48\,637$ is $48\,600$.

To estimate a calculation, round each number to $1$ significant figure first, then work out the easy calculation. $\frac{58.7\times4.12}{0.198}\approx\frac{60\times4}{0.2}=1200$. The calculator value is $1221.43\ldots$, so the estimate is reasonable. Estimating is also a check: if your calculator answer is far from the estimate, you may have made a keying error. Do not round the divisor to $0$.

The percentage error measures how far an approximate value is from the exact value: $\text{percentage error}=\frac{|\text{approximate}-\text{exact}|}{\text{exact}}\times100.$ Divide by the exact value. Example: the population is $48\,637$, rounded to $49\,000$. Error $=363$, so percentage error $=\frac{363}{48637}\times100=0.75\%$. For the estimate $1200$ and exact value $1221.43$: $\frac{21.43}{1221.43}\times100=1.75\%$.

Sometimes the context decides how to round. A builder needs $\frac{29.14}{1.5}=19.43$ packs of tiles, but only whole packs are sold, and $19$ packs would not cover the floor, so round up to $20$. A number of complete deliveries is rounded down. Keep the full calculator value during a multi-step calculation and round only at the end, and give the final answer to a sensible accuracy (such as $3$ s.f.).

A good estimate has a small percentage error. Rounding the pack price R$245$ to $1$ s.f. gives R$200$, an estimate of $\text{R}4000$ for $20$ packs against the exact $\text{R}4900$: error $=\frac{900}{4900}\times100=18.4\%$. Rounding to $2$ s.f. gives R$250$, an estimate of $\text{R}5000$, with error $2.04\%$. The $2$ s.f. estimate is much better. To justify, compare the two errors and say which is smaller and why that matters in the real situation.