Approximation and accuracy

Fluency

(a) Exact (b) Approximate (c) Exact (d) Approximate.
(a) $7.3$ (b) $7.35$ (c) $7.35$ (3 s.f.).
(a) $7.3$ (b) $7.34$ .
$|\pi - 3.14| = 0.001\,592\ldots \approx 0.0016$ .
$\sqrt{3} = 1.732\,050\ldots$ Absolute error $= |1.73 - 1.732\,050\ldots| = 0.002\,05\ldots$ Relative error $= \dfrac{0.002\,05}{1.732\,05} \times 100\% \approx 0.118\%$ .
$\dfrac{2}{3} = 0.666\,666\ldots$ repeating. Absolute error $= |0.666\,666\,7 - 0.666\,666\ldots| \approx 3.3 \times 10^{-8}$ .
$0.0050$ .
$\dfrac{5}{6} = 0.8333\ldots$ Rounded to 4 d.p.: $0.8333$ . Absolute error $= 0.0000\overline{3} \approx 3.3 \times 10^{-5}$ .
$\dfrac{7}{3} = 2.333\ldots$ Absolute error $= |2.33 - 2.333\ldots| = 0.003\ldots$ Relative error $= \dfrac{0.003\overline{3}}{2.333\ldots} \times 100\% \approx 0.143\%$ .
False. For example, truncating $3.6478$ to 2 d.p. gives $3.64$ (error $0.0078$ ), while rounding gives $3.65$ (error $0.0022$ ). Truncation gave the larger error.

Reasoning

Side is between $5.15$ and $5.25$ . Area range: $5.15^2 = 26.5225$ to $5.25^2 = 27.5625$ . Range is approximately $26.52$ to $27.56$ cm $^2$ .
Exact: $1000 \times 1.065^{10} = 1000 \times 1.877\,14\ldots = 1877.14$ . Rounded: $1000 \times 1.07^{10} = 1000 \times 1.967\,15\ldots = 1967.15$ . Difference $\approx 90.01$ .
Exact amount per batch: $\dfrac{2}{3} = 0.666\ldots$ cup. Measured: $0.67$ . Error per batch: $0.67 - 0.666\ldots = 0.003\overline{3}$ . Tripled: total error $= 3 \times 0.003\overline{3} = 0.01$ cup.
Radius between $3.35$ and $3.45$ cm. Circumference range: $2\pi(3.35) = 6.7\pi \approx 21.05$ to $2\pi(3.45) = 6.9\pi \approx 21.68$ cm.
Exact: $2.45 \times 3.15 \times 1.85 = 14.276\,625$ . Rounded factors: $2.5 \times 3.2 \times 1.9 = 15.2$ . Error $= |15.2 - 14.276\,625| = 0.923\,375$ .
Surds are exact representations. Rounding introduces error that compounds through further operations. For example, $\sqrt{2} \times \sqrt{2} = 2$ exactly, but $1.414 \times 1.414 = 1.999\,396$ .
$1.4^5 = 5.378\,24$ . $1.41^5 = 5.584\,059\ldots$ Absolute error $= |5.584\,06 - 5.378\,24| \approx 0.206$ .
$0.5\%$ of $12.0 = 0.06$ . True value is between $12.0 - 0.06 = 11.94$ and $12.0 + 0.06 = 12.06$ cm.

Reasoning

$(x + \epsilon)^2 = x^2 + 2x\epsilon + \epsilon^2$ . For small $\epsilon$ , $\epsilon^2$ is negligible, so the error $\approx 2x\epsilon$ . Similarly $(x - \epsilon)^2 \approx x^2 - 2x\epsilon$ , confirming maximum absolute error $\approx 2x\epsilon$ .
Distance between $141.5$ and $142.5$ km. Fuel: $\dfrac{141.5 \times 8.3}{100} = 11.7445$ to $\dfrac{142.5 \times 8.3}{100} = 11.8275$ L. Using $142$ : $11.786$ L. Maximum error $\approx 0.083$ L.
Min volume: $3.15 \times 4.45 \times 6.05 = 84.835\ldots$ Max volume: $3.25 \times 4.55 \times 6.15 = 90.966\ldots$ Using rounded values: $3.2 \times 4.5 \times 6.1 = 87.84$ . Percentage range $= \dfrac{90.97 - 84.84}{87.84} \times 100\% \approx 6.98\%$ .
Catastrophic cancellation: if $a \approx 1000.3$ and $b \approx 1000.1$ (each accurate to 4 s.f.), then $a - b \approx 0.2$ , which has only 1 significant figure. The relative error jumps from $\sim 0.01\%$ in each value to potentially $50\%$ in their difference.

Reasoning

Exact monthly rate: $r = 1 + \dfrac{0.048}{12} = 1.004$ . After 240 months: $50\,000 \times 1.004^{240} = 50\,000 \times 2.6051\ldots = 130\,255.20$ . Rounded to 4 d.p. the multiplier is already $1.0040$ , so there is no rounding error at 4 d.p. in this case. With a cruder rounding (e.g. 3 d.p. giving $1.004$ ) the result is the same. If rounded to 2 d.p. as $1.00$ , the balance would be $50 000 — a discrepancy of roughly $80 255.
$\phi^{10} = \left(\dfrac{1+\sqrt{5}}{2}\right)^{10} = \dfrac{123 + 55\sqrt{5}}{2^{10}} \cdot 2^{10}$ . Exact value $= 122.991\,869\ldots$ Using $1.618^{10} = 122.966\ldots$ Percentage error $= \dfrac{|122.992 - 122.966|}{122.992} \times 100\% \approx 0.021\%$ .
$a - b$ ranges from $(10.0 - 0.05) - (9.9 + 0.05) = 0.0$ to $(10.0 + 0.05) - (9.9 - 0.05) = 0.2$ . Best estimate of $a - b = 0.1$ . Error up to $\pm 0.1$ , relative error $= \dfrac{0.1}{0.1} = 100\%$ , while individual relative errors are $\dfrac{0.05}{10.0} = 0.5\%$ and $\dfrac{0.05}{9.9} \approx 0.505\%$ .
With 7 significant digits: $\sqrt{10\,000\,001} = 3162.277\,8\ldots$ and $\sqrt{10\,000\,000} = 3162.277\,7\ldots$ The difference $\approx 0.000\,016$ , but both stored values agree in their first 7 digits, so the subtraction leaves at most 1-2 correct digits. Rearrangement: multiply by the conjugate: $\sqrt{10\,000\,001} - \sqrt{10\,000\,000} = \dfrac{1}{\sqrt{10\,000\,001} + \sqrt{10\,000\,000}} \approx \dfrac{1}{6324.555} \approx 1.581 \times 10^{-4}$ , which retains full precision.

Year 10 core - answers