Sec 1 Maths: Approximation & Estimation — practice questions & worked solutions

Values rounding to $12.7$ to 1 d.p. lie in $12.65 \le t < 12.75$.

With 3 decimal places, the smallest measurable value is:

$$\begin{aligned}t_{\min} = 12.650 \text{ s}\end{aligned}$$

and the largest measurable value below $12.75$ is:

$$\begin{aligned}t_{\max} = 12.749 \text{ s}\end{aligned}$$

(i)(a)

$$\begin{aligned}895.899 \approx 900\end{aligned}$$

(i)(b)

$$\begin{aligned}0.044772 \approx 0.045\end{aligned}$$

(ii) Using the 2 s.f. estimates:

$$\begin{aligned}\frac{895.899}{0.044772 \times 100} &\approx \frac{900}{0.045 \times 100} \\ &= \frac{900}{4.5} \\ &= 200\end{aligned}$$

(a) Round each value:

$$\begin{aligned}\sqrt[3]{65.125} &\approx \sqrt[3]{64} = 4\end{aligned}$$
$$\begin{aligned}(10.92)^{2} &\approx 11^{2} = 121\end{aligned}$$
$$\begin{aligned}9.81 &\approx 10\end{aligned}$$
$$\begin{aligned}\frac{4 + 121}{10} &= \frac{125}{10} = 12.5 \approx 13\end{aligned}$$

(b) Round each price up:

$$\begin{aligned}5 \times 3 &= 15\end{aligned}$$
$$\begin{aligned}2 \times 4 &= 8\end{aligned}$$
$$\begin{aligned}3 \times 2 &= 6\end{aligned}$$
$$\begin{aligned}15 + 8 + 6 &= 29\end{aligned}$$

Estimated total $\approx \$29 > \$27$, so she does not have enough money.

Rounding to convenient values:

$$\begin{aligned}\text{Total} &\approx 20\times14+10\times10\\ &= 280+100\\ &= \$380\end{aligned}$$

(a) Approximate values:

$$\begin{aligned}\sqrt{12} &\approx 3.46\\ -\frac{3\pi}{4} &\approx -2.36\\ 1.654 &= 1.654\\ -2 &= -2\end{aligned}$$

Order on the number line from left to right:

$$-\frac{3\pi}{4},\ -2,\ 1.654,\ \sqrt{12}$$

(b) Round each value:

$$\begin{aligned}2.054 &\approx 2\\ (-0.98)^2 &\approx 1\\ \sqrt{102} &\approx 10\end{aligned}$$

Estimate:

$$\begin{aligned}\frac{2+1}{\frac{10}{4}} &= \frac{3}{2.5}\\ &= 1.2\\ &\approx 1\end{aligned}$$

(a)

$$\begin{aligned}\sqrt{\frac{76+4.37^{3}}{1.82-0.195}} &= \sqrt{\frac{159.453\ldots}{1.625}}\\ &= 9.906\end{aligned}$$

(b) Estimate by rounding each value to 1 significant figure:

$$\begin{aligned}150.91\times29.8 &\approx 150\times30\\ &= 4500\end{aligned}$$

Mike’s answer is not reasonable, as $8407.136$ is far from the estimated value of $4500$.

Largest:

$$\begin{aligned}728\,649\,999 \text{ m}^2\end{aligned}$$

Smallest:

$$\begin{aligned}728\,550\,000 \text{ m}^2\end{aligned}$$