Bounds and error intervals

Number

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A length is $7.3$ cm, measured to $1$ decimal place. Write the error interval.

2 marks
A mass is $400$ g, to $1$ significant figure. Write the error interval.

2 marks
Two lengths are $4.6$ cm and $3.2$ cm, both to $1$ decimal place. Find the lower and upper bounds of their sum.

3 marks
$x = 6.0$ to $1$ decimal place. Write the error interval for $x$.

3 marks
A rectangle measures $8$ cm by $5$ cm, both to the nearest centimetre. Find the lower and upper bounds of the area.

4 marks
A distance is $100$ m to the nearest metre. A time is $12.5$ seconds to $1$ decimal place. Find the upper bound of the average speed, to $3$ significant figures.

4 marks
$y = 20$ to the nearest $10$, and $z = 5.0$ to $1$ decimal place. Find the error interval for $y$. Then find the lower and upper bounds of $\dfrac{y}{z}$, each to $3$ significant figures.

5 marks
A cuboid measures $8$ cm by $5$ cm by $4$ cm, all to the nearest centimetre. Find the lower and upper bounds of the volume.

6 marks

Mark scheme

Answer. $7.25 \le l < 7.35$
- The measurement is halfway to the next tenth.
- $7.25 \le l < 7.35$
Answer. $350 \le m < 450$
- Numbers from $350$ up to $450$ round to $400$ to $1$ s.f.
- $350 \le m < 450$
Answer. Lower $7.7$ cm, upper $7.9$ cm
- Lower sum $= 4.55 + 3.15 = 7.7$.
- Upper sum $= 4.65 + 3.25 = 7.9$.
Answer. $5.95 \le x < 6.05$
- Half of $0.1$ is $0.05$.
- $5.95 \le x < 6.05$
Answer. Lower $33.75$ cm$^{2}$, upper $46.75$ cm$^{2}$
- Lower area $= 7.5 \times 4.5 = 33.75$.
- Upper area $= 8.5 \times 5.5 = 46.75$.
Answer. $8.07$ m/s
- Upper speed uses the greatest distance and the least time.
- $\dfrac{100.5}{12.45} = 8.072\ldots$
- $8.07$ m/s to $3$ s.f.
Answer. $15 \le y < 25$; lower $2.97$, upper $5.05$
- $15 \le y < 25$ and $4.95 \le z < 5.05$.
- Lower value $= \dfrac{15}{5.05} = 2.970\ldots = 2.97$ to $3$ s.f.
- Upper value $= \dfrac{25}{4.95} = 5.050\ldots = 5.05$ to $3$ s.f.
Answer. Lower $118.125$ cm$^{3}$, upper $210.375$ cm$^{3}$
- Lower volume $= 7.5 \times 4.5 \times 3.5 = 118.125$.
- Upper volume $= 8.5 \times 5.5 \times 4.5 = 210.375$.