Upper and Lower Bounds

Worked example

1. 1

   $6.3\text{ kg, 1 d.p.}$ — The last digit is in the tenths column, so it was rounded to the nearest 0.1.

2. 2

   $\frac{1}{2}\times 0.1=0.05$ — Halve the unit. The real weight is at most half a unit away from 6.3, or it would have rounded to 6.2 or 6.4.

3. 3

   $6.3-0.05=6.25$ — Take half a unit away. This is the lower bound: the smallest weight that still rounds to 6.3.

4. 4

   $6.3+0.05=6.35$ — Add half a unit. This is the upper bound: 6.35 itself rounds up to 6.4, so the weight stays just below it.

5. 5

   $6.25\le w<6.35$ — Put both together. This range is called the error interval.

Worked example

1. 1

   $7.8\text{ cm, 1 d.p.}$ — The last digit is tenths, so the unit is 0.1 and half a unit is 0.05.

2. 2

   $7.8-0.05=7.75$ — Take half a unit away for the lower bound.

3. 3

   $7.8+0.05=7.85$ — Add half a unit for the upper bound.

4. 4

   $260\text{, 2 s.f.}$ — Count two figures from the left: 2, then 6. The 6 is in the tens column, so the unit is 10.

5. 5

   $\frac{1}{2}\times 10=5$ — Half of the unit 10 is 5.

6. 6

   $260-5=255,260+5=265$ — Take 5 away for the lower bound, add 5 for the upper bound.

7. 7

   $40\text{, nearest }5$ — Here the unit is 5 itself, so half a unit is 2.5.

8. 8

   $40-2.5=37.5,40+2.5=42.5$ — Take 2.5 away and add 2.5. Both bounds sit 2.5 from 40.

Worked example

1. 1

   $100\text{ g}=0.1\text{ kg}$ — The accuracy is in grams but the answer is in kg, so change the unit first. 1000 g = 1 kg.

2. 2

   $\frac{1}{2}\times 0.1=0.05$ — Halve the unit, in kg.

3. 3

   $5.6-0.05=5.55$ — Take half a unit away: the lower bound. It goes on the left.

4. 4

   $5.6+0.05=5.65$ — Add half a unit: the upper bound. It goes on the right.

5. 5

   $5.55\le m<5.65$ — Use ≤ on the left, because 5.55 rounds up to 5.6. Use < on the right, because 5.65 rounds up to 5.7.

Worked example

1. 1

   $\text{mass: }60+5=65$ — Nearest 10 means half a unit is 5. Mass is on the top of mass ÷ volume, so use its upper bound: a bigger top gives a bigger answer.

2. 2

   $\text{volume: }8.0-0.05=7.95$ — 1 d.p. means half a unit is 0.05. Volume is on the bottom, so use its lower bound: a smaller bottom gives a bigger answer.

3. 3

   $\frac{65}{7.95}=8.1761\dots$ — Density = mass ÷ volume. Keep all the digits for now.

4. 4

   $=8.18{\text{ g/cm}}^3$ — Round only at the end, here to 3 significant figures.

Worked example

1. 1

   $7.4-0.05=7.35$ — For the smallest area, take the shortest length: its lower bound.

2. 2

   $3.6-0.05=3.55$ — Take the narrowest width too: its lower bound.

3. 3

   $7.35\times 3.55=26.0925$ — Area = length × width. Multiply the two lower bounds.

4. 4

   $\text{lower bound}=26.0925{\text{ cm}}^2$ — Give this exact value. It is short, so there is nothing to round.

Worked example

1. 1

   $2.4+0.05=2.45$ — The upper bound of the plank: the longest it could be.

2. 2

   $0.85-0.005=0.845$ — 2 d.p. means half a unit is 0.005. Take the lower bound of the piece: cutting off less leaves more.

3. 3

   $2.45-0.845=1.605$ — Upper − lower gives the upper bound of what is left: 1.605 m.

4. 4

   $2.35-0.855=1.495$ — For the lower bound, swap both: shortest plank − longest piece. So 1.495 m.