Changing the subject of a formula
5.1 Length, mass, capacity and time
Prefix Meaning
milli-
centi-kilo- 1000
1 1000
---1 100
---The common conversions for length are:
10 mm = 1 cm (i.e. 10 millimetres = 1 centimetre)
100 cm = 1 m (i.e. 100 centimetres = 1 metre)
1000 m = 1 km (i.e. 1000 metres = 1 kilometre)
The common conversions for mass are:
1000 mg = 1 g (i.e. 1000 milligrams = 1 gram)
1000 g = 1 kg (i.e. 1000 grams = 1 kilogram)
1000 kg = 1 t (i.e. 1000 kilograms = 1 tonne)
The common conversions for capacity are:
1000 mL = 1 L (i.e. 1000 millilitres = 1 litre)
1000 L = 1 kL (i.e. 1000 litres = 1 kilolitre)
1000 kL = 1 ML (i.e. 1000 kilolitres = 1 megalitre)
■ Time
We use time to order the events that take place in our everyday lives. Without time, it would not be possible to say which event came before or after another event. We often measure the degree of change in a particular situation according to the amount of time that passes, for example when calculating the speed of a moving object. The speed is, in fact, a measure of the change in distance with respect to the elapsed time.
We use instruments such as watches and clocks to tell the time. These are either analog or digital. Time pieces with rotating hands are called analog, whereas those that display digits only are called digital. Many digital watches and clocks operate in 24-hour time, that is from 00:00 to 24:00 hours, rather than in am or pm time.
Many time calculations can be more easily performed with the use of the degrees and minutes, or and keys on the calculator. It may first be necessary to express one of the given times in 24-hour time.
Example 1 Convert:
a 8 cm to mm b 5.2 m to cm c 0.04 km to m
d 70 mm to cm e 129 cm to m f 2300 m to km Solutions
a 8 cm= (8 × 10) mm b 5.2 m= (5.2 × 100) cm c 0.04 km= (0.04 × 1000) m
= 80 mm = 520 cm = 40 m
d 70 mm= (70 ÷ 10) cm e 129 cm= (129 ÷ 100) m f 2300 m= (2300 ÷ 1000) km
= 7 cm = 1.29 m = 2.3 km
To convert to a smaller unit, multiply by the conversion factor.
To convert to a larger unit, divide by the conversion factor.
The common conversions for time are:
60 s = 1 min (i.e. 60 seconds = 1 minute)
60 min = 1 h (i.e. 60 minutes = 1 hour)
24 h = 1 day (i.e. 24 hours = 1 day)
To convert from 12-hour time to 24-hour time:
add 12 hours to the time if it is 1 pm or greater
write the time using 4 digits.
DMS ° ′ ′′
EG+S
Example 2 Convert:
a 0.57 m to mm b 98 000 cm to km
Solutions
a 0.57 m = (0.57 × 100) cm b 98 000 cm = (98 000 ÷ 100) m
= 57 cm = 980 m
= (57 × 10) mm = (980 ÷ 1000) km
= 570 mm = 0.98 km
Example 3 Convert:
a 5 L to mL b 6.8 kL to L c 910 L to kL
Solutions
a 5 L = (5 × 1000) mL b 6.8 kL = (6.8 × 1000) L c 910 L = (910 ÷ 1000) kL
= 5000 mL = 6800 L = 0.91 kL
Example 4 Convert:
a 4 kg to g b 3.72 g to mg c 9100 g to kg d 384 kg to t
a 4 kg = (4 × 1000) g b 3.72 g = (3.72 × 1000) mg
= 4000 g = 3720 mg
c 9100 g = (9100 ÷ 1000) kg d 384 kg = (384 ÷ 1000) t
= 9.1 kg = 0.384 t
Example 5
Use the degrees and minutes key on the calculator to convert:
a 1.25 h to hours and minutes b 3 h 21 min to hours Solutions
a Press 1.25 . The display of 1°15′ is then interpreted as 1 h 15 min.
b Press 3 21 . Therefore, 3 h 21 min = 3.35 h.
1 Choose the most appropriate unit (mm, cm, m, km) that could be used to measure:
a the length of a fly b the height of a 4-year-old girl c the length of a caravan d the distance between two towns EG+S
EG +S
EG +S
Solutions
EG +S
2nd F DMS
DMS DMS 2nd F DMS
Exercise 5.1
e the height of a table f the distance run in a sprint race g the width of a postage stamp h the distance between two bus depots 2 Choose the most appropriate unit (mg, g, kg, t) that could be used to measure the mass of:
a an orange b a bee’s wing c a railway carriage
d a television set e a clump of hair f a calculator
g a baby h a truck i a box of pencils
3 Choose the most appropriate unit (mL, L, kL) that could be used to measure the capacity of:
a a glass of water b a backyard fountain c a swimming pool
d Sydney Harbour e a bird bath f a tea cup
g a teaspoon h the petrol tank of a bus i a small dam 4 Convert:
a 6 km to m b 300 cm to m c 9 cm to mm d 2500 m to km
e 0.46 m to cm f 4 mm to cm g 178 m to km h 2.3 cm to mm i 0.8 km to m j 0.1 m to cm k 200 cm to mm l 16 m to km m 30 m to cm n 0.07 mm to cm o 2 cm to m p 0.3 m to km 5 Complete each of the following conversions.
a 5 m = mm b 2 km = cm c 4000 mm = m
d 900 000 cm = km e 3.8 km = mm f 1 650 000 mm = km 6 Complete each of these conversions.
a 4 g = mg b 8000 kg = t c 1.5 kg = g
d 14 500 mg = g e 2790 g = kg f 70 000 kg = t
g 12.4 g = mg h 1.82 kg = g i 375 g = kg
j 140 mg = g k 0.87 t = kg l 0.046 kg = g
m 20 kg = t n 6 mg = g o 0.005 47 g = mg
7 Complete each of these conversions.
a 4 L = mL b 3000 mL = L c 8 kL = L
d 7500 L = kL e 2.4 L = mL f 1950 L = kL
g 3610 mL = L h 5.07 kL = L i 0.73 L = mL
j 195 L = kL k 11 mL = L l 0.0068 kL = L
8 Convert:
a 1 min = s b 1 h = min c 1 day = h
d 3 h = min e 2 days = h f 5 min = s
g h = min h min = s i day = h
j 1 min = s k 3 days = h l 2 h = min
m 180 s = min n 72 h = days o 420 s = min
p 90 s = min q 75 min = h r 32 h = days
9 a Explain why 1.25 h does not mean 1 h 25 min.
b Express 1.25 h in hours and minutes.
1
2--- 3
4--- 2
3 ---1
2--- 1
4--- 5
6
---■ Consolidation
10 Express each time in minutes and seconds, without the use of a calculator.
a 1.1 min b 2.4 min c 3.25 min d 4.75 min
11 Use the degrees and minutes key on your calculator to express each time in hours and minutes.
a 1.9 h b 0.35 h c 3.45 h d 2.8 h
12 Use the degrees and minutes key on your calculator to express each time in hours.
a 1 h 24 min b 2 h 42 min c 4 h 45 min d 36 min
13 Express each of these in 24-hour time.
a 2 am b 7 pm c 12 midnight d 12 noon
e 4:30 am f 1:45 pm g 11:59 pm h 12:24 am
14 Express each of these in standard 12-hour time.
a 04:00 b 07:30 c 13:00 d 15:20
e 08:15 f 16:35 g 20:00 h 23:47
15 Simplify, giving the answer in metres.
a 1 m + 37 cm + 9 mm b 3.6 m + 228 cm + 15 mm
c 12.7 km + 83 m + 54 cm d 1 km + 455 m + 38 cm
16 a Which distance is greater, 15.8 m or 14 950 mm, and by how many metres?
b How many toothpicks of length 65 mm can be cut from a 1.3 m strip of wood?
c A snooker table is to have 6 legs made and each leg is to be 72 cm long. How many metres of wood are needed?
d How many laps of a 400 m running track must an athlete complete in order to finish a 10 km run?
e From a 3.6 m piece of timber, 5 pieces of equal length are cut, leaving 28 cm. What lengths of timber were cut?
f The average length of Lucy’s walking stride is 38 cm. How far, in kilometres, would Lucy walk if she took 9500 strides?
g Fourteen cars each of width 1.65 m are parked side by side in a car park. The distance between each car is 85 cm. Find, in metres, the total distance taken up by the cars.
17 a Find, in kilograms, the mass of 24 tins of soup, if each tin has a mass of 535 g.
b The total mass of 8 small cars in a shipping container is 7.6 t. What is the mass, in kilograms, of each car?
c Find the mass, in kilograms, of 3000 pumpkin seeds if each seed weighs 450 mg.
d A bunch of 64 grapes has a mass of 430 g. Find the average mass of each grape, correct to 1 decimal place.
e How many 225 g bags of lollies can be filled completely from a container that holds 12 kg of lollies?
f A builder wants to construct a brick wall consisting of 18 layers, with 15 bricks in each layer. Calculate, in kilograms, the total mass of bricks needed for the job if each brick weighs 2150 g.
g A box of 15 fresh pineapples has a mass of 58.6 kg and each pineapple has a mass of 3680 g. Determine the mass of the box when empty. Give your answer in kilograms.
h If 7 containers of wheat have a total mass of 980 kg, find, in tonnes, the mass of 19 containers of wheat.
18 a A 300 mL bottle of salad dressing contains enough dressing for 15 serves. What is the serving size?
b i How many 250 mL glasses can be filled from a juice container that holds 20 L?
ii How many 300 mL glasses can be filled completely?
c Cary purchased a 250 mL bottle of cough medicine. The adult dosage is 10 mL, three times a day. How many full days will the cough medicine last?
d The local council pool has a capacity of 1500 kL. During a hot spell, 15 000 L of water was lost due to evaporation. How many kilolitres of water remain?
e The dam on old Henry’s farm has a capacity of 8.3 ML. How many litres is this?
(1 ML = 1000 kL)
f A recipe requires cup of water for each person. Find, in litres, the amount of water that is needed for 11 people if 1 metric cup is equivalent to 250 mL.
g i Amber’s garden tap is dripping at the rate of 16 drops per minute, with each drop of water having a volume of 0.5 mL. How many litres of water will be lost in one day?
ii If on a subsequent day the tap drips at the rate of 10 drops per minute and loses 36 L over the course of a day, find the volume of water in each drop.
19 How long is it, in hours and minutes, from:
a 8 am to 2:15 pm? b 4:45 am to 10:30 am? c 11:19 am to 10:08 pm?
20 Calculate the time difference between:
a 7:20 pm Saturday and 2:05 am Sunday b 9:12 am Thursday and 12 noon Friday 21 What will the time be:
a 9 h 26 min after 12:57 am? b 3 h 10 min before 8:05 pm?
■ Further applications
22 The carat is a unit of mass that is used to measure precious stones and some expensive metals such as gold. If a certain amount of gold is described as 24 carat, it means that the entire mass is composed of 100% pure gold with no impurities. The purity of the gold can be worked out by expressing the number of carats as a fraction of 24. For example, a 10 kg ingot of 12 carat gold is only 50% pure, because 12 is one-half or 50% of 24. Therefore, in this 10 kg ingot, 5 kg of the mass is pure gold and the other 5 kg is made up of impurities (such as other metals).
How many grams of gold are there in:
a a 1 kg ingot of 12 carat gold? b a 2 kg ingot of 18 carat gold?
c a 1.5 kg ingot of 8 carat gold? d a 6 kg ingot of 14 carat gold?
1 2
---It is not possible to measure any length, mass, time, temperature or other quantity exactly. The value read off a measuring instrument is affected by physical factors, such as the thickness of the ink that is used in the markings on the instrument. However, we know that the exact value of the quantity being measured lies between two adjacent markings. The accuracy of a measurement refers to how close the reading is to the exact value of the quantity. The degree of accuracy in any measurement depends very much on the accuracy of the measuring instrument that is being used. Before measuring something, we need to consider the following questions:
1 Why are we conducting the measurement?
2 How precise does the measurement have to be?
3 What measuring instrument would be the most appropriate to use?
The precision of a measuring instrument refers to the smallest unit that is marked on it. For example, a metre ruler that is marked in 1 cm intervals has a precision of 1 cm. It is not possible to be more precise than the smallest unit that is marked on the instrument.
Example 1
State the limits of accuracy for each of the following measurements.
a The temperature of a sick child is 39°C, correct to the nearest 1°C.
b The height of a man is 180 cm, correct to the nearest 10 cm.
Solutions
a The temperature is given correct to the nearest 1°C, so the possible error is , i.e. ±0.5°C.
Therefore, the limits of accuracy are 39 ± 0.5°C, i.e. 38.5°C and 39.5°C. The actual temperature of the child must lie between 38.5°C and 39.5°C.
b The height is given correct to the nearest 10 cm, so the possible error is , i.e. ±5 cm.
Therefore, the limits of accuracy are 180 ± 5 cm, i.e. 175 cm and 185 cm. The actual height of the man must lie between 175 cm and 185 cm.