Mathematics notes, unit 5

Length, Mass and Volume of Liquid

Textbook Mathematics Year 5 (DLP), Part 1 and Part 2, pages Part 2, 177–216

What this chapter covers

You can change between mm, cm, m and km, using decimals and fractions (5.1.1, 5.1.2)
You can add, subtract, multiply and divide lengths, including mixed forms and units (5.1.3–5.1.6)
You can change between g and kg, using decimals and fractions (5.2.1)
You can add, subtract, multiply and divide masses (5.2.2–5.2.5)
You can change between mℓ and ℓ, using decimals and fractions (5.3.1)
You can add, subtract, multiply and divide volumes of liquid (5.3.2–5.3.5)
You can solve problems about length, mass and volume of liquid in daily life (5.4.1)

In Year 5 you measure length, mass and volume of liquid using fractions and decimals as well as whole numbers. You change between units, then add, subtract, multiply and divide. The big idea is the same every time: make the units the same, and write the numbers in the same form (all decimals or all fractions), before you calculate.

Facts to remember (textbook pp. 177 to 199): 1 cm = 10 mm, 1 m = 100 cm, 1 km = 1 000 m. 1 kg = 1 000 g. 1 ℓ = 1 000 mℓ. Going to a smaller unit, multiply. Going to a bigger unit, divide.

5.1Length

Converting units of length

Units of length: millimetre (mm), centimetre (cm), metre (m) and kilometre (km).
1 cm = 10 mm 1 m = 100 cm 1 km = 1 000 m

cm to mm: multiply by 10; mm to cm: divide by 10 — cm and mm

m to cm: multiply by 100; cm to m: divide by 100 — m and cm

km to m: multiply by 1 000; m to km: divide by 1 000 — km and m

Conversion rules after textbook pp. 177 to 179.

The answer can be written as a decimal or as a fraction. Do whichever the question asks for.

Change	Working	Answer
9.2 cm to mm	9.2 × 10	92 mm
3612 cm to mm	36.5 × 10	365 mm
47 mm to cm	47 ÷ 10	4.7 cm or 4710 cm
1.37 m to cm	1.37 × 100	137 cm
114 m to cm	1 m + (14 × 100) cm = 100 cm + 25 cm	125 cm
39 cm to m	39 ÷ 100	0.39 m
42.195 km to m	42.195 × 1 000	42 195 m
10 300 m to km	10 300 ÷ 1 000	10.3 km or 10310 km

Worked example: mixed units

a) 20 cm 6 mm = ? cm
Step 1: 6 mm = 6 ÷ 10 = 0.6 cm.
Step 2: 20 cm + 0.6 cm = 20.6 cm (or 2035 cm).

b) 5 m 80 cm = ? m
Step 1: 80 cm = 80 ÷ 100 = 0.8 m.
Step 2: 5 m + 0.8 m = 5.8 m (or 545 m).

c) 68 km 4 m = ? km
Step 1: 4 m = 4 ÷ 1 000 = 0.004 km.
Step 2: 68 km + 0.004 km = 68.004 km.

Number line from 0 to 1 000 m with quarter, half and three-quarter kilometre marks — A number line from 0 to 1 km. 14 km = 250 m, 12 km = 500 m, 34 km = 750 m (textbook p. 179).

Mind the zeros. 68 km 4 m is 68.004 km, not 68.4 km. A metre is a thousandth of a kilometre, so the metres need three decimal places: 004.

a) 3.4 cm = ___ mm b) 2 m 5 cm = ___ m c) 6 km 40 m = ___ m

Show answerHide answer

a) 3.4 × 10 = 34 mm
b) 5 ÷ 100 = 0.05, so 2 m + 0.05 m = 2.05 m
c) 6 × 1 000 + 40 = 6 040 m

Adding and subtracting length

Check that all the lengths are in the same unit and the same form. Then line up the decimal points and calculate.

Worked example: adding

The strings of three planet models are 26.3 cm, 18.4 cm and 3012 cm long. Find the total length in mm. (textbook p. 181)
Step 1: 3012 cm = 30.5 cm.
Step 2: 26.3 + 18.4 + 30.5 = 75.2 cm.
Step 3: change to mm: 75.2 × 10 = 752 mm.

Worked example: mixed forms

74 cm + 1.08 m + 234 m = ? m
Step 1: 74 cm = 0.74 m and 234 m = 2.75 m.
Step 2: 0.74 + 1.08 + 2.75 = 4.57 m.

Worked example: subtracting

The Penang Bridge is 13.5 km long and the Terengganu Drawbridge is 638 m long. How much longer is the Penang Bridge? (textbook p. 184)
Step 1: 638 m = 0.638 km.
Step 2: 13.500 − 0.638 = 12.862 km.

Do not work out 13.5 km − 638 m as 13.5 − 638. The units are different. Change 638 m to 0.638 km first (or 13.5 km to 13 500 m), then subtract.

4.75 m + 62 cm = ___ m and 62.7 cm − 54.9 cm = ___ cm

Show answerHide answer

62 cm = 0.62 m, so 4.75 + 0.62 = 5.37 m.
62.7 − 54.9 = 7.8 cm.

Multiplying and dividing length

Worked examples

a) A frame is made of 6 cards, each 2.8 cm wide. Total width = 6 × 2.8 = 16.8 cm.
b) 14 cards, each 1.9 cm long, in mm: 14 × 1.9 = 26.6 cm = 266 mm.
c) 100 × 4812 m: change 12 to 0.5 so the number is 48.5 m. 100 × 48.5 = 4 850 m = 4.85 km.
d) A stack of 8 ten-sen coins is 12 mm high. One coin: 12 mm ÷ 8 = 1.5 mm thick.
e) 315 m ÷ 16 in cm: 315 m = 320 cm, and 320 ÷ 16 = 20 cm.

Multiplying or dividing by 10, 100 and 1 000: move the decimal point 1, 2 or 3 places. 100 × 0.46 km = 46 km. 76.9 m ÷ 1 000 = 0.0769 m.

9 × 1.5 m = ___ m and 7.5 cm ÷ 5 = ___ mm

Show answerHide answer

9 × 1.5 = 13.5 m.
7.5 ÷ 5 = 1.5 cm = 15 mm.

5.2Mass

Mass is measured in grams (g) and kilograms (kg).
1 kg = 1 000 g

kg to g: multiply by 1 000; g to kg: divide by 1 000 — kg to g: × 1 000. g to kg: ÷ 1 000 (textbook p. 189).

Change	Working	Answer
1.8 kg to g	1.8 × 1 000	1 800 g
234 kg to g	2 000 g + (34 × 1 000) g = 2 000 g + 750 g	2 750 g
3.07 kg to g	3.07 × 1 000	3 070 g
6 900 g to kg	6 900 ÷ 1 000	6.9 kg or 6910 kg
5 kg 800 g to kg	5 kg + 0.8 kg	5.8 kg

A dial scale with the pointer at 2 kg 300 g for a jackfruit — Read the dial: each small mark is 100 g. The jackfruit is 2 300 g, which is 2.3 kg.

Worked examples

a) The bag has a mass of 3.505 kg and the present has a mass of 0.48 kg. Total = 3.505 + 0.480 = 3.985 kg (textbook p. 191).
b) 318 kg + 270 g in kg: 318 = 3.125 kg and 270 g = 0.27 kg. 3.125 + 0.27 = 3.395 kg.
c) Khairul is 34.9 kg and Divesh is 41.2 kg. Difference = 41.2 − 34.9 = 6.3 kg.
d) How much more is 14 kg than 0.1 kg, in g? 14 kg = 250 g and 0.1 kg = 100 g, so 250 − 100 = 150 g.
e) 6 042 g − 1.07 kg − 980 g in kg: 6.042 − 1.070 − 0.980 = 3.992 kg.
f) 7 packets of 0.25 kg each: 7 × 0.25 = 1.75 kg.
g) 1 000 tins of 0.85 kg: 1 000 × 0.85 = 850 kg = 850 000 g.
h) One chocolate bar if 7 bars are 0.35 kg: 0.35 ÷ 7 = 0.05 kg.
i) 100 macaroons weigh 112 kg. One macaroon: 1 500 g ÷ 100 = 15 g.

In a column sum, write the numbers one under the other with the decimal points lined up. Add zeros to fill the empty places: 3.505 + 0.480. When there are three numbers, subtract one at a time, one after another.

Change units before you add. For 7.42 kg + 845 g, do not add 7.42 + 845. Change 845 g to 0.845 kg first: 7.42 + 0.845 = 8.265 kg.

a) 0.04 kg = ___ g b) 7 250 g = ___ kg c) 1.6 kg + 450 g = ___ kg
d) 5 kg − 2 350 g = ___ g e) 12 × 0.75 kg = ___ kg f) 4.8 kg ÷ 6 = ___ g

Show answerHide answer

a) 40 g
b) 7.25 kg
c) 1.6 + 0.45 = 2.05 kg
d) 5 000 − 2 350 = 2 650 g
e) 9 kg
f) 0.8 kg = 800 g

5.3Volume of liquid

Volume of liquid is how much liquid there is. We measure it in millilitres (mℓ) and litres (ℓ).
1 ℓ = 1 000 mℓ

litres to millilitres: multiply by 1 000; millilitres to litres: divide by 1 000 — ℓ to mℓ: × 1 000. mℓ to ℓ: ÷ 1 000 (textbook p. 199).

Change	Working	Answer
0.3 ℓ to mℓ	0.3 × 1 000	300 mℓ
12 ℓ to mℓ	12 × 1 000	500 mℓ
2 750 mℓ to ℓ	2 750 ÷ 1 000	2.75 ℓ or 234 ℓ
42 ℓ 9 mℓ to ℓ	42 ℓ + (9 ÷ 1 000) ℓ	42.009 ℓ
6 025 mℓ to ℓ	6 025 ÷ 1 000	6.025 ℓ or 6140 ℓ

A measuring jug marked to 2 litres with the water level at 1 750 millilitres — The water is at 1 750 mℓ, which is 1.75 ℓ or 134 ℓ.

Worked examples

a) 1.875 ℓ + 512 ℓ = 1.875 + 5.5 = 7.375 ℓ (textbook p. 201).
b) 1.875 ℓ + 125 mℓ in mℓ: 1.875 ℓ = 1 875 mℓ, so 1 875 + 125 = 2 000 mℓ.
c) 5 ℓ − 3.3 ℓ = 1.7 ℓ.
d) 1014 ℓ − 2.5 ℓ = 10.25 − 2.50 = 7.75 ℓ.
e) 12.09 ℓ − 780 mℓ in mℓ: 12 090 − 780 = 11 310 mℓ.
f) 5 × 0.08 ℓ = 0.4 ℓ.
g) 100 bottles of 0.4 ℓ in mℓ: 100 × 0.4 = 40 ℓ = 40 000 mℓ.
h) 2.5 ℓ of soup into 4 bowls: 2.5 ÷ 4 = 0.625 ℓ each.
i) 2.5 ℓ of honey into 100 mini bottles, in mℓ: 2 500 mℓ ÷ 100 = 25 mℓ each.

Check your answer by doing the opposite operation. If 2.5 ℓ ÷ 4 = 0.625 ℓ, then 4 × 0.625 ℓ must give 2.5 ℓ.

0.5 ℓ is 500 mℓ, not 50 mℓ and not 5 mℓ. Multiply by 1 000, which moves the decimal point three places.

a) 0.6 ℓ = ___ mℓ b) 3 750 mℓ = ___ ℓ (fraction) c) 4.2 ℓ + 850 mℓ = ___ ℓ
d) 9 ℓ − 234 ℓ = ___ ℓ e) 8 × 0.45 ℓ = ___ ℓ f) 3.6 ℓ ÷ 8 = ___ mℓ

Show answerHide answer

a) 600 mℓ
b) 3 750 ÷ 1 000 = 3.75 = 334 ℓ
c) 4.2 + 0.85 = 5.05 ℓ
d) 9 − 2.75 = 6.25 ℓ
e) 3.6 ℓ
f) 0.45 ℓ = 450 mℓ

5.4Solving problems

Use the four steps (textbook p. 210): understand the problem, plan the strategy, solve, then check. Underline the important information, change all measurements to the same unit and form, and draw a bar model when it helps.

Path from home to library 1 250 m and library to school 2 750 m — Home to library 1 250 m, library to school 2 750 m. Total = 4 000 m = 4 km.

Worked example: three ropes (textbook p. 210)

Cliff has ropes of 12 cm, 0.16 m and 15 m. Find (a) the total length in cm and (b) the difference between the longest and the shortest rope in m.
Step 1: change all to cm: 12 cm, 0.16 m = 16 cm, 15 m = 20 cm.
Step 2: (a) 12 + 16 + 20 = 48 cm.
Step 3: (b) longest 20 cm, shortest 12 cm. 20 − 12 = 8 cm = 0.08 m.

Worked example: "times as much" (textbook p. 211)

Ashley's home is 814 km from school. That is 3 times the distance of Jeera's home. How far is Jeera's home, in m?
Step 1: 814 km = 8 250 m.
Step 2: the distance for Jeera = 8 250 ÷ 3 = 2 750 m.
Step 3: check: 3 × 2 750 = 8 250. Correct!

Worked example: mass of paper (textbook p. 211)

Pupils collected 4.67 kg of cans. The paper is 12 times as heavy. Mass of paper in g?
4.67 kg = 4 670 g. 12 × 4 670 = 56 040 g.

Always finish with the unit asked for in the question: "in m", "in kg" or "in mℓ". Many marks are lost by giving the right number in the wrong unit.