Metric conversions: worked examples and method

The length chain

10 mm = 1 cm, 100 cm = 1 m, 1 000 m = 1 km.

To go to a smaller unit, multiply. To go to a larger unit, divide.

Metres and centimetres

A metre is 100 centimetres. 2.5 m = 250 cm.

Moving the digits two places is the same as × 100 or ÷ 100.

Millimetres

1 cm = 10 mm. A 6 cm pencil is 60 mm long.

Do not use 100 here. That factor belongs to centimetres in a metre.

Kilometres

1 km = 1 000 m. A 3 km walk is 3 000 m.

A 1 500 m walk is 1.5 km, because 1 500 ÷ 1 000 = 1.5.

Mass and capacity

1 kg = 1 000 g and 1 litre = 1 000 ml. The factor is the same as kilometres to metres.

250 g is 0.25 kg. 750 ml is 0.75 litres.

Sensible units

Use mm for a coin’s thickness, cm for a hand, m for a room, km for a town.

A person’s mass is kilograms, not grams or tonnes.

Grams, kilograms, millilitres and litres

Mass uses 1000. 2.75 kg = 2.75 × 1000 = 2750 g. You can split it: 2 kg = 2000 g and 0.75 kg = 750 g, then 2000 + 750 = 2750 g. The other way, 2750 g = 2.75 kg. The point moves three places because 1000 has three zeros. 250 g = 0.25 kg, not 2.5 kg.

Capacity uses the same 1000. 1.25 litres = 1250 ml. 400 ml = 0.4 litres. A bottle of 1.25 litres with 400 ml poured out is easiest in millilitres: 1250 - 400 = 850 ml, which is 0.85 litres. Subtracting 1.25 - 400 as if both were litres gives a nonsense negative.

Do not borrow the length factors for mass. There are not 100 grams in a kilogram, and there are not 10 millilitres in a litre. If the unit is grams, kilograms, millilitres or litres, the step is 1000 unless a question gives a different rate, which a standard conversion does not.

Method: 1. Kilograms to grams is multiply by 1000. 2. Litres to millilitres is also multiply by 1000. 3. Divide by 1000 to go back to the larger unit.

Multiply towards the smaller unit

Ask which unit is smaller before you pick multiply or divide. Centimetres are smaller than metres, so metres to centimetres multiplies. Kilograms are larger than grams, so kilograms to grams multiplies, and grams to kilograms divides. The number gets bigger when the unit gets smaller, because you are counting smaller pieces.

A check is the size of a familiar object. Maya is about 1.6 m tall, which is 160 cm. An answer of 16 cm is far too short for a person, so the factor was 10 instead of 100. An answer of 1600 cm is 16 m, far too tall, so an extra zero crept in. The familiar object catches the wrong link in the chain.

Decimal answers need the point in the right column. 2500 ÷ 1000 = 2.5, so 2500 g = 2.5 kg. Moving the point three places because 1000 has three zeros is the same division. Moving it two places gives 25 kg, which would be a heavy suitcase rather than a bag of flour. Count the zeros in the factor you meant to use.

Method: 1. Decide which unit is smaller before you choose multiply or divide. 2. Expect a larger count when the unit is the smaller one. 3. Check the answer against a familiar size, such as a person who is about one and a half metres tall.

Convert before you add

Two lengths written in different units are not ready to add. 1.6 km and 850 m: convert the kilometres to metres, 1.6 × 1000 = 1600, then add 1600 + 850 = 2450 m. If the question wants kilometres, divide that total by 1000 to get 2.45 km. Adding 1.6 + 850 = 851.6 mixes a kilometre number with a metre number.

The same rule is for mass. A parcel of 1.4 kg and a letter of 250 g: convert the kilograms first, 1.4 × 1000 = 1400 g, then 1400 + 250 = 1650 g. In kilograms that is 1.65 kg. Either unit is fine if the whole answer uses it. Do not leave the answer as 1.4 kg + 250 g.

Small length mixes use the 10. 3 cm + 40 mm: convert the centimetres, 3 × 10 = 30 mm, then 30 + 40 = 70 mm, which is 7 cm. Adding 3 + 40 = 43 keeps the digits and loses the units. 43 cm would be a long ruler. 7 cm is a short label. The converted total should look like the objects in the story.

Method: 1. Convert both measurements into the same unit first. 2. Add or subtract those converted numbers. 3. Convert the total at the end only if the question asks for the other unit.

Answers in two units, such as kg and g

Some questions say "Give your answer in kilograms and grams". The calculation is easier in one unit, so change everything to grams, do the sum, then split the answer at the end. A parcel of 4 kg 650 g is 4650 g. If the range of masses is 900 g, the heaviest parcel is 4650 + 900 = 5550 g.

To write 5550 g as kilograms and grams, divide by 1000. The whole number of thousands is the kilograms and the remainder is the grams: 5 kg 550 g. Never write a mixed answer as a decimal such as 5.550 g, because the unit is wrong. The same idea works for metres and centimetres, or hours and minutes, with the factor changing to 100 or 60.

Method: 1. Change every amount into the smaller unit, for example grams. 2. Do the addition or subtraction in that unit. 3. Divide by 1000 to find the whole kilograms, and keep the remainder as grams. 4. Write both units in the answer, kilograms first.

Tip: Do not mix a point and a unit: 5 kg 550 g and 5.55 kg are the same mass, but 5.55 g is not.

Ordering amounts written in different units

A typical question lists three distances such as 2.3 km, 2150 m and 21/4 km and asks for them in order of size. You cannot compare the numbers as they stand. Choose one unit, usually the smaller one, and write all three in it: 2300 m, 2150 m and 2250 m.

Rank the converted values, then write the original amounts in that order. The exam asks for the amounts as they were given, with their own units, so do not replace them with the converted numbers in the final answer. A fraction of a kilometre is converted by finding the decimal first: 21/4 = 2.25, then 2.25 × 1000 = 2250. A list of converted values written beside the originals earns the method mark.

Method: 1. Pick one unit for all of the amounts. 2. Convert each amount, turning mixed numbers into decimals if that helps. 3. Put the converted values in order. 4. Write the original amounts, with their units, in the final order.

How many fit, and what is left over

Worded questions ask how many glasses can be filled from a jug, or how many pieces can be cut from a length of ribbon. The container is in one unit and the small amount in another. First change both to the same unit. A 3.5 litre jug is 3500 ml. The glasses hold 200 ml, so divide: 3500 ÷ 200 = 17.5.

The answer is the number of whole glasses, 17, not 17.5 and not 18. The leftover liquid is 3500 - 17 × 200 = 100 ml. Questions often ask for both pieces of information, so read the last line carefully. If a question asks how many are needed to hold an amount, you round up instead. That is a different question, so ask yourself whether you are filling up or holding everything.

Method: 1. Convert the big amount and the small amount into the same unit. 2. Divide the big amount by the small amount. 3. Decide whether the question wants whole pieces (round down) or enough to hold everything (round up). 4. Find the amount left over by taking the used amount away from the total.

Tip: Check by multiplying back: whole pieces times piece size, plus the leftover, must equal the original amount.

Converting with a rate that is given to you

Some metric questions give you a link between two units, such as "Use 8 km = 5 miles", or "1 gallon = 4.5 litres". You do not need to remember it. Work out what one unit is worth, or scale up, whichever is easier. For 8 km = 5 miles, one kilometre is 5 ÷ 8 = 0.625 miles, so 36 km is 36 × 0.625 = 22.5 miles.

Check which way the conversion goes. Miles are bigger than kilometres, so there must be fewer miles than kilometres. If your answer has more miles than kilometres, you have multiplied when you should have divided. Write the unit with the final answer. Calculators are usually allowed for these, but the setting out of 36 × 5 ÷ 8 still earns the method mark.

Method: 1. Write the given link as a statement with both units. 2. Decide whether the answer should be bigger or smaller than the starting number. 3. Scale up or down using multiplication or division. 4. Write the unit and check that the size of the answer makes sense.