The same length, a different unit
Here is a rope 3 meters long. Measured in meters, it is 3 m. Now measure the very same rope in centimeters. The rope does not grow or shrink. Only the unit changes, and so the number that counts the units changes.
The rope runs from 0 to 3 on a ruler marked in meters, so it is 3 m long.
Each meter is 100 centimeters
Every meter is 100 cm, so each meter of the rope is 100 cm of it. The first meter is 100 cm, the second is 100 more, and the third is 100 more again: 100 + 100 + 100 = 3 × 100 = 300 cm.
So 3 m = 300 cm. The reason for the × 100 is the fact 1 m = 100 cm, and nothing else.
Meters on top, centimeters below. Each meter sits over another 100 cm, so 3 m sits over 300 cm.
Smaller unit, bigger number
A centimeter is much smaller than a meter, so it takes many more of them to cover the same rope. That is why the number grows, from 3 to 300, when the unit gets smaller.
Going the other way, from a small unit to a big one, the number gets smaller, so divide. 500 cm is , because every 100 cm make 1 m.
Look up the unit, then multiply
Each unit is worth a fixed amount. 1 km is 1000 m, 1 m is 100 cm, and 1 cm is 10 mm. Find the fact that joins the two units, then multiply by it.
3 km = 3 × 1000 = 3000 m. 4 cm = 4 × 10 = 40 mm. The same works with decimals: 3.5 m = 3.5 × 100 = 350 cm.
Each kilometer sits over another 1000 m, so 3 km is 3000 m.
Meters and centimeters together
A length is often given in two units at once, such as 2 m 85 cm. To write it in centimeters, change the meters first: 2 m = 200 cm. Then add the centimeters: 200 + 85 = 285 cm.
To split centimeters into meters and centimeters, take out every full 100 cm as a meter. 410 cm holds 4 full hundreds, which are 4 m, and 10 cm are left: 410 cm = 4 m 10 cm.
Two slips are common. Adding the bare numbers, 2 + 85 = 87, mixes meters with centimeters. And writing 12 m as 120 cm uses 10 cm to the meter: a meter is 100 cm, so 12 m = 12 × 100 = 1200 cm.
Worked example: Lengths in Meters and Centimeters Cut from a Roll
Question A roll of wire is 12 m long. Mr Tan cuts off one piece that is 2 m 85 cm long and another piece that is 75 cm long. He cuts the rest of the wire into 8 equal pieces. (a) How many centimeters of wire are left after the first two pieces are cut off? (b) How long is each of the 8 equal pieces?
1.Change every length to centimeters so that the lengths can be added and subtracted: 12 m = 1200 cm and 2 m 85 cm = 285 cm.
Every length is in centimeters: the roll is 1200 cm, and the two pieces cut off are 285 cm and 75 cm. 2.Add the two pieces that were cut off: 285 + 75 = 360 cm.
The two pieces cut off are 285 + 75 = 360 cm. 3.(a) The wire that is left is 1200 − 360 = 840 cm.
(a) 1200 − 360 = 840 cm of wire is left. 4.The 840 cm is cut into 8 equal pieces, so one piece is 840 ÷ 8 = 105 cm.
(b) Each of the 8 equal pieces is 840 ÷ 8 = 105 cm. 5.(b) Each equal piece is 105 cm long, which is 1 m 5 cm. Check: 8 × 105 + 360 = 1200.
Answer: (a) 840 cm; (b) 105 cm
Common mistakes
- Working out 12 − 285 − 75 or 12 − 2.85 − 75 with the numbers as they are given. A number of meters and a number of centimeters cannot be added or subtracted until they are in the same unit.
- Writing 12 m as 120 cm. There are 100 centimeters in a meter, so 12 m is 12 × 100 = 1200 cm.