Unit conversion — length, area and time

Top left is the length in centimetres. The top row converts it step by step to the right: cm → m → km. In the bottom row, the left card goes from cm to km in one step, and the right card turns the metres back into centimetres. Each card shows whether it divides or multiplies, such as "÷100" or "×100".

To a bigger unit you divide; to a smaller unit you multiply

A metre is 100 cm. Writing 350 cm in metres means splitting it into groups of 100 cm and counting the groups. 350 ÷ 100 = 3.5, so it is 3.5 m.

350 cm and 3.5 m are the same length. Counted in a bigger unit, it takes fewer of them, so the number gets smaller. Counted in a smaller unit, the number gets bigger. That is why converting to a bigger unit is a divide, and converting to a smaller unit is a multiply.

The bottom-right card on Board 1 turns 3.5 m back into centimetres. It shows "×100" and gives 350 cm again. One pairing such as "1 m = 100 cm" fixes the direction: 3.5 m holds 3.5 metres, each metre is 100 cm, so 3.5 × 100 = 350 cm.

Chaining two steps multiplies the factors

The top-right card converts metres to kilometres. A kilometre is 1,000 m, so it shows "÷1,000", and 350 cm is 0.0035 km.

The chain divides by 100 (cm → m) and then by 1,000 (m → km). Dividing twice like that is the same as dividing once by 100 × 1,000 = 100,000. The bottom-left card, cm → km (in one go), shows "÷100,000" and lands on the same 0.0035 km. However many units sit in between, the overall factor is the product of the factors along the way.

You can pick different units inside a convert card, and its swap button (⇄) reverses the direction of the conversion.

Area: both sides get ×100

For length, 1 m = 100 cm. For area, 1 m² is 10,000 cm². Board 2 shows why it is not 100.

The middle row has the height and width in metres, with the area between them. The bottom row converts the height and width to centimetres and works out the area in centimetres. The card at the top converts the middle area from m² to cm².

1 m² is the area of a square 1 m high and 1 m wide. Measured in centimetres, the same square is 100 cm high and 100 cm wide. Area is height × width, so it is 100 × 100 = 10,000 cm². The convert card at the top shows "×10,000" and gives the same 10,000 cm².

A length runs in one direction; an area has two, height and width. Switching to centimetres makes the height's number 100 times bigger and the width's number 100 times bigger too. Multiplied together, they make the area's number 100 × 100 = 10,000 times bigger.

100 cm² is a square 10 cm on each side

1 m = 100 cm does not make 1 m² equal to 100 cm². 100 cm² is the area on Board 2 with the height and width set to 0.1 m: 10 cm by 10 cm, or 0.01 m².

A 1 m square holds 10 of these 10 cm squares across and 10 down, 100 in all. 100 lots of 100 cm² make 10,000 cm². For volume, the same factor applies three times: height, width and depth.

Time comes in sixties

Metric units of length step up by 10, 100 or 1,000 (mm, cm, m, km). Time does not: an hour is 60 minutes and a minute is 60 seconds. Board 3 takes a film that runs 142 minutes and converts it to hours.

On the left is the running time in minutes. The top card converts it to hours (÷60). Below it, Whole hours counts how many full 60-minute blocks fit, and Minutes left is what remains. The two cards on the right turn the leftover minutes into hours and add them to the whole hours.

142 minutes is two blocks of 60 with 22 minutes left over: 2 h 22 min. In the formulas, quot gives the whole-number part of a division and rem gives the remainder.

As a single number, 142 ÷ 60 is about 2.37 hours (the board shows 2.366667 hr). Hourly pay and speeds in km/h work with time in this form, as one number of hours.

142 minutes is not 1.42 hours

Dividing by 100, as with cm → m, would turn 142 minutes into 1.42 hours. But an hour is 60 minutes, not 100. With the running time on Board 3 set to 85.2 minutes, min → hr shows 1.42 hr: 1.42 hours is 1 h 25.2 min.

Nor is the ".37" in 2.37 hours 37 minutes. As the Left, in hours card shows, it is the 22 leftover minutes divided by 60: 0.366667 hr. "2 h 22 min" and "about 2.37 hours" are the same length of time, written two ways.

Other numbers

Different lengths on Board 1:

Whatever the length, the number in metres is a hundredth of the number in centimetres, and the number in kilometres is a hundred-thousandth. The last row is a marathon (42.195 km).

Different heights and widths on Board 2:

Whatever the shape, the number in cm² is 10,000 times the number in m². A desk top 0.6 m by 1.2 m is 0.72 m², or 7,200 cm².

Different running times on Board 3:

90 minutes is exactly 1.5 hours because the 30 minutes left over are half of 60. 45 minutes is three quarters of 60, so 0.75 hours. The 20 minutes left from 200 are a third of 60, so the decimal never ends: 3.333333 hours. The part after the point in decimal hours is always the leftover minutes divided by 60.

Read next — Average speed for a round trip — total distance over total time. It calculates the average speed of a round trip by adding up the time taken each way and dividing the total distance by the total time.

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