Decimal scaling and unit conversion

Keep track of each digit’s value

2.375 × 100 = 237.5. Every digit has one hundred times its former value: 2 ones become 2 hundreds. Conversely 237.5÷100 = 2.375. Multiplication by a number greater than one increases this positive amount; division reduces it.

For 4.06÷1000 = 0.00406, the zero between 4 and 6 remains important. Check by multiplying the answer by 1000. Do not simply remove every zero you see.

Multiplying by 10 makes each digit worth ten times as much; multiplying by 1000 makes it worth one thousand times as much. Dividing by these numbers reverses that change. The decimal point still separates ones and tenths: it is the digits’ place values that change.

Check your understanding 1 Keep the empty places

Without a calculator, find 0.508 × 10 and 0.508÷1000. Say which result is greater than 0.508. Explain why the zero between 5 and 8 must stay.

Your two results, size check and zero explanation

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0.508 × 10 = 5.08, which is greater than 0.508. 0.508÷1000 = 0.000508, which is smaller. In 5.08 the zero holds the tenths place; without it, 5.8 would have eight tenths instead of eight hundredths. In 0.000508 the zero between 5 and 8 holds the hundred-thousandths place. The extra zeros before 5 hold places emptied by division.

Unit size tells you the direction

1 km is 1000 m, so 1.265 km is 1265 m. The distance is unchanged, but the smaller unit is counted more times. One metre is 100 cm, so 3.08 m is 308 cm. Metres and centimetres use a factor of 100, not 1000.

One kilogram is 1000 g: 45 g is 0.045 kg. One litre is 1000 ml: 2.4 ℓ is 2400 ml. Always keep the unit with the numerical value.

Check your understanding 2 Which factor belongs here?

Convert 75 cm to metres, and 75 ml to litres. Why do the decimal answers differ?

Your two conversions and factor explanation

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75÷100 = 0.75 m, but 75÷1000 = 0.075 ℓ. The unit pairs have different conversion factors; the initial number 75 does not choose the factor.

Check a measurement claim

Check your understanding 3 Correct a bottle label

A bottle marked 0.35 ℓ is described as holding 3500 ml. Correct the description and explain a reverse check.

Your corrected amount and reverse check

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0.35 ℓ is 350 ml, since 0.35 × 1000 = 350. Dividing 350 by 1000 returns 0.35. The proposed 3500 ml would be 3.5 ℓ, ten times too much.