Scale decimals and convert units

Follow the value, not a moving dot

In 3.047 × 100 = 304.7, each digit has one hundred times its old value. The 3 changes from 3 ones to 3 hundreds. In 3.047÷100 = 0.03047, every digit has one hundredth its old value. The decimal point still separates ones and tenths.

Multiplication by 10, 100 or 1000 makes each digit worth 10, 100 or 1000 times as much; division makes it worth one tenth, one hundredth or one thousandth as much. Keep zeros that hold empty places. For example, 0.406÷1000 = 0.000406, not 0.406 with its zero removed.

Check your understanding 1 Predict the size, then calculate

Without a calculator, find 0.607 × 10 and 0.607÷1000. Which result should be larger than 0.607? Use place value to explain a zero in the smaller result.

Your two results and place-value explanation

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0.607 × 10 = 6.07, which is larger than 0.607. 0.607÷1000 = 0.000607, which is smaller. The original 6 tenths become 6 ten-thousandths, so the tenths, hundredths and thousandths places before that 6 must be zero. The zero between 6 and 7 still holds an empty place.

Split a multiple into useful factors

For a multiple of ten, use factors: 1.24 × 300 = 1.24 × 3 × 100 = 372. Multiplying by 3 and then by 100 has the same effect as multiplying by 300.

For division, 37.2÷200 = 37.2÷100÷2 = 0.186. You must divide by both factors; multiplying by 2 would undo part of the reduction. Check 0.186 × 200 = 37.2.

Check your understanding 2 Repair the second operation

Mina writes 6.4÷400 = (6.4÷100) × 4 = 0.256. Identify the wrong step, find the correct quotient without a calculator and check it by multiplication.

Your correction, quotient and reverse check

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Since 400 = 100 × 4, divide by 100 and then by 4: 6.4÷100÷4 = 0.064÷4 = 0.016. Check 0.016 × 400 = 6.4. Mina’s multiplication by 4 makes her answer sixteen times the correct quotient.

Decide whether the numerical measure grows

One kilometre is 1000 metres. A distance of 2.375 km contains 2.375 × 1000 = 2375 m. The distance did not grow; metres are smaller units, so more of them are counted.

Do not use 1000 for every pair: metres and centimetres use 100. Write the unit after the converted value.

Check your understanding 3 Which distance is longer?

Compare 1.08 km and 1080 m. Then find how much longer 1.8 km is than 1080 m.

Your comparison and difference with units

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1.08 km equals 1080 m, so the first pair are equal. 1.8 km is 1800 m, which is 720 m longer than 1080 m. The zero in 1.08 changes the value.

Choose a sensible operation

Check your understanding 4 Find one total in two ways

Four bottles each hold 0.375 ℓ. Find the total in millilitres. Check by converting each bottle first and comparing with a calculation in litres.

Your total and two calculations with units

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4 × 0.375 = 1.5 ℓ, or 1500 ml. Alternatively each bottle contains 375 ml, and four contain 1500 ml. Both routes preserve the amount.