Multiply and divide decimal amounts

Count the units in a decimal product

You can multiply by a decimal even when it is less than one. Taking 0.3 of an amount means taking three tenths of it.

For 0.24 × 0.3, 0.24 is 24 hundredths and 0.3 is three tenths:

One tenth of 24 hundredths is 24 thousandths. Three such parts contain 72 thousandths.

The product is less than 0.24, as expected when taking only three tenths of it. Counting 24 times 3 gives 72, but those 72 units are thousandths, not tenths or hundredths.

A factor greater than one has a different effect. For 2.4 × 1.5, take one whole amount and another half:

Alternatively, 24 × 15 = 360 counts hundredths, giving 3.60. The two methods agree. A positive factor below one reduces a positive amount; a factor above one increases it. Multiplying by one preserves it, and multiplying by zero gives zero.

Check your understanding 1 Check the size before the digits

Find 0.36 × 0.4 and 0.36 × 1.4. Decide which product should be below 0.36 and which should be above it. Explain the units in one calculation.

Your two products, size checks and units

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0.36 × 0.4 = 0.144, which is below 0.36. Multiplying 36 hundredths by four tenths gives 144 thousandths.

0.36 × 1.4 = 0.504, which is above 0.36. You can check by adding one whole amount and four tenths of it: 0.36 + 0.144 = 0.504.

Change both measures to the same unit

How many 0.6 ℓ bottles can be filled from 4.8 ℓ? Counting tenths of a litre, the question becomes how many groups of six fit in 48:

Eight bottles check: 8 × 0.6 = 4.8 ℓ. The amount of liquid has not changed; you expressed both measures in smaller units.

The number being divided is called the dividend; the number you divide by is the divisor. Multiplying both by the same positive number preserves the quotient. If this scaling needs practice, revisit Decimal scaling and unit conversion.

A quotient need not be a whole number. For 0.84÷0.2, multiply both values by ten:

Check 4.2 × 0.2 = 0.84. Dividing by 0.2 asks how many fifths fit, so the numerical quotient is larger than the dividend. By comparison, 8.4÷1.2 = 84÷12 = 7: a divisor greater than one gives a smaller quotient for a positive dividend. Division needs a nonzero divisor.

Check your understanding 2 Keep the quotient unchanged

Asha replaces 0.96÷0.4 with 0.96÷4 and obtains 0.24. Which quantity did she leave unscaled? Find the correct quotient and check it by multiplication.

Your corrected division and multiplication check

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She multiplied the divisor by ten but left the dividend unchanged. Scale both: 0.96÷0.4 = 9.6÷4 = 2.4. Check 2.4 × 0.4 = 0.96. The answer 0.24 belongs to a different division, 0.96÷4.

Keep the exact result until rounding is requested

For 1.0÷0.6, equal scaling gives 10÷6 = 5/3 = 1.666…. To two decimal places this is 1.67. The rounded value checks only approximately: 1.67 × 0.6 = 1.002, rather than exactly 1.0.

Use the exact value or a calculator’s unrounded stored result in a later calculation. Round the final answer to the requested place. A displayed approximation should not be used as though it were the exact original amount.

Check your understanding 3 Choose two operations from a situation

A 3.6 m ribbon is cut into pieces 0.45 m long, with none left over. Each piece costs 80 cents. Find the total cost in dollars. Check your result by grouping two pieces at a time.

Your piece count, total cost and grouped check

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The piece count is 3.6÷0.45 = 360÷45 = 8. Eight pieces cost 8 × 0.80 = 6.40 dollars.

Two pieces use 0.90 m and cost 1.60 dollars. The ribbon makes 3.6÷0.90 = 4 such groups, costing 4 × 1.60 = 6.40 dollars. Both methods account for the same eight pieces; neither rounds an intermediate amount.