Division, fractions and decimals

Sharing can produce a fraction

Three identical cakes shared equally among four people give each person 3÷4 = 3/4 of a cake. Cut each cake into four equal pieces. Each person receives one quarter from each of three cakes: three quarters altogether.

The dividend becomes the numerator and the non-zero divisor becomes the denominator: a÷ b = a/b. Both describe how much one person receives when a wholes are shared among b people.

For 7÷4, each person gets one whole and three quarters: 7/4 = 13/4. A fraction can exceed one; the denominator tells the size of each part, not an upper limit on the answer.

If the sharing is exact, the quotient can be a whole number: 8÷4 = 8/4 = 2. Sharing zero among a positive number of people gives zero each. Dividing by zero is undefined.

Two ways to find a decimal

For 3/4, make hundredths: 3/4 = 75/100 = 0.75. Alternatively divide 3.00 by 4. Three ones become thirty tenths. Four groups get seven tenths each, leaving two tenths, or twenty hundredths. Those share into five hundredths per group. Each group gets 0.75.

For 3/8, 3÷8 = 0.375. Check 0.375 × 8 = 3. Do not write 3/8 = 0.38: that is a rounded approximation, not an exact equality.

Check your understanding 1 Keep the unit whole clear

Five litres of juice are shared equally among eight bottles. Give the amount in each bottle as a fraction of a litre and as a decimal in litres.

Your fraction, decimal and check with units

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Each bottle contains 5÷8 = 5/8 litre, or 0.625 litre. Check eight lots of 0.625 litre make 5 litres. The fraction is of one litre, not of one bottle’s capacity.

When an exact decimal does not stop

1/3 = 0.333… continues. If a problem asks for two decimal places, write 1/3 ≈ 0.33, using an approximation sign. Keep the fraction when an exact answer is needed.

Check your understanding 2 Distinguish an exact value from a rounded one

Jin writes 2/3 = 0.67. Correct the statement and give the decimal to two places. What happens if you multiply the rounded value by 3?

Your corrected statement, rounding and multiplication check

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2/3 = 0.666… ≈ 0.67 to two decimal places. The finite 0.67 is an approximation, not an exact equality. Its multiplication check gives 0.67 × 3 = 2.01, whereas the exact fraction gives 2/3 × 3 = 2. Keep the exact fraction for a later calculation unless rounding is requested.

Check your understanding 3 Represent a quotient greater than one

Express 9÷5 as a fraction and as a decimal. Check the result by multiplication.

Your fraction, decimal and reverse check

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9/5 = 14/5 = 1.8. Five groups of 1.8 make 9.