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A whole number times a fraction

Fractions, grade 6 Lesson 6 of 22Learning

A whole number times a fraction

Ten times a fifth, and a fifth of ten
Ten times a fifth – two stories
10×15=215×10=210×15=15×10=2
10×15\displaystyle 10\times\frac{1}{5} – ten times a fifth. This calculation has two stories.
The first story: 10 times a fifth. We have 10 cakes, and from each cake we take one fifth.
Gather the ten fifths: every five fifths make a whole cake – that is 2 cakes.
The second story: a fifth of 10. Split the 10 cakes into 5 equal groups.
Each group has 2 cakes, and one group is a fifth of all the cakes: 15\displaystyle \frac{1}{5} of 10 is 2 cakes.
Why is it the same? The ten fifths of the first story – a fifth from each cake – exactly fill the two cakes of one group. Taking a fifth of each cake is the same as taking a fifth of all the cakes together.
Two stories – and the same result. In multiplication, the order does not matter.
Step 1 of 7
Repeated addition

3×25\displaystyle 3\times\frac{2}{5} means three times 25\displaystyle \frac{2}{5}: 25+25+25=65\displaystyle {\frac{2}{5}+\frac{2}{5}+\frac{2}{5}}={\frac{6}{5}}. Only the numerator is multiplied – the parts stay fifths: 3×25=3×25\displaystyle {3\times\frac{2}{5}}={\frac{3\times2}{5}}.

A whole number times a mixed number

2×134\displaystyle 2\times1\frac{3}{4} can be worked out in two ways.

Way A
Write the mixed number as a fraction, then multiply.

2×134=2×74=144=324=312\displaystyle {2\times1\frac{3}{4}}={2\times\frac{7}{4}}={\frac{14}{4}}={3\frac{2}{4}}={3\frac{1}{2}}

Way B
The distributive law: multiply the whole part and the fraction separately, and add.

2×134=2×1+2×34=2+64=2+124=312\displaystyle {2\times1\frac{3}{4}}={2\times1+2\times\frac{3}{4}}={2+\frac{6}{4}}={2+1\frac{2}{4}}={3\frac{1}{2}}

Exercise 1

Work out 3×235\displaystyle 3\times2\frac{3}{5} both ways.
Did you get the same result?

Way A

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

    Way B

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

      Exercise 2

      Calculate in your head first, then write the calculation.

      a

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

        b
        A quarter of 20, three times.

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

          c

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power