Mixed Numbers and Fractions of Amounts

Improper Fractions #

A fraction bigger than one whole.

A fraction can be bigger than one whole

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$\frac{4}{4}+\frac{3}{4}=\frac{7}{4}$4/4 + 3/4 = 7/4. A fraction past a whole is called an improper fraction.

Mixed Numbers #

Whole ones plus a leftover piece.

The same amount can be written as whole ones plus a leftover

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Here is one full bar, plus 3 quarters of the next one.

A whole and a part together is a mixed number. We write this one as $1\frac{3}{4}$1 3/4.

Mixed Numbers and Improper Fractions #

Break whole ones back into pieces.

Whole ones can be broken back into pieces whenever you need them

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Here is 1 and 3 quarters — a whole and a part, so a mixed number.

Break that whole bar into 4 quarters too.

4 quarters and 3 quarters make $\frac{7}{4}$7/4: the same amount, as an improper fraction.

Fractions as Division #

Three shared among four is three quarters.

A fraction is a division: the top shared among the bottom

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Share 3 bars among 4 people: cut each into quarters, and take one piece from each.

One share is 3 quarter-pieces: $3÷4=\frac{3}{4}$3 ÷ 4 = 3/4. The fraction bar is a division sign.

Any division writes as a fraction — and past a whole, the mixed number takes over.

Adding and Subtracting Mixed Numbers #

Wholes with wholes, parts with parts.

Wholes add with wholes and parts with parts, trading a whole when the parts need it

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Add the wholes, add the parts: 2 + 1 = 3 and $\frac{1}{5}+\frac{3}{5}=\frac{4}{5}$1/5 + 3/5 = 4/5.

When the parts pass a whole, trade: $\frac{6}{4}$6/4 is 1 whole and $\frac{2}{4}$2/4, so the 3 becomes 4.

$\frac{1}{4}$1/4 cannot give up $\frac{3}{4}$3/4 — break a whole: $3\frac{1}{4}$3 1/4 is $2\frac{5}{4}$2 5/4, and now the parts subtract.

A Fraction of an Amount #

Divide by the bottom, multiply by the top.

Divide by the denominator, then multiply by the numerator

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Split 12 into 3 equal shares: one share is 12 ÷ 3 = 4.

Two shares are 2 × 4 = 8, so $\frac{2}{3}$2/3 of 12 = 8.

Fractions of a Changing Whole #

Every step shrinks the whole the next one acts on.

Each fraction takes its share of what the last step left, not of the start

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$\frac{1}{3}$1/3 of the 24 marbles are red: 8. That leaves 24 − 8 = 16.

$\frac{1}{4}$1/4 of the remainder are blue: a quarter of 16 is 4 — not a quarter of 24.

24 at the start, 16 after red, 12 after blue.