Mixed Numbers and Improper Fractions

Pieces of two sizes

Here is $1\frac{3}{4}$1 3/4: one whole bar and 3 quarters of another. It is a mixed number, a whole and a part.

To write it as one fraction, every piece has to be the same size. The $\frac{3}{4}$3/4 is already in quarters, but the whole is still one big piece, so the whole and the quarters cannot be counted together yet.

The whole is one uncut piece, and the part is 3 quarters.

Break the whole into quarters

Cut the whole bar into 4 equal pieces. Each piece is a quarter, the same size as the pieces of the $\frac{3}{4}$3/4, and the whole is now $\frac{4}{4}$4/4. Nothing was added or taken away: the whole bar is exactly as big as before, only cut up.

Drag the handle to cut the whole bar into more pieces. At 4 pieces they match the quarters below, and every piece gets a number: $1\frac{3}{4}=\frac{4}{4}+\frac{3}{4}=\frac{7}{4}$1 3/4 = 4/4 + 3/4 = 7/4.

Count every quarter

Now every piece is a quarter, so count them all: 4 quarters in the whole and 3 more in the part make 4 + 3 = 7 quarters. $1\frac{3}{4}=\frac{7}{4}$1 3/4 = 7/4, the same amount written as an improper fraction.

The denominator does not change. The pieces are quarters before and after, so it stays 4.

Quarters 1 to 4 make the whole, and quarters 5, 6 and 7 make the part: $\frac{7}{4}$7/4.

Several wholes: multiply, then add

Write $3\frac{2}{5}$3 2/5 as an improper fraction. Each whole breaks into 5 fifths, so the 3 wholes make 3 × 5 = 15 fifths. The $\frac{2}{5}$2/5 adds 2 more fifths: 15 + 2 = 17. So $3\frac{2}{5}=\frac{17}{5}$3 2/5 = 17/5.

That is the rule: multiply the whole number by the denominator, add the numerator, and keep the denominator.

Three whole bars hold 3 × 5 = 15 fifths, and 2 more make 17: $3\frac{2}{5}=\frac{17}{5}$3 2/5 = 17/5.

And back again: divide

To turn an improper fraction back into a mixed number, ask how many wholes the pieces fill. For $\frac{17}{5}$17/5, every 5 fifths fill one whole, so divide: 17 ÷ 5 = 3 remainder 2. The 3 is the number of full wholes, and the remainder, 2, is the fifths left over. $\frac{17}{5}=3\frac{2}{5}$17/5 = 3 2/5.

For $\frac{11}{4}$11/4, 11 ÷ 4 = 2 remainder 3, so $\frac{11}{4}=2\frac{3}{4}$11/4 = 2 3/4. The remainder stays over the same denominator, because the pieces left over are still quarters.

Counted in quarters from 0, $\frac{11}{4}$11/4 lands past 2, three quarters of the way to 3: $\frac{11}{4}=2\frac{3}{4}$11/4 = 2 3/4.

The usual mistakes

Dropping the whole: $1\frac{3}{4}$1 3/4 is not $\frac{3}{4}$3/4, because the whole bar holds 4 quarters of its own. Changing the denominator: $1\frac{3}{4}$1 3/4 is not $\frac{7}{8}$7/8. The pieces are still quarters, so the denominator stays 4. Using the wrong numbers: for $5\frac{3}{4}$5 3/4, adding 5 + 3 = 8 or multiplying 5 × 3 = 15 does not count the quarters. The 5 wholes are 5 × 4 = 20 quarters, and 20 + 3 = 23, so $5\frac{3}{4}=\frac{23}{4}$5 3/4 = 23/4.