🍱Mixed Numbers & Improper Fractions

The fractions in fractions were mostly small: numerator below denominator, less than one whole. But real life rarely stops at one pizza — a pizza and a half, two and three quarters... How do we write those? Mathematics keeps two costumes for every such number, and this lesson teaches you to change between them freely.

Three names for fractions

Sorted by who is bigger, the numerator or the denominator, fractions come in three grades:

• Proper fractions: numerator smaller than the denominator, like $\frac{3}{4}$ — less than one whole;
• Improper fractions: numerator equal to or bigger than the denominator, like $\frac{7}{4}$ — a whole and then some;
• Mixed numbers: a whole number with a fraction riding along, like $1\frac{3}{4}$, read "one and three quarters".

Three names, but only one number: $1\frac{3}{4}$ and $\frac{7}{4}$ stand on the very same spot of the number line.

Improper → mixed: do one division

$\frac{7}{4}$ means 7 quarter-slices. Every 4 slices rebuild one whole pizza, so 7 slices make 1 whole with 3 slices left over:

$\frac{7}{4} = 1\frac{3}{4}$

So the conversion is simply divide the numerator by the denominator: the quotient becomes the whole part, the remainder becomes the new numerator, the denominator stays put. Since $11 \div 4$ gives 2 remainder 3, we get $\frac{11}{4} = 2\frac{3}{4}$.

Mixed → improper: multiply, then add

Walking the other way, $2\frac{1}{3}$ is 2 whole pizzas plus one third. Each whole pizza is cut into 3 slices, so 2 wholes give $2 \times 3 = 6$ slices; add the lone slice and you have 7:

$2\frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{7}{3}$

The chant: multiply the whole by the denominator, then add the numerator, denominator unchanged. Multiply, then add — two steps and done.

3/4

Numerator

−3+

Denominator

−4+

3/4 > 5/8

A greater than B

5/8

Numerator

−5+

Denominator

−8+

Compare with: 3/4 = 6/8 —— Equivalent multiplier ×2

Arrange it on the two pizzas: $\frac{7}{4}$ and $1\frac{3}{4}$ build the very same picture — only the costume changed, not a crumb of pizza.

Which costume, when

Each form owns its home turf:

• Reading, comparing, daily life — mixed numbers feel natural: $2\frac{1}{2}$ kg is instantly "two and a half kilos";
• Computing — improper fractions save work: to find $1\frac{1}{2} \times \frac{2}{3}$, transform into $\frac{3}{2} \times \frac{2}{3} = 1$ and let the rules from fraction arithmetic take over.

Comparing mixed numbers has a shortcut too: compare the whole parts first. The bigger whole wins outright — $3\frac{1}{6} > 2\frac{5}{6}$, even though $\frac{1}{6}$ is far smaller than $\frac{5}{6}$. Only when the whole parts tie does the fraction part step in: $2\frac{1}{3} < 2\frac{3}{4}$.

Check yourself

✅

Quick quiz

↺ Try again

1. 1. Write the improper fraction 9/4 as a mixed number.

   1 1/42 1/42 3/4

2. 2. Write the mixed number 3 2/5 as an improper fraction.

   11/515/517/5

3. 3. Which is bigger, 2 1/3 or 2 3/4?

   2 1/32 3/4They are equal