Table of Contents
Understanding Mixed Numbers
Mixed numbers appear often in everyday situations, such as measuring lengths, cooking, and telling time. In this chapter we focus on what mixed numbers are, how to move between mixed numbers and improper fractions, and how to place them on a number line.
You should already be familiar with proper and improper fractions from the previous chapter. Here we will use those ideas without re-explaining them in full.
What a Mixed Number Is
A mixed number combines a whole number and a proper fraction (a fraction less than 1). It has the form
$$
\text{(whole number)}\ \ \text{(fraction)}.
$$
Examples:
- $2 \dfrac{1}{3}$ (read “two and one third”)
- $5 \dfrac{7}{8}$ (read “five and seven eighths”)
- $11 \dfrac{1}{4}$ (read “eleven and one quarter”)
Key features:
- The whole number tells how many complete “ones” you have.
- The fraction shows the extra part of one more that you have.
- The fraction part of a mixed number is always a proper fraction (its value is less than $1$).
So $2 \dfrac{1}{3}$ means: $2$ whole units, plus another $\dfrac{1}{3}$ of a unit.
Visualizing Mixed Numbers
Mixed numbers can be pictured as “a number of whole objects plus some part of one more.”
Example: $1 \dfrac{3}{4}$
Think of 1 whole pizza and $\dfrac{3}{4}$ of another pizza:
- First pizza: complete.
- Second pizza: divided into $4$ equal slices; you have $3$ of the $4$ slices.
So $1 \dfrac{3}{4}$ is more than $1$ but less than $2$.
Another example: $3 \dfrac{2}{5}$
Imagine $3$ full bars of chocolate plus $\dfrac{2}{5}$ of another bar:
- $3$ full bars.
- One more bar cut into $5$ equal pieces; you keep $2$ of them.
So $3 \dfrac{2}{5}$ is more than $3$ but less than $4$.
Mixed Numbers and the Number Line
Mixed numbers fit naturally on the number line between whole numbers.
To place a mixed number $a \dfrac{b}{c}$ (where $0 < b < c$) on a number line:
- Find the two whole numbers it lies between:
- It is between $a$ and $a+1$.
- Divide the interval from $a$ to $a+1$ into $c$ equal parts.
- Starting at $a$, count $b$ parts to the right; that mark is $a \dfrac{b}{c}$.
Example: Place $2 \dfrac{3}{5}$ on the number line.
- It is between $2$ and $3$.
- Divide the segment from $2$ to $3$ into $5$ equal parts.
- Count $3$ parts to the right from $2$.
- That point is $2 \dfrac{3}{5}$.
This shows that mixed numbers are just another way of writing certain points on the number line.
Converting Mixed Numbers to Improper Fractions
Often it is easier to do calculations with improper fractions rather than mixed numbers. So it is important to be able to convert back and forth.
A mixed number $a \dfrac{b}{c}$ can be written as an improper fraction with denominator $c$.
Rule:
$$
a \dfrac{b}{c} = \dfrac{ac + b}{c}.
$$
Explanation:
- $a$ whole units each equal $\dfrac{c}{c}$, so $a$ wholes equal $\dfrac{ac}{c}$.
- Then you have the extra $\dfrac{b}{c}$.
- Together this is $\dfrac{ac}{c} + \dfrac{b}{c} = \dfrac{ac + b}{c}$.
Step-by-step method:
- Multiply the whole number by the denominator: $a \times c$.
- Add the numerator: $(a \times c) + b$.
- Place this sum over the original denominator $c$.
Examples:
- Convert $3 \dfrac{2}{5}$ to an improper fraction.
- Whole number $a = 3$, numerator $b = 2$, denominator $c = 5$.
- Multiply: $3 \times 5 = 15$.
- Add the numerator: $15 + 2 = 17$.
- Put over $5$: $3 \dfrac{2}{5} = \dfrac{17}{5}$.
- Convert $1 \dfrac{3}{4}$ to an improper fraction.
- $1 \times 4 = 4$.
- $4 + 3 = 7$.
- So $1 \dfrac{3}{4} = \dfrac{7}{4}$.
- Convert $6 \dfrac{1}{3}$.
- $6 \times 3 = 18$.
- $18 + 1 = 19$.
- So $6 \dfrac{1}{3} = \dfrac{19}{3}$.
Converting Improper Fractions to Mixed Numbers
Sometimes it is more natural to express an amount as a mixed number, especially when measuring or talking about quantities in real life. To do this, you start from an improper fraction and separate out the whole number part.
Given an improper fraction $\dfrac{n}{d}$ (with $n \ge d$), you can write it as a mixed number.
Method (using division):
- Divide the numerator by the denominator:
- $n \div d = q$ with remainder $r$, where $0 \le r < d$.
- The quotient $q$ becomes the whole number part.
- The remainder $r$ becomes the numerator of the fraction part.
- The denominator stays as $d$.
- So $\dfrac{n}{d} = q \dfrac{r}{d}$.
Examples:
- Convert $\dfrac{17}{5}$ to a mixed number.
- Divide $17 \div 5$:
- $5$ goes into $17$ three times ($3 \times 5 = 15$), remainder $2$.
- Quotient $q = 3$ (whole number part).
- Remainder $r = 2$ (fraction numerator).
- Denominator stays $5$.
- So $\dfrac{17}{5} = 3 \dfrac{2}{5}$.
- Convert $\dfrac{7}{4}$.
- $7 \div 4 = 1$ with remainder $3$.
- So $\dfrac{7}{4} = 1 \dfrac{3}{4}$.
- Convert $\dfrac{25}{6}$.
- $25 \div 6 = 4$ with remainder $1$ (since $4 \times 6 = 24$).
- So $\dfrac{25}{6} = 4 \dfrac{1}{6}$.
Note: If the remainder is $0$, there is no fraction part; the improper fraction is actually a whole number. For example,
$$
\dfrac{18}{3} = 6 \dfrac{0}{3} = 6.
$$
Comparing Mixed Numbers
To compare mixed numbers (decide which is larger), it helps to use both the whole number and fraction parts.
Given two mixed numbers:
$$
a \dfrac{b}{c} \quad \text{and} \quad d \dfrac{e}{f},
$$
you can compare them as follows:
- Compare the whole number parts $a$ and $d$.
- If $a > d$, then $a \dfrac{b}{c}$ is larger.
- If $a < d$, then $a \dfrac{b}{c}$ is smaller.
- If $a = d$, then compare the fraction parts $\dfrac{b}{c}$ and $\dfrac{e}{f}$:
- Use methods for comparing fractions (such as common denominators or cross-multiplication) from the fractions chapter.
Examples:
- Compare $3 \dfrac{1}{4}$ and $2 \dfrac{5}{6}$.
- Whole numbers: $3$ and $2$. Since $3 > 2$, $3 \dfrac{1}{4}$ is larger.
- No need to compare the fraction parts.
- Compare $4 \dfrac{2}{3}$ and $4 \dfrac{5}{8}$.
- Whole numbers: both $4$. Compare fraction parts $\dfrac{2}{3}$ and $\dfrac{5}{8}$.
- Use cross-multiplication:
- $2 \times 8 = 16$
- $5 \times 3 = 15$
- Since $16 > 15$, $\dfrac{2}{3} > \dfrac{5}{8}$, so
\dfrac{2}{3} > 4 \dfrac{5}{8}$.
Mixed Numbers in Everyday Contexts
Mixed numbers are especially common when units are divided into standard fractions:
- Length:
- $2 \dfrac{1}{2}$ meters, $5 \dfrac{3}{4}$ inches.
- Time:
- $1 \dfrac{1}{2}$ hours (one and a half hours).
- Cooking:
- $3 \dfrac{1}{4}$ cups of flour.
- Money (when expressed in fractional dollars instead of cents):
- $4 \dfrac{1}{2}$ dollars.
Often, instructions or measurements are given as mixed numbers because they are easier to read and understand than improper fractions like $\dfrac{9}{4}$ or $\dfrac{11}{2}$.
When to Use Mixed Numbers vs Improper Fractions
Both mixed numbers and improper fractions represent the same values; they are just different forms.
- Mixed numbers are usually:
- Preferred in spoken language and in measurement (for clarity).
- Used in word problems to describe amounts.
- Improper fractions are usually:
- Preferred when performing most arithmetic operations (addition, subtraction, multiplication, division).
- Easier to use in formulas and algebraic calculations.
You will practice arithmetic with mixed numbers in the “Operations with fractions” chapter. Here, the main skill to remember is how to move flexibly between mixed numbers and improper fractions depending on what the situation requires.