---
title: "First Fractions"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 24
exercises: 11
source: https://one-course.com/books/math/1/en/chapter/24-first-fractions
---

# Chapter 24 — First Fractions

[Half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) an apple, a [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of an hour, three [quarters](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of the class: [fractions](#def-g4-fractions-def) name the pieces of things. This chapter introduces the writing $\frac{a}{b}$, reads [fractions](#def-g4-fractions-def) on pictures and number lines, and compares the simplest ones. ([Fractions](#def-g4-fractions-def) grow up in [Chapter 39](https://one-course.com/books/math/1/en/chapter/39-fractions-first-steps#ch-g6-fractions).)

## 24.1 Naming the pieces

**Definition 24.1 (Fraction).**

Cut a whole into equal parts and take some — that is a *fraction*:

$$
\frac{3}{4}
\quad
\begin{array}{l}
3 \text{: how many parts are taken (the \emph{numerator});}\\
4 \text{: how many equal parts the whole is cut into (the
\emph{denominator}).}
\end{array}
$$

$\frac12$ is a *[half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half)*, $\frac13$ a *third*, $\frac14$ a *[quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half)*, $\frac{1}{10}$ a *tenth*.

![Reading fractions on pictures. The parts must be equal: three unequal pieces do not make thirds!](https://one-course.com/images/onecourse/chapters/math-1/g4-fractions/fig-baa66ff816d3.svg)

*Reading [fractions](#def-g4-fractions-def) on pictures. The parts must be *equal*: three unequal pieces do not make thirds!*

**Example 24.2 (Fractions of a collection).**

$\frac13$ of $12$ marbles: share the $12$ marbles into $3$ equal groups of $4$; one group is $\frac13$, so $\frac13$ of $12$ is $4$ — and $\frac23$ of $12$ is two groups, $8$.

## 24.2 Fractions on the number line

**Method 24.3 (Placing ab\frac{a}{b}ba​).**

Cut each unit of the line into $b$ equal steps; from $0$, walk $a$ steps. If $a$ is bigger than $b$, the walk passes $1$: [fractions](#def-g4-fractions-def) can be bigger than one whole!

![The line cut in quarters: 2/4 lands on the same point as 1/2; 5/4 is one quarter past 1; 11/4 is 2 and 3/4.](https://one-course.com/images/onecourse/chapters/math-1/g4-fractions/fig-0247cd195ef4.svg)

*The line cut in [quarters](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half): $\frac24$ lands on the same point as $\frac12$; $\frac54$ is one [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) past $1$; $\frac{11}{4}$ is $2$ and $\frac34$.*

**Example 24.4 (Whole numbers hiding in fractions).**

$\frac44 = 1$ (four [quarters](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) make the whole); $\frac82 = 4$; $\frac{12}{3} = 4$. And $\frac74$ is $1 + \frac34$: one whole and three [quarters](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) more.

## 24.3 Comparing and adding simple fractions

**Proposition 24.5 (Same denominator).**

With the same [denominator](#def-g4-fractions-def), the pieces have the same size — so just count them:

$$
\frac{2}{8} < \frac{5}{8},
\qquad
\frac{3}{8} + \frac{4}{8} = \frac{7}{8}.
$$

Comparing to one whole is also easy: $\frac{5}{8} < 1 < \frac{9}{8}$ (fewer, then more, than $8$ eighths).

**Why, on a picture.** Shading $3$ eighths and then $4$ more eighths of the same bar shades $7$ eighths in total. ∎

**Example 24.6 (Different denominators, same point).**

On the number line, $\frac12$, $\frac24$ and $\frac{5}{10}$ all land on the same point: they are three names of the same number. Cutting each [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) into two makes [quarters](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half); into five, tenths. (The general rule is in [Chapter 39](https://one-course.com/books/math/1/en/chapter/39-fractions-first-steps#ch-g6-fractions).)

**Example 24.7 (Quarters of an hour).**

An hour has $60$ minutes, so a [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of an hour is $60 \div 4 = 15$ minutes, and three [quarters](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of an hour is $45$ minutes. “Half past two” is $\frac12$ hour after two o’clock.

## 24.4 Exercises

**Exercise 24.1 ★.**

Write the [fraction](#def-g4-fractions-def) shown: a pizza cut in $6$ with $5$ slices left; a chocolate bar of $8$ squares with $3$ eaten ([fraction](#def-g4-fractions-def) eaten); a flag divided in $3$ equal vertical bands, $1$ colored.

**Solution of Exercise 24.1.**

Pizza: $\frac56$ left. Chocolate: $\frac38$ eaten. Flag: $\frac13$ colored.

**Exercise 24.2 ★.**

Draw a bar cut into $5$ equal parts and shade $\frac35$. Then shade $\frac{2}{5}$ of another equal bar. Which shading is bigger?

**Solution of Exercise 24.2.**

$\frac35$ shades three parts, $\frac25$ only two: $\frac35$ is the bigger shading.

**Exercise 24.3 ★.**

Compute: $\frac12$ of $18$; $\frac14$ of $20$; $\frac13$ of $21$; $\frac34$ of $20$ (three groups of a [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half)).

**Solution of Exercise 24.3.**

[Half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of $18$: $9$. [Quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of $20$: $5$. Third of $21$: $7$. $\frac34$ of $20$: three [quarters](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half), $3 \times 5 = 15$.

**Exercise 24.4 ★.**

Place on a number line cut in thirds: $\frac13$; $\frac33$; $\frac53$; $2$. Which [fraction](#def-g4-fractions-def) lands exactly on $2$?

**Solution of Exercise 24.4.**

$\frac13$: one step; $\frac33$: on $1$; $\frac53$: two steps past $1$. The [fraction](#def-g4-fractions-def) landing exactly on $2$ is $\frac63$.

**Exercise 24.5 ★.**

Copy and complete with $<$, $>$ or $=$:

$$
\frac38 \;?\; \frac68, \qquad
\frac55 \;?\; 1, \qquad
\frac74 \;?\; 1, \qquad
\frac12 \;?\; \frac24 .
$$

**Solution of Exercise 24.5.**

$\frac38 < \frac68$; $\frac55 = 1$; $\frac74 > 1$; $\frac12 = \frac24$.

**Exercise 24.6 ★.**

Compute: $\frac26 + \frac36$; $\frac58 - \frac28$; $\frac14 + \frac24$; $1 - \frac13$ (how many thirds make $1$?).

**Solution of Exercise 24.6.**

$\frac26 + \frac36 = \frac56$; $\frac58 - \frac28 = \frac38$; $\frac14 + \frac24 = \frac34$; $1 = \frac33$, so $1 - \frac13 = \frac23$.

**Exercise 24.7 ★.**

Write as a whole number plus a [fraction](#def-g4-fractions-def) smaller than $1$: $\frac54$; $\frac73$; $\frac{9}{2}$.

**Solution of Exercise 24.7.**

$\frac54 = 1 + \frac14$; $\frac73 = 2 + \frac13$; $\frac92 = 4 + \frac12$.

**Exercise 24.8 ★.**

How many minutes are: [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) an hour; a [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of an hour; three [quarters](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of an hour; $\frac13$ of an hour?

**Solution of Exercise 24.8.**

[Half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) an hour: $30$ min. [Quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half): $15$ min. Three [quarters](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half): $45$ min. Third: $20$ min.

**Exercise 24.9 ★.**

In a class of $24$ students, $\frac14$ wear glasses. How many students wear glasses? How many do not?

**Solution of Exercise 24.9.**

$\frac14$ of $24$ is $6$ students with glasses; $24 - 6 = 18$ without.

**Exercise 24.10 ★★.**

Nina ate $\frac38$ of a pizza and Sam ate $\frac48$ of the same pizza. What [fraction](#def-g4-fractions-def) did they eat together? What [fraction](#def-g4-fractions-def) is left? Who ate more?

**Solution of Exercise 24.10.**

Together: $\frac38 + \frac48 = \frac78$. Left: $\frac18$. Sam ate more ($4$ eighths against $3$).

**Exercise 24.11 ★★.**

True or false, with a picture or an example: “$\frac12$ of a small pizza is less than $\frac14$ of a very big pizza is possible”. What must one always know before comparing two [fractions](#def-g4-fractions-def) of something?

**Solution of Exercise 24.11.**

Possible indeed: [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of a small pizza can be less food than a [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of a giant one. [Fractions](#def-g4-fractions-def) compare parts *of the same whole*; before comparing $\frac12$ and $\frac14$ “of something”, one must know that the two somethings are equal.