---
title: "Fractions as Numbers"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 31
exercises: 11
source: https://one-course.com/books/math/1/en/chapter/31-fractions-as-numbers
---

# Chapter 31 — Fractions as Numbers

A [fraction](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) is not just a picture of a pie: it is a genuine *number*, with a place on the number line — sometimes the very same place as a decimal number. This chapter connects the two worlds. (The full rules of [fraction](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) arithmetic come in [Chapter 39](https://one-course.com/books/math/1/en/chapter/39-fractions-first-steps#ch-g6-fractions) and beyond.)

## 31.1 Fractions on the line, again

**Example 31.1 (Placing and reading).**

On a line cut in fifths, $\frac{7}{5}$ is $7$ steps from $0$: past $1$ (which is $\frac55$), at $1 + \frac25$. Every [fraction](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) lives somewhere on the line; bigger [numerator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) (same [denominator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def)) means further right.

**Proposition 31.2 (Same point, many names).**

Cutting every part in two (or three …) does not move the point:

$$
\frac{1}{2} = \frac{2}{4} = \frac{3}{6} = \frac{5}{10},
\qquad
\frac{2}{3} = \frac{4}{6} = \frac{20}{30}.
$$

Multiplying (or dividing) the [numerator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) and the [denominator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) by the same number gives another name of the same [fraction](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def).

**Proof.** *Admitted at this level.* ∎

![Three bars, three names, one amount: 1/2 = 2/4 = 5/10.](https://one-course.com/images/onecourse/chapters/math-1/g5-fractions/fig-4f3e64597041.svg)

*Three bars, three names, one amount: $\frac12 = \frac24 =
\frac{5}{10}$.*

## 31.2 Fractions and decimals

**Proposition 31.3 (Fractions with decimal names).**

[Fractions](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) of tenths, hundredths, thousandths *are* decimal numbers:

$$
\frac{3}{10} = 0.3, \qquad \frac{27}{100} = 0.27 .
$$

Some other [fractions](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) also have decimal names, found by switching to tenths or hundredths:

$$
\frac{1}{2} = \frac{5}{10} = 0.5, \qquad
\frac{1}{4} = \frac{25}{100} = 0.25, \qquad
\frac{3}{4} = 0.75, \qquad
\frac{1}{5} = \frac{2}{10} = 0.2 .
$$

But not all: $\frac13 = 0.333\dots$ never ends — the [fraction](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) is its only exact name.

**Proof.** *Admitted at this level.* ∎

![One line, two languages: each point can carry a fraction name and a decimal name.](https://one-course.com/images/onecourse/chapters/math-1/g5-fractions/fig-b13d61484ddb.svg)

*One line, two languages: each point can carry a [fraction](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) name and a decimal name.*

**Example 31.4 (Choosing the handier name).**

To pay three [quarters](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of $12$: the [fraction](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) name is handier, $\frac34$ of $12$ is $9$. To compare $\frac34$ and $0.8$: the decimal name is handier, $0.75 < 0.8$. Being bilingual pays off.

## 31.3 Comparing fractions

**Method 31.5 (Three comparison tools).**

1. *Same [denominator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def)* : more parts wins, $\frac57 > \frac37$ ;
2. *same [numerator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def)* : bigger parts win, so *smaller* [denominator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) wins: $\frac35 > \frac38$ (fifths are bigger than eighths);
3. *landmarks* : compare each to $\frac12$ or to $1$ : $\frac38 < \frac12 < \frac59$ , so $\frac38 < \frac59$ .

**Example 31.6.**

Who ate more pizza: Ali with $\frac58$ or Bea with $\frac54$? Bea’s [fraction](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) is bigger than $1$ (five [quarters](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) is more than one whole pizza), Ali’s is smaller than $1$: Bea ate more — more than a whole pizza, in fact.

## 31.4 Exercises

**Exercise 31.1 ★.**

Draw a number line cut in fifths from $0$ to $2$, and place: $\frac25$; $\frac55$; $\frac85$; $\frac{10}{5}$.

**Solution of Exercise 31.1.**

$\frac25$: two steps from $0$; $\frac55$: on $1$; $\frac85$: three steps past $1$; $\frac{10}{5}$: on $2$.

**Exercise 31.2 ★.**

Complete the equal names: $\frac12 = \frac{?}{8}$; $\frac34 = \frac{?}{8}$; $\frac{2}{5} = \frac{4}{?}$; $\frac{6}{9} = \frac{2}{?}$.

**Solution of Exercise 31.2.**

$\frac12 = \frac48$; $\frac34 = \frac68$; $\frac25 = \frac{4}{10}$; $\frac69 = \frac23$.

**Exercise 31.3 ★.**

Which [fractions](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) name whole numbers? Give the whole number when so: $\frac82$; $\frac{9}{4}$; $\frac{15}{3}$; $\frac{20}{5}$.

**Solution of Exercise 31.3.**

$\frac82 = 4$; $\frac94$ is not whole ($9$ is not a multiple of $4$); $\frac{15}{3} = 5$; $\frac{20}{5} = 4$.

**Exercise 31.4 ★.**

Write as decimals: $\frac{7}{10}$; $\frac{41}{100}$; $\frac12$; $\frac34$; $\frac15$.

**Solution of Exercise 31.4.**

$0.7$; $0.41$; $0.5$; $0.75$; $0.2$.

**Exercise 31.5 ★.**

Write as [fractions](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) (tenths or hundredths): $0.9$; $0.13$; $2.5$; $0.75$ (then give its [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) name).

**Solution of Exercise 31.5.**

$0.9 = \frac{9}{10}$; $0.13 = \frac{13}{100}$; $2.5 = \frac{25}{10}$; $0.75 = \frac{75}{100} = \frac34$.

**Exercise 31.6 ★.**

Compare, naming the tool used ([Method 31.5](#met-g5-fractions-compare)):

$$
\frac47 \;?\; \frac67, \qquad
\frac35 \;?\; \frac37, \qquad
\frac25 \;?\; \frac{7}{12}, \qquad
\frac98 \;?\; \frac{11}{12} .
$$

**Solution of Exercise 31.6.**

$\frac47 < \frac67$ (same [denominator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def)). $\frac35 > \frac37$ (same [numerator](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def): fifths are bigger). $\frac25 < \frac12 < \frac{7}{12}$, so $\frac25 < \frac{7}{12}$ (landmark $\frac12$). $\frac98 > 1 > \frac{11}{12}$ (landmark $1$).

**Exercise 31.7 ★.**

Order with the decimal names: $\frac12$; $0.4$; $\frac34$; $0.6$; $\frac15$.

**Solution of Exercise 31.7.**

Decimals: $0.5$; $0.4$; $0.75$; $0.6$; $0.2$. Order:

$$
\frac15 < 0.4 < \frac12 < 0.6 < \frac34 .
$$

**Exercise 31.8 ★.**

A [quarter](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of the students of a class of $28$ play an instrument, and [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) play a sport. How many students is each? Which group is bigger?

**Solution of Exercise 31.8.**

Instrument: $\frac14$ of $28 = 7$. Sport: $\frac12$ of $28 = 14$. The sport group is bigger.

**Exercise 31.9 ★.**

Match each [fraction](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) with a time: $\frac14$ h, $\frac12$ h, $\frac34$ h, $\frac{1}{60}$ h — to $30$ min, $45$ min, $1$ min, $15$ min.

**Solution of Exercise 31.9.**

$\frac14$ h $= 15$ min; $\frac12$ h $= 30$ min; $\frac34$ h $= 45$ min; $\frac{1}{60}$ h $= 1$ min.

**Exercise 31.10 ★★.**

Tom drank $\frac35$ of his $50$ cL bottle; Ana drank $\frac12$ of her $60$ cL bottle. Who drank more? (Compute both amounts in cL — comparing the bare [fractions](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) is not enough here, and say why.)

**Solution of Exercise 31.10.**

Tom: $\frac35$ of $50 = 30$ cL. Ana: $\frac12$ of $60 = 30$ cL. They drank the same amount! The bare [fractions](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) ($\frac35 > \frac12$) compare shares of *different* bottles — only the actual amounts can be compared.

**Exercise 31.11 ★★.**

Give a [fraction](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) strictly between $\frac12$ and $1$; then one strictly between $\frac12$ and $\frac34$. (Decimal names may help.)

**Solution of Exercise 31.11.**

Between $\frac12$ and $1$: for instance $\frac34$ (i.e. $0.75$). Between $\frac12 = 0.5$ and $\frac34 = 0.75$: for instance $0.6 =
\frac35$.