Primary & Middle School Mathematics · Grades 1–9

31Fractions as Numbers

A fraction 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 arithmetic come in Chapter 39 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 lives somewhere on the line; bigger numerator (same denominator) 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 and the denominator by the same number gives another name of the same fraction.

Proof. Admitted at this level. ∎

Three bars, three names, one amount: 1/2 = 2/4 = 5/10. — Three bars, three names, one amount: $\frac12 = \frac24 = \frac{5}{10}$ .

31.2 Fractions and decimals

Proposition 31.3 (Fractions with decimal names)

Fractions of tenths, hundredths, thousandths are decimal numbers:

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

Some other fractions 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 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.

Example 31.4 (Choosing the handier name)

To pay three quarters of $12$ : the fraction 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)

Same denominator: more parts wins, $\frac57 > \frac37$ ;
same numerator: bigger parts win, so smaller denominator wins: $\frac35 > \frac38$ (fifths are bigger than eighths);
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 is bigger than $1$ (five quarters 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

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

Solution of Exercise 31.2.

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

Exercise 31.3 ★

Which fractions name whole numbers? Give the whole number when so: $\frac82$ ; $\frac{9}{4}$ ; $\frac{15}{3}$ ; $\frac{20}{5}$ .

Solution

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

Solution of Exercise 31.4.

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

Exercise 31.5 ★

Write as fractions (tenths or hundredths): $0.9$ ; $0.13$ ; $2.5$ ; $0.75$ (then give its quarter name).

Solution

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):

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

Solution

Solution of Exercise 31.6.

$\frac47 < \frac67$ (same denominator). $\frac35 > \frac37$ (same numerator: 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

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 of the students of a class of $28$ play an instrument, and half play a sport. How many students is each? Which group is bigger?

Solution

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 with a time: $\frac14$ h, $\frac12$ h, $\frac34$ h, $\frac{1}{60}$ h — to $30$ min, $45$ min, $1$ min, $15$ min.

Solution

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 is not enough here, and say why.)

Solution

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 ( $\frac35 > \frac12$ ) compare shares of different bottles — only the actual amounts can be compared.

Exercise 31.11 ★★

Give a fraction strictly between $\frac12$ and $1$ ; then one strictly between $\frac12$ and $\frac34$ . (Decimal names may help.)

Solution

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 $0.6 = \frac35$ .