Primary & Middle School Mathematics · Grades 1–9

25Tenths and Hundredths

Between $3$ and $4$ the whole numbers are silent — but a runner can be $3$ meters and a bit ahead. Cutting the unit into ten, then a hundred, gives the decimal fractions, and a wonderful shorthand for them: the decimal point. (Decimal numbers get their full chapter in Chapter 29.)

25.1 Decimal fractions

Definition 25.1 (Tenths and hundredths)

Cutting a unit into $10$ equal parts gives tenths, $\frac{1}{10}$ ; cutting each tenth into $10$ again gives hundredths, $\frac{1}{100}$ . So

1 = \frac{10}{10} = \frac{100}{100}, \qquad \frac{1}{10} = \frac{10}{100}.

The hundred-square: ten columns of ten cells. One column is a tenth, one cell a hundredth.

Example 25.2

Shading $3$ columns and $7$ extra cells of the hundred-square shades

\frac{3}{10} + \frac{7}{100} = \frac{30}{100} + \frac{7}{100} = \frac{37}{100}

of the square — thirty-seven hundredths.

25.2 The decimal point

Definition 25.3 (Decimal writing)

The number $2 + \frac{3}{10} + \frac{7}{100}$ is written

2.37

— the decimal point separates the whole part ( $2$ ) from the decimal part; the first digit after the point counts the tenths, the second the hundredths. Reading: “two point three seven”, or “two and thirty-seven hundredths”.

Example 25.4

\frac{7}{10} = 0.7, \qquad \frac{43}{100} = 0.43, \qquad \frac{5}{100} = 0.05, \qquad 3 + \frac{4}{10} = 3.4 .

Careful with $\frac{5}{100}$ : five hundredths need a $0$ in the tenths place — $0.05$ , not $0.5$ .

Zooming between 2 and 2.8: big marks every tenth, small marks every hundredth. The number 2.37 sits between 2.3 and 2.4, at the 7th small mark. — Zooming between $2$ and $2.8$ : big marks every tenth, small marks every hundredth. The number $2.37$ sits between $2.3$ and $2.4$ , at the $7$ th small mark.

Example 25.5 (Money is hundredths)

One cent is one hundredth of one euro (or dollar, or pound): a price of $4.85$ means $4$ whole coins and $85$ hundredths. Money is the decimal point you already use every day: $10$ cents $= 0.10$ ; two euros fifty $= 2.50$ ; and $0.05$ is a five-cent coin, not fifty!

25.3 Comparing with tenths

Method 25.6 (Comparing decimals (first contact))

Compare the whole parts first: $3.2 > 2.9$ .
Same whole parts: compare the tenths, then the hundredths: $2.37 < 2.5$ because $3$ tenths $< 5$ tenths.
Writing both with two decimals helps: $2.5 = 2.50$ , and $37 < 50$ hundredths.

The trap: $2.37$ has more digits than $2.5$ , but is smaller — more digits does not mean bigger!

Example 25.7

Order $0.4$ , $0.35$ and $0.09$ : with hundredths, $0.40$ , $0.35$ , $0.09$ ; so

0.09 < 0.35 < 0.4 .

25.4 Exercises

Exercise 25.1 ★

On a hundred-square, how much is shaded if one shades: $5$ columns; $2$ columns and $3$ cells; $45$ cells? Answer with fractions.

Solution

Solution of Exercise 25.1.

$5$ columns: $\frac{5}{10}$ (i.e. $\frac{50}{100}$ ). $2$ columns and $3$ cells: $\frac{23}{100}$ . $45$ cells: $\frac{45}{100}$ .

Exercise 25.2 ★

Write with a decimal point: $\frac{9}{10}$ ; $\frac{27}{100}$ ; $\frac{3}{100}$ ; $5 + \frac{6}{10}$ ; $\frac{60}{100}$ .

Solution

Solution of Exercise 25.2.

$0.9$ ; $0.27$ ; $0.03$ ; $5.6$ ; $0.60 = 0.6$ .

Exercise 25.3 ★

Write as a fraction (tenths or hundredths): $0.3$ ; $0.51$ ; $0.07$ ; $1.9$ .

Solution

Solution of Exercise 25.3.

$0.3 = \frac{3}{10}$ ; $0.51 = \frac{51}{100}$ ; $0.07 = \frac{7}{100}$ ; $1.9 = 1 + \frac{9}{10} = \frac{19}{10}$ .

Exercise 25.4 ★

In $6.58$ : what does the digit $5$ count? The digit $8$ ? Decompose the number as in Definition 25.3.

Solution

Solution of Exercise 25.4.

The $5$ counts the tenths, the $8$ the hundredths:

6.58 = 6 + \frac{5}{10} + \frac{8}{100}.

Exercise 25.5 ★

Place on a number line graduated in tenths from $4$ to $5$ : $4.2$ ; $4.75$ (between which two marks?); $4.9$ .

Solution

Solution of Exercise 25.5.

$4.2$ : on the second mark after $4$ . $4.75$ : between the marks $4.7$ and $4.8$ , halfway. $4.9$ : one mark before $5$ .

Exercise 25.6 ★

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

0.6 \;?\; 0.60, \qquad 2.8 \;?\; 2.75, \qquad 0.09 \;?\; 0.1, \qquad 5.3 \;?\; 5.13 .

Solution

Solution of Exercise 25.6.

$0.6 = 0.60$ ; $2.8 > 2.75$ (compare $2.80$ ); $0.09 < 0.1$ ( $9$ hundredths against $10$ ); $5.3 > 5.13$ .

Exercise 25.7 ★

Order from smallest to biggest: $1.05$ ; $1.5$ ; $0.95$ ; $1.45$ .

Solution

Solution of Exercise 25.7.

$0.95 < 1.05 < 1.45 < 1.5$ .

Exercise 25.8 ★

Write in decimal writing: three euros and five cents; twelve euros and fifty cents; ninety-nine cents. Then order the three prices.

Solution

Solution of Exercise 25.8.

$3.05$ ; $12.50$ ; $0.99$ . Order: $0.99 < 3.05 < 12.50$ .

Exercise 25.9 ★

Which decimal numbers with one decimal digit lie strictly between $3$ and $4$ ? List them all. How many are there?

Solution

Solution of Exercise 25.9.

$3.1$ , $3.2$ , $3.3$ , $3.4$ , $3.5$ , $3.6$ , $3.7$ , $3.8$ , $3.9$ — nine numbers.

Exercise 25.10 ★★

Tom says: “ $0.25$ is bigger than $0.5$ because $25$ is bigger than $5$ ”. Show him his mistake with the hundred-square (how many cells does each shade?).

Solution

Solution of Exercise 25.10.

On the hundred-square, $0.25$ shades $25$ cells but $0.5 = 0.50$ shades $50$ cells: $0.5$ is the bigger one. Tom compared $25$ and $5$ as whole numbers; he should compare $25$ and $50$ hundredths.

Exercise 25.11 ★★

A runner’s three jumps measured $3.6$ m, $3.58$ m and $3.65$ m. Order the jumps from shortest to longest. Which two differ by exactly $\frac{5}{100}$ of a meter?

Solution

Solution of Exercise 25.11.

Order: $3.58 < 3.6 < 3.65$ . The jumps $3.6$ and $3.65$ differ by $0.05 = \frac{5}{100}$ of a meter (five centimeters).