---
title: "Tenths and Hundredths"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 25
exercises: 11
source: https://one-course.com/books/math/1/en/chapter/25-tenths-and-hundredths
---

# Chapter 25 — Tenths 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](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def)*, and a wonderful shorthand for them: the [decimal point](#def-g4-decimals-point). (Decimal numbers get their full chapter in [Chapter 29](https://one-course.com/books/math/1/en/chapter/29-decimal-numbers#ch-g5-decimals).)

## 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.](https://one-course.com/images/onecourse/chapters/math-1/g4-decimals/fig-2508e95698f8.svg)

*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.](https://one-course.com/images/onecourse/chapters/math-1/g4-decimals/fig-f01fe489abe6.svg)

*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](#def-g4-decimals-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)).**

1. Compare the whole parts first: $3.2 > 2.9$ .
2. Same whole parts: compare the tenths, then the hundredths: $2.37 < 2.5$ because $3$ tenths $< 5$ tenths.
3. 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](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def).

**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](#def-g4-decimals-point): $\frac{9}{10}$; $\frac{27}{100}$; $\frac{3}{100}$; $5 + \frac{6}{10}$; $\frac{60}{100}$.

**Solution of Exercise 25.2.**

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

**Exercise 25.3 ★.**

Write as a [fraction](https://one-course.com/books/math/1/en/chapter/24-first-fractions#def-g4-fractions-def) (tenths or hundredths): $0.3$; $0.51$; $0.07$; $1.9$.

**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](#def-g4-decimals-point).

**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 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 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 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 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 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 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 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).