Fractions: 5th grade math course with PDF lesson.

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1 I. Fractional writing
- 1.1 1. quotient
- 1.2 2. proportions
2 II. Equalities of quotients and fractions
3 III. Comparison of fractions

A math course on fractions in fifth grade is very interesting for the student. Thus, this math course on fractions involves the following concepts:

definition of a quotient and a fraction;
fractional writing;
rules for simplifying a fraction;
reduction to the same denominator;
sum of two fractions;
difference of two fractions;
product of two fractions.

The student will need to know the definition of a fraction and how to represent it on a graduated line. They will need to develop computational skills and know how to reduce two fractions to the same denominator in order to compare them. We will end this course on fractions with problem solving in the fifth grade.

I. Fractional writing

1. quotient

Definition:

Consider a and b as two relative numbers such that $b\neq\,0$ .

The quotient of a by b, noted $\frac{a}{b}$ is the number such that $\frac{a}{b}\times \,b=a$ .

The number a is the numerator of $\frac{a}{b}$ and the number b is the denominator of $\frac{a}{b}$ .

When a or b is a decimal number, $\frac{a}{b}$ is called fractional writing.
When a and b are whole numbers, $\frac{a}{b}$ is called a fraction.
When b is equal to 10,100,1,000, …, $\frac{a}{b}$ is called a decimal fraction.

Example:

The quotient of 3 by 4 is noted $\frac{3}{4}$ .

This number verifies the equality $\frac{3}{4}\times \,4=3$ .

Remark:

Any decimal number has an infinite number of fractional entries.

For example $3,05=\frac{3,05}{1}=\frac{30,5}{10\,}=\frac{305}{100}$

2. proportions

Example:

The disk below is divided into six equal parts.

Each part represents $\frac{1}{8}$ of the disk.

The proportion of the disc colored in blue is therefore $\frac{3}{8}$ .

The proportion of the disc not colored is $\frac{5}{8}$ .

$fraction of a fraction$

II. Equalities of quotients and fractions

1. equal quotients

Ownership:

Consider a, b and k three relative numbers such as $b\neq\,0$ and $k\neq\,0$ . If we multiply (or divide) the numerator and the denominator of a quotient by the same non-zero number k, then we do not change the value of the quotient.

$\frac{a}{b}=\frac{a\times \,k\,}{b\times \,k}=\frac{ak}{bk}$ and $\frac{a}{b}=\frac{a\,:\,k\,}{b\,\:\,k}$ .

Example:

$A=\frac{0,2}{1,2}=\frac{0,2\times \,5\,}{1,2\,\times \,5}=\frac{1}{6}$

$B=\frac{24}{18}=\frac{24:6}{18:6}=\frac{4}{3}$

2. simplifying fractions

Ownership:

Consider a and b two relative integers such that $b\neq\,0$ .

In order to simplify a fraction, we use the previous property by dividing the numerator and denominator by a common divisor.

When we can no longer simplify a fraction, it is said to be irreducible.

Example:

$0,75=\frac{3}{4}=\frac{1,5}{2}$

These are three writings of the same number.

The first is the decimal writing, the second is a fraction and the third is a fractional writing.

$\frac{25}{15}=\frac{5\times \,5}{5\times \,3}=\frac{5}{5}\times \,\frac{5}{3}=1\times \,\frac{5}{3}=\frac{5}{3}$

Remark:

To simplify a fraction, we use the criteria of divisibility.

3.division of two decimal numbers

Ownership:

The quotient of two decimal numbers can be written as a fraction.

Example:

$9: \,0,4=\frac{9}{0,4}=\frac{9\times \,5}{0,4\times \,5}=\frac{45}{2}$

4. decimal fraction

Ownership:

Consider a and b two relative integers such that $b\neq\,0$ .

We call decimal fraction, any fraction $\frac{a}{b}$ such that b is equal to 10; 100; 1 000; etc..

Example:

$\frac{5}{10};\frac{12}{100};\frac{1394}{10000}$ are decimal fractions.

Ownership:

Any decimal number n can be written as a decimal fraction.

Example:

$A=4,75=\frac{475}{100}$

5. proportion and percentage

Definition:

When a proportion is written as a quotient that has a denominator of 100, we get a percentage proportion.

Example:

a city of 50,000 inhabitants is crossed by a canal.

18,250 people live on the right bank of the canal.

The proportion of residents with their homes on the right bank is $\frac{18250}{50000}$ .

$\frac{18250}{50000}=\frac{18250:1000}{50000:1000}=\frac{18,25}{50}=\frac{18,25\times \,2}{50\times \,2}=\frac{36,5}{100}$ .

It is said that the percentage of residents with their homes on the right bank is 36.5%.

III. Comparison of fractions

Definition:

To compare two fractions, you must first reduce them to the same denominator and then compare the numerators.

Example:

Let’s arrange the following fractions in ascending order: $\frac{5}{7};\frac{9}{7};\frac{8}{7};\frac{4}{7}$ .

All these fractions have 7 as a common denominator, they will be ordered in ascending order of their denominator.

$\frac{4}{7}<\frac{5}{7}<\frac{8}{7}<\frac{9}{7}$ .

Remark:

To compare two fractions that do not have the same denominator, we reduce them to the same denominator and then apply the previous definition.

Example:

Compare $\frac{4}{3}$ and $\frac{5}{4}$ .

$\frac{4}{3}=\frac{4\times \,4}{3\times \,4}=\frac{16}{12}$

$\frac{5}{4}=\frac{5\times \,3}{4\times \,3}=\frac{15}{12}$

$\frac{4}{3}>\frac{5}{4}$

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Fractions: 5th grade math lessons to download in PDF.

I. Fractional writing

1. quotient

2. proportions

II. Equalities of quotients and fractions

1. equal quotients

2. simplifying fractions

3.division of two decimal numbers

4. decimal fraction

5. proportion and percentage

III. Comparison of fractions

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