Lesson Objectives

Dear learner,

At the end of this lesson, you will be able to:

• Define the concept of rational numbers
• Classify different types of fractions (proper, improper, and mixed) based on the relationship between the numerator and the denominator.
• Convert repeating and terminating decimals into their corresponding fraction forms and simplify the results.
• Locate rational numbers on a number line and differentiate between their decimal forms.

Key Terms

• Rational number
• Proper fraction
• Improper fraction
• Repeating decimal
• Mixed number
• Terminating decimals

Brainstorming Question

1. A farmer in Addis Ababa sells 7.5 kilograms of teff for 150 birr. What is the price per kilogram of teff? Express the price as a fraction and a decimal.

2.A bus travels from Bahir Dar to Gondar, a distance of 180 km. If the bus has already traveled 2/3 of the total distance, how many kilometers has it traveled? Represent the distance traveled as a fraction and a decimal.

3.In a recipe, you need 3/4 of a liter of milk, but you only have a 250 mL measuring cup. How many cups of milk do you need to use? Express the answer as a fraction and as a decimal.

Solution:

1. Price per kilogram:

$Price\; per\; kg} = \frac{150 \text{ birr}}{7.5 \text{ kg}}$.

• To simplify:

$Price \;per\; kg = \frac{150}{7.5} = \frac{150 \times 10}{7.5 \times 10} = \frac{1500}{75} = 20$ birr per kg.

Decimal form: The price per kilogram is 20.0 birr.

2.Distance traveled:

Distance traveled $= \frac{2}{3} \times 180 \text{ km}$

• To calculate:

Distance traveled $= \frac{2 \times 180}{3} = \frac{360}{3} = 120 \text{ km}$

• Decimal form: The bus has traveled 120.0 km.

3.Convert $\frac{3}{4}$ liter to milliliters:

$\frac{3}{4} \text{ liter} = \frac{3}{4} \times 1000 \text{ mL} = 750 \text{ mL}$

Number of 250 mL cups needed:

$Cups \; needed = \frac{750 \text{ mL}}{250 \text{ mL/cup}} = 3 \text{ cups}$

• Decimal form: You need 3.0 cups of milk.

2.1. Definition of rational numbers

Definition

• Rational numbers are numbers denoted by Q.
• Rational numbers are numbers that can be expressed as a ratio of two integers, where the denominator is not zero.
• They can be written in the form of $Q = {\frac{a}{b}, \;where \;a,b\varepsilon Z \;and \;b\neq 0}$.
• Rational numbers include natural numbers, whole numbers, integers, fractions, and terminating or repeating decimals.

Note

Suppose $x=\frac{a}{b} \epsilon Q $, 𝑥 is a fraction with numerator 𝑎 and denominator 𝑏,
i) If 𝑎 < 𝑏, then 𝑥 is called proper fraction.
ii) If 𝑎 ≥ 𝑏, then 𝑥 is called improper fraction.
iii) If $y=c\frac{a}{b}$, where 𝑐 𝜖 ℤ and $\frac{a}{b}$ is a proper fraction, then 𝑦 is called a mixed fraction (mixed number).
iv) 𝑥 is said to be in simplest (lowest form) if 𝑎 and 𝑏 are relatively prime or $GCF(a, b) = 1$.

Example 1:

Categorize each of the following as proper, improper or mixed fraction $\frac{2}{3}$, $\frac{7}{6}$, $12$, $5\frac{2}{3}$.

Solution:

$\frac{2}{3}$ is proper fraction, $\frac{7}{6}$ is improper fraction, $12$ is improper fraction, and $5\frac{2}{3}$ is mixed fraction.

Example 2:

Express the improper fraction $\frac{15}{7}$ as mixed fraction?

Solution:

$\frac{15}{7} = \frac{(14+1)}{7}= \frac{14}{7}\ast \frac{1}{7} = \frac{2}{1}\ast \frac{1}{7} = 2\frac{1}{7}$.

Example 3:

Express the mixed fraction $5\frac{4}{7}$ as improper fraction?

Solution:

$5\frac{4}{7} = \frac{5}{1}\ast \frac{4}{7}$ to make denominator the same multiply

$\frac{5}{1}$ by $\frac{7}{7}$ we will get $\frac{35}{7}$.

Thus add $\frac{35}{7}$ and $\frac{4}{7}$ we will get $\frac{39}{7}$.

2.2 Representation of Rational Numbers by Decimals

Example 4:

Perform each of the following divisions. a) $\frac{4}{5}$ b) $\frac{15}{4}$ c) $-\frac{8}{3}$

Solution:

a) $\frac{4}{5}$ = 4 divided by 5 = 0.8

b) $\frac{15}{4}$ = $15 \div 4 = 3.75$

c) $-\frac{8}{3}$ = $-8 \div 3 = – 2.666…$

Example 5:

Write the numbers 0.5 and 2.125 as a fraction form.

Solution:

$0.5 = \frac{0.5}{1} = \frac{5}{10} = \frac{1}{2}$

$2.125 = \frac{2.125}{1} = \frac{2125}{1000} = \frac{425}{200} = \frac{85}{40} = \frac{17}{8}$

When we change a rational number $\frac{a}{b}$ into decimal form, one of the following cases will occur

i. Repeating decimals: are decimal numbers in which one or more digits repeat infinitely after the decimal point. If the denominator of a fraction can be expressed as a product of numbers other than 2 and 5, then its decimal form is a repeating decimal.

Example 6:

The decimal representation of 1/3 is 0.3333…, where the digit 3 repeats indefinitely.

To represent the repetition of a digit or digits, we use a bar notation above the repeating digit or digits.

Thus

$0.3333….. = 0.\overline{3}$, $0.23454545454….. = 0.23\overline{45}$

ii. Terminating decimals: on the other hand, are decimal numbers that have a finite number of digits after the decimal point. If the denominator of a fraction can be expressed only as product of numbers 2 and 5, then its decimal form is a terminating decimal. For instance, the decimal representation of

$\frac{1}{4}$ = $\frac{1}{2\ast 2}$ is 0.25,

where there are only two digits after the decimal point.

iii. Converting Terminating Decimals to Fractions

Every terminating decimal can be expressed as a fraction with a denominator that is a power of 10, such as 10, 100, 1000, and so on, depending on the number of decimal places.

Examples 7:

Convert each of the following decimals to fraction form? a) 7.5 b) – 0.25 c) 4.125

Solution:

a. $7.5 = \frac{7.5}{1} = \frac{75}{10}$ and after simplification, we obtain $\frac{15}{2}$

b. $ – 0.25 = -\frac{0.25}{1} = -\frac{25}{100}$ and after simplification, we obtain $-\frac{1}{4}$

c. $ 4.125 = \frac{4.125}{1} = \frac{4125}{1000} = \frac{825}{200} = \frac{165}{40} = \frac{33}{8}$

Therefore, after simplification, we obtain $\frac{33}{8}$.

iv. Representing rational numbers on the number line

Example 8:

Locate the rational numbers $-2, 3, \frac{3}{5}, – \frac{7}{6}$ on the number line.

Solution:

2.3. Conversion of Repeating Decimals into Fractions

Conversion of repeating decimals into fractions is the process of expressing a decimal number that has a repeating pattern of digits after the decimal point as a fraction in the form of a ratio of two integers. 

Example 9:

Represent each of the following decimals as a simplest fraction form (ratio of two integers).
a. $0.\overline{5}$

b. $-3.98\overline{6}$.

Solution:

a. $0.\overline{5}$, let $x = 0.\overline{5}$ (after a decimal point there is only one number), so, $10x = 5.\overline{5}$.

then subtract $x$ from $10x$, $10x – x = 5.\overline{5} – 0.\overline{5}$

$9x = 5$

$x = \frac{5}{9}$.

Therefore, $0.\overline{5} = \frac{5}{9}$.

b. $-3.98\overline{6}$ reject the negative sign and express $3.98\overline{6}$ in to fraction, finally add negative sign is good.

Thus, $3.98\overline{6}$.

Let

$x = 3.98\overline{6}$ (the number is written in 3 decimal places), so, $1000x = 3986.\overline{6}$

From 3 decimal places two digits is non – repeating, so

$100x = 398\overline{6}$. then subtract $100x$ from $1000x$, $1000x – 100x = 3986.\overline{6} – 398.\overline{6}$ $900x = 3588$

$x = \frac{3588}{900}.= \frac{1196}{300} = \frac{598}{150} = \frac{299}{75}$

Therefore, $-3.98\overline{6} = -\frac{299}{75}$.