MATH 9
unit 3.1: what is a rational number?
what do we need to know?
Comparing fractions
Equivalent fractions
How to convert fractions to decimals and viceversa
Equivalent fractions
How to convert fractions to decimals and viceversa
More information
In this unit, we will be dealing with rational numbers. Here is a short introduction on the concept of rational numbers, and a short summary on what we will be learning in this unit.
What is the difference between rational and irrational numbers?
Take a look at this explanation of negative rational numbers:
which type of numbers are rational numbers?
Rational numbers are numbers (positive and negative) that can be made into a fraction. They are:
 INTEGERS: All integers have a DENOMINATOR of 1, and therefore considered fractions.
 DECIMALS: As you already know, decimals can be made into fractions.
 MIXED NUMBERS: These numbers are easily converted unto improper fractions.
 SQUARE ROOT OF PERFECT SQUARE numbers.
 INTEGERS: All integers have a DENOMINATOR of 1, and therefore considered fractions.
 DECIMALS: As you already know, decimals can be made into fractions.
 MIXED NUMBERS: These numbers are easily converted unto improper fractions.
 SQUARE ROOT OF PERFECT SQUARE numbers.
the irrational numbers
comparing rational numbers
When comparing rational numbers:
 Make sure all numbers are of the same type. That is, convert them all to fractions OR to decimals.
 Remember that POSITIVE NUMBERS ARE ALWAYS GREATER THAN NEGATIVE NUMBERS.
 The closer a negative number is to zero, the greater it is.
 FRACTIONS WITH SAME DENOMINATOR: The greater the numerator, the greater the value.
 FRACTIONS WITH SAME NUMERATOR: The greater the denominator, the greater the value.
 When comparing decimals, make sure the decimal numbers have the same amount of decimal places (you can always add a 0), and compare them from right to left.
 Make sure all numbers are of the same type. That is, convert them all to fractions OR to decimals.
 Remember that POSITIVE NUMBERS ARE ALWAYS GREATER THAN NEGATIVE NUMBERS.
 The closer a negative number is to zero, the greater it is.
 FRACTIONS WITH SAME DENOMINATOR: The greater the numerator, the greater the value.
 FRACTIONS WITH SAME NUMERATOR: The greater the denominator, the greater the value.
 When comparing decimals, make sure the decimal numbers have the same amount of decimal places (you can always add a 0), and compare them from right to left.
finding numbers between two rational numbers
If m and n be two rational numbers such that m < n then 1/2 (m + n) is a rational number between m and n. This is called the AVERAGE.
Let's do some examples!
1. Find out a rational number lying halfway between 2/7 and 3/4.
 Solution: Find the average: (2/7 + 3/4)/2 = 29/56
 Hence, 29/56 is a rational number lying halfway between 2/7 and 3/4.
Let's prove it: 2/7 = 0.285
3/4 = 0.75
29/56 = 0.517
0.75 > 0.517 > 0.285
2. Find out a rational number lying between 1/3 and 1/2.
 Solution: Find the average: (1/3 + 1/2)/2 = 1/12
 Hence, 1/12 is a rational number lying between 1/3 and 1/2.
3. Find out ten rational numbers lying between 3/11 and 8/11.
 Solution: Notice that the denominator is the same, and because
3 < 2 < 1 < 0 < 1 < 2 < 3 < 4 < 5 < 6 < 7 < 8, then,
2/11, 1/11, 0/11, 1/11, 2/11, 3/11, 4/11, 5/11, 6/11 and 7/11 are the ten rational numbers lying between 3/11 and 8/11.
4. Find out three rational numbers lying between 3 and 4.
 Solution: Find the average: 1/2 (3 + 4) = 7/2. Then, 3 < 7/2 < 4
 Now, find the average of 3 and 7/2: (3 + 7/2)/2 = 13/4
 Do the same to find a rational number between 7/2 and 4: (7/2 + 4)/2 = 15/4
3 < 13/4 < 7/2 < 15/4 < 4
 Hence, 13/4, 7/2 and 15/4 are the three rational numbers lying between 3 and 4.
Let's do some examples!
1. Find out a rational number lying halfway between 2/7 and 3/4.
 Solution: Find the average: (2/7 + 3/4)/2 = 29/56
 Hence, 29/56 is a rational number lying halfway between 2/7 and 3/4.
Let's prove it: 2/7 = 0.285
3/4 = 0.75
29/56 = 0.517
0.75 > 0.517 > 0.285
2. Find out a rational number lying between 1/3 and 1/2.
 Solution: Find the average: (1/3 + 1/2)/2 = 1/12
 Hence, 1/12 is a rational number lying between 1/3 and 1/2.
3. Find out ten rational numbers lying between 3/11 and 8/11.
 Solution: Notice that the denominator is the same, and because
3 < 2 < 1 < 0 < 1 < 2 < 3 < 4 < 5 < 6 < 7 < 8, then,
2/11, 1/11, 0/11, 1/11, 2/11, 3/11, 4/11, 5/11, 6/11 and 7/11 are the ten rational numbers lying between 3/11 and 8/11.
4. Find out three rational numbers lying between 3 and 4.
 Solution: Find the average: 1/2 (3 + 4) = 7/2. Then, 3 < 7/2 < 4
 Now, find the average of 3 and 7/2: (3 + 7/2)/2 = 13/4
 Do the same to find a rational number between 7/2 and 4: (7/2 + 4)/2 = 15/4
3 < 13/4 < 7/2 < 15/4 < 4
 Hence, 13/4, 7/2 and 15/4 are the three rational numbers lying between 3 and 4.
HERE is THE ANSWER TO THE WORKSHEET GIVEN IN CLASS
9_unit_3.1_ws_1_solutions.pdf  
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review  workbook
workbook__unit_3.1.pdf  
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