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Number (8Nf)

Mathematics Stage 8 Chapter Notes

What this chapter covers

Number - Fractions, decimals, percentages, ratio and proportion
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Number (8Nf) notes

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In this chapter

  1. 1The Hierarchy of Numbers
  2. 2Defining Rational Numbers
  3. 3Adding and Subtracting Fractions
  4. 4Multiplying and Dividing Fractions
  5. 5Using Fractions in Functions

1. The Hierarchy of Numbers

In mathematics, we group numbers into different sets based on their properties. The three fundamental sets you need to know are Natural Numbers, Integers, and Rational Numbers. Think of them like nesting dolls: the smallest set is inside the next, which is inside the largest.

  1. Natural Numbers (N): These are the positive whole numbers you use for counting. They start at 1 and go on forever. Examples: 1, 2, 3, 100, 543.
  1. Integers (Z): This set includes all the natural numbers, zero, and the negative versions of the natural numbers. Integers are whole numbers. Examples: -10, -3, 0, 1, 45.
  1. Rational Numbers (Q): This is a much larger set. It includes all integers and any number that can be written as a fraction, where the top and bottom are both integers (and the bottom is not zero). This means fractions, terminating decimals (like 0.5), and recurring decimals (like 0.666...) are all rational numbers. Examples: -4.5, -2, 0, 5, 1/2, 8.12.

Key term

Integer: A whole number (not a fraction or decimal) that can be positive, negative, or zero.

Examiner insight

Examiners look for a clear understanding that these number sets are nested. A good answer will show that a number like '5' is simultaneously a natural number, an integer, and a rational number.

Common pitfall

Students often forget that integers are also part of the rational number set. Any integer 'n' can be written as the fraction n/1.

Fun fact

The symbol 'Z' for integers comes from the German word 'Zahlen', which means 'numbers'. 'Q' for rational numbers comes from the Italian 'quoziente', meaning 'quotient'.

Worked example 13 marks

Consider the following numbers: -7, 1/4, 5, 0, 2.8, -1/2, 100. List which of these numbers are:a) Natural Numbers,b) Integers,c) Rational Numbers.

  1. 1

    a) Natural Numbers are positive whole numbers used for counting. From the list, the only natural numbers are 5 and 100.

  2. 2

    b) Integers are positive or negative whole numbers, including zero. From the list, the integers are -7, 5, 0, and 100.

  3. 3

    c) Rational Numbers include all integers and any number that can be written as a fraction. All the numbers in the list are rational numbers. -7 can be written as -7/1. 1/4 is a fraction. 5 is 5/1. 0 is 0/1. 2.8 is 28/10. -1/2 is a fraction. 100 is 100/1. So, all are rational: -7, 1/4, 5, 0, 2.8, -1/2, 100.

Recap

  • Natural numbers are the positive counting numbers (1, 2, 3...).
  • Integers are all whole numbers, including positive, negative, and zero (...-2, -1, 0, 1, 2...).
  • Rational numbers are any number that can be written as a fraction, including all integers and many decimals.
  • All natural numbers are also integers.
  • All integers are also rational numbers.

Quick check

  1. Is -10 an integer? Is it a natural number?1 mark
  2. Explain why 4.5 is a rational number.1 mark

2. Defining Rational Numbers

A rational number is any number that can be expressed in the form of a fraction a/b, where 'a' and 'b' are both integers and 'b' is not zero. This is the formal definition. If a number can be written as one integer divided by another (non-zero) integer, it is rational. This includes several types of numbers you're already familiar with:

  • Integers: e.g., 8 can be written as 8/1.
  • Terminating Decimals: e.g., 0.75 can be written as 75/100, which simplifies to 3/4.
  • Recurring Decimals: e.g., 0.333... can be written as 1/3.

Numbers that cannot be written in this form, like pi (π) or the square root of 2, are called irrational numbers.

Rational Number = a/b, where a and b are integers and b ≠ 0.

Key term

Rational Number: Any number that can be expressed as a fraction (or quotient) of two integers, where the denominator is not zero.

Common pitfall

Confusing terminating and recurring decimals (which are rational) with non-terminating, non-recurring decimals (which are irrational).

Worked example 12 marks

Write the terminating decimal 1.35 as a fraction in its simplest form.

  1. 1

    Step 1: Write the decimal as a fraction with a denominator of 10, 100, 1000, etc. Since 1.35 has two decimal places, we use 100. So, 1.35 = 135/100.

  2. 2

    Step 2: Simplify the fraction by finding the highest common factor (HCF) of the numerator and denominator. The HCF of 135 and 100 is 5.

  3. 3

    Step 3: Divide both the numerator and the denominator by the HCF. 135 ÷ 5 = 27. 100 ÷ 5 = 20.

  4. 4

    The simplified fraction is 27/20. This can also be written as a mixed number: 1 7/20.

Worked example 23 marks

Show that the recurring decimal 0.777... is a rational number by converting it to a fraction.

  1. 1

    Step 1: Let x be equal to the recurring decimal. So, x = 0.777...

  2. 2

    Step 2: Multiply x by 10 so that the decimal part aligns. Since one digit is recurring, we multiply by 10. 10x = 7.777...

  3. 3

    Step 3: Subtract the first equation from the second. 10x = 7.777...

    • x = 0.777...

    ---------------- 9x = 7

  4. 4

    Step 4: Solve for x. If 9x = 7, then x = 7/9.

  5. 5

    Since 0.777... can be written as the fraction 7/9, it is a rational number.

Recap

  • A rational number can be written as a fraction a/b where b is not zero.
  • Integers are rational because they can be written with a denominator of 1.
  • Terminating and recurring decimals are always rational.
  • To convert a decimal to a fraction, use a denominator of 10, 100, or 1000.
  • Numbers like pi (π) that cannot be written as a fraction are irrational.

Quick check

  1. Write -12 as a fraction.1 mark
  2. Is 0.12345... (with no repeating pattern) a rational number? Explain.1 mark

3. Adding and Subtracting Fractions

To add or subtract fractions, they must have the same denominator. This is called a common denominator. The most important rule is: whatever you do to the denominator, you must also do to the numerator to keep the fraction's value the same.

Method:

  1. Find the Lowest Common Multiple (LCM) of the denominators. This will be your common denominator.
  2. Convert each fraction into an equivalent fraction with the new common denominator.
  3. Add or subtract the numerators only. The denominator stays the same.
  4. Simplify the resulting fraction if possible.

When dealing with mixed numbers (like 3 1/4), it is often easiest to convert them into improper ('top-heavy') fractions first.

a/b + c/d = (ad + bc)/bd

a/b - c/d = (ad - bc)/bd

Key term

Common Denominator: A shared multiple of the denominators of two or more fractions, which is necessary for addition and subtraction.

Common pitfall

A very common mistake is to add or subtract the denominators as well as the numerators (e.g., writing 1/2 + 1/3 = 2/5). This is incorrect. Only the numerators are added or subtracted once the denominators are the same.

Worked example 13 marks

Calculate 3 1/4 - 1 2/3. Give your answer as a mixed number.

  1. 1

    Step 1: Convert the mixed numbers to improper fractions. 3 1/4 = (3*4 + 1)/4 = 13/4. 1 2/3 = (1*3 + 2)/3 = 5/3.

  2. 2

    Step 2: Find a common denominator for 4 and 3. The lowest common multiple (LCM) is 12.

  3. 3

    Step 3: Convert each fraction to an equivalent fraction with a denominator of 12. For 13/4, we multiply the denominator by 3, so we must multiply the numerator by 3: 13*3 = 39. So, 13/4 = 39/12. For 5/3, we multiply the denominator by 4, so we must multiply the numerator by 4: 5*4 = 20. So, 5/3 = 20/12.

  4. 4

    Step 4: Perform the subtraction with the new fractions. 39/12 - 20/12 = 19/12.

  5. 5

    Step 5: Convert the improper fraction back to a mixed number. 19 ÷ 12 is 1 with a remainder of 7. So, the answer is 1 7/12.

Recap

  • You must find a common denominator before adding or subtracting fractions.
  • The best common denominator is the Lowest Common Multiple (LCM) of the original denominators.
  • Convert mixed numbers to improper fractions to make calculations easier.
  • Only add or subtract the numerators; the denominator stays the same.
  • Always simplify your final answer and convert back to a mixed number if required.

Quick check

  1. What is the lowest common denominator for the fractions 1/6 and 3/8?1 mark
  2. Calculate 2/5 + 1/5.1 mark

4. Multiplying and Dividing Fractions

Multiplying and dividing fractions have different rules from addition and subtraction. Luckily, they are often more straightforward!

Multiplying Fractions: Simply multiply the numerators together and multiply the denominators together. Simplify the result if possible. (Numerator 1 × Numerator 2) / (Denominator 1 × Denominator 2)

Dividing Fractions: To divide by a fraction, you multiply by its reciprocal. The reciprocal is the fraction 'flipped' upside down. A popular way to remember the process is KCF: Keep the first fraction, Change the division sign to multiplication, and Flip the second fraction.

As with addition, if you have mixed numbers, convert them to improper fractions first.

a/b × c/d = ac/bd

a/b ÷ c/d = a/b × d/c = ad/bc

Key term

Reciprocal: The reciprocal of a fraction is found by inverting it; for example, the reciprocal of 3/4 is 4/3.

Examiner insight

Examiners award marks for correctly identifying and using the reciprocal in a division problem. Writing down the 'flipped' fraction is a key step that scores points.

Worked example 12 marks

Calculate 6 ÷ 3/4.

  1. 1

    Step 1: Write the integer as a fraction. 6 = 6/1.

  2. 2

    Step 2: The problem is now 6/1 ÷ 3/4. Use the KCF method (Keep, Change, Flip).

  3. 3

    Step 3: Keep 6/1. Change ÷ to ×. Flip 3/4 to its reciprocal, 4/3.

  4. 4

    Step 4: The calculation is now 6/1 × 4/3.

  5. 5

    Step 5: Multiply the numerators and denominators: (6 × 4) / (1 × 3) = 24/3.

  6. 6

    Step 6: Simplify the result. 24 ÷ 3 = 8.

Worked example 23 marks

Calculate 2 1/2 × 4/5.

  1. 1

    Step 1: Convert the mixed number 2 1/2 to an improper fraction: (2*2 + 1)/2 = 5/2.

  2. 2

    Step 2: The calculation is now 5/2 × 4/5.

  3. 3

    Step 3: Multiply the numerators and denominators: (5 × 4) / (2 × 5) = 20/10.

  4. 4

    Step 4: Simplify the result. 20 ÷ 10 = 2.

  5. 5

    Alternative Step 3: You can 'cross-cancel' common factors before multiplying. The 5 on top and 5 on bottom cancel to 1. The 4 on top and 2 on bottom can be divided by 2, leaving 2 on top and 1 on bottom. The calculation becomes 1/1 × 2/1 = 2.

Recap

  • To multiply fractions, multiply the numerators and multiply the denominators.
  • To divide by a fraction, multiply by its reciprocal (Keep, Change, Flip).
  • Always convert mixed numbers to improper fractions before multiplying or dividing.
  • Look for opportunities to 'cross-cancel' to simplify numbers before you multiply.
  • An integer 'n' can always be written as the fraction n/1.

Quick check

  1. What is the reciprocal of 7?1 mark
  2. Calculate 1/3 × 6/7.1 mark

5. Using Fractions in Functions

A function is a rule that takes an input number, performs one or more operations on it, and produces a single output number. These functions can involve fractions as inputs, outputs, or as part of the operations themselves. There are two main tasks:

  1. Finding the output: If you are given the input and the function rule, you simply substitute the input into the rule and calculate the result. Remember the order of operations (BODMAS/PEMDAS).
  1. Finding the input: If you are given the output and the function rule, you need to work backwards. You do this by applying the inverse operations in the reverse order. For example, the inverse of 'add 7' is 'subtract 7', and the inverse of 'multiply by 4' is 'divide by 4'.

Key term

Inverse Operation: An operation that reverses the effect of another operation (e.g., division is the inverse of multiplication).

Examiner insight

Examiners look for clear working when you find an input. Show the output, then the result of the first inverse operation, then the final input. This demonstrates your method clearly.

Common pitfall

When finding an input, students often perform the inverse operations in the original order instead of the reverse order. You must reverse the sequence of events, like rewinding a video.

Worked example 12 marks

A function machine has the rule: INPUT × 4 + 7 → OUTPUT. Find the output if the input is -1 1/2.

  1. 1

    Step 1: Convert the input to an improper fraction. -1 1/2 = -3/2.

  2. 2

    Step 2: Apply the first operation: multiply by 4. -3/2 × 4 = -3/2 × 4/1 = -12/2 = -6.

  3. 3

    Step 3: Apply the second operation: add 7. -6 + 7 = 1.

  4. 4

    The output is 1.

Worked example 22 marks

Using the same function rule (INPUT × 4 + 7 → OUTPUT), find the input if the output is -9.

  1. 1

    Step 1: Start with the output, which is -9.

  2. 2

    Step 2: Reverse the last operation. The last operation was '+ 7', so the inverse is '- 7'. -9 - 7 = -16.

  3. 3

    Step 3: Reverse the first operation. The first operation was '× 4', so the inverse is '÷ 4'. -16 ÷ 4 = -4.

  4. 4

    The input is -4.

Recap

  • To find the output of a function, substitute the input and follow the order of operations.
  • To find the input, work backwards from the output.
  • When working backwards, use inverse operations in the reverse order.
  • Convert mixed numbers to improper fractions to avoid errors in calculation.
  • Be careful with negative signs when performing operations.

Quick check

  1. A function is defined by the formula y = 6x - 1. What is the output 'y' when the input x = 1/2?2 marks

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Place these numbers in the correct column of a table with headings 'Integer' and 'Rational Number'. Some may fit in both. -3, 2.5, 8/3, 0, 10, -1/2.3 marks
  2. Calculate 4 1/5 - 2 3/4. Show your working and give your answer as a mixed number.3 marks
  3. A cake recipe requires 2/3 of a cup of sugar. You only want to make half of the recipe. What fraction of a cup of sugar do you need?2 marks
  4. Calculate 5/8 ÷ 1 1/4. Give your answer as a fraction in its simplest form.3 marks
  5. A function is described by the formula y = (x/2) + 5. Find the value of y when x = 3/4.3 marks
  6. Convert the recurring decimal 0.242424... into a fraction in its simplest form.3 marks
  7. A plank of wood is 3 3/4 metres long. A carpenter cuts off a piece that is 1 1/2 metres long. They then cut the REMAINING piece into 3 equal smaller pieces. How long is each smaller piece?4 marks
  8. A function machine's rule is 'Subtract 5, then multiply by 3'. The output is -4 1/2. What was the input?4 marks
  9. Find a fraction that is exactly halfway between 1/4 and 2/3. Show your method.4 marks
  10. Arrange the following numbers in ascending order (smallest to largest): 3/5, 0.65, 2/3, 5/8.3 marks

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