Section A — Chapters 1–22

Rational Numbers I

Junior Cycle — 1st Year

✓By the end of this lesson students will be able to define rational numbers and identify examples.
✓By the end of this lesson students will be able to add and subtract fractions with different denominators.
✓By the end of this lesson students will be able to multiply fractions and simplify the results.
✓By the end of this lesson students will be able to apply the commutative, associative, and distributive properties to rational numbers.
✓By the end of this lesson students will be able to use a calculator to perform operations with rational numbers.

Key concepts

Rational Number

A rational number is any number that can be expressed as a fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \) is not equal to zero. Examples include \( \frac{1}{2}, -\frac{3}{4}, 5 \) (which can be written as \( \frac{5}{1} \)), and \( 0.75 \) (which can be written as \( \frac{3}{4} \)).

Fraction

A fraction represents a part of a whole. It is written as \( \frac{\text{numerator}}{\text{denominator}} \). The numerator tells us how many parts we have, and the denominator tells us how many equal parts the whole is divided into.

Equivalent Fractions

Equivalent fractions are different fractions that represent the same value. For example, \( \frac{1}{2} \) and \( \frac{2}{4} \) are equivalent fractions. You can create equivalent fractions by multiplying or dividing both the numerator and denominator by the same non-zero number.

Lowest Common Denominator (LCD)

The LCD is the smallest common multiple of the denominators of two or more fractions. It is essential for adding and subtracting fractions because fractions must have the same denominator before these operations can be performed.

Adding and Subtracting Fractions

To add or subtract fractions, you must first find a common denominator, ideally the LCD. Convert each fraction to an equivalent fraction with the LCD. Then, add or subtract the numerators and keep the common denominator. Always simplify your answer to its lowest terms.

Multiplying Fractions

To multiply fractions, multiply the numerators together and multiply the denominators together. Simplify the resulting fraction to its lowest terms. For example, \( \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \).

Commutative Property

This property states that the order in which you add or multiply rational numbers does not change the result.

\( a + b = b + a \text{ and } a \times b = b \times a \)

Associative Property

This property states that the way in which rational numbers are grouped when adding or multiplying does not change the result.

\( (a + b) + c = a + (b + c) \text{ and } (a \times b) \times c = a \times (b \times c) \)

Distributive Property

This property connects multiplication and addition (or subtraction). It states that multiplying a sum (or difference) by a number is the same as multiplying each term in the sum (or difference) by the number and then adding (or subtracting) the products.

\( a \times (b + c) = (a \times b) + (a \times c) \)

Key facts to remember

1A rational number is any number that can be written as a fraction \( \frac{p}{q} \) where \( p \) and \( q \) are integers and \( q \neq 0 \).
2All integers, terminating decimals, and recurring decimals are rational numbers.
3To add or subtract fractions, you must first find a common denominator (ideally the LCD).
4To multiply fractions, multiply the numerators and multiply the denominators.
5Always simplify fractions to their lowest terms by dividing the numerator and denominator by their greatest common factor.
6Rational numbers obey the commutative, associative, and distributive properties for addition and multiplication.
7Your calculator has a dedicated fraction button that can help with operations and simplification.

Worked examples

Example 1

Evaluate \( \frac{2}{3} + \frac{1}{5} \).

IFind the Lowest Common Denominator (LCD) of 3 and 5. The multiples of 3 are 3, 6, 9, 12, 15, 18... The multiples of 5 are 5, 10, 15, 20... The LCD is 15.

IIConvert each fraction to an equivalent fraction with a denominator of 15.\n\( \frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15} \)\n\( \frac{1}{5} = \frac{1 \times 3}{5 \times 3} = \frac{3}{15} \)

IIIAdd the equivalent fractions.\n\( \frac{10}{15} + \frac{3}{15} = \frac{10 + 3}{15} = \frac{13}{15} \)

IVCheck if the answer can be simplified. 13 and 15 have no common factors other than 1, so it is in its lowest terms.

Answer

\( \frac{13}{15} \)

Remember to always find a common denominator before adding or subtracting fractions.

Example 2

Calculate \( \frac{3}{4} \times \frac{8}{9} \).

IMultiply the numerators together.\n\( 3 \times 8 = 24 \)

IIMultiply the denominators together.\n\( 4 \times 9 = 36 \)

IIIWrite the new fraction.\n\( \frac{24}{36} \)

IVSimplify the fraction to its lowest terms. Find the greatest common factor (GCF) of 24 and 36. The GCF is 12.\nDivide both the numerator and denominator by 12.\n\( \frac{24 \div 12}{36 \div 12} = \frac{2}{3} \)

Answer

\( \frac{2}{3} \)

You can often simplify before multiplying by 'cancelling' common factors diagonally or vertically, which can make the numbers smaller and easier to work with. For example, \( \frac{\cancel{3}^1}{_1\cancel{4}} \times \frac{\cancel{8}^2}{\cancel{9}_3} = \frac{1 \times 2}{1 \times 3} = \frac{2}{3} \).

Example 3

Evaluate \( \frac{1}{2} \times (\frac{4}{5} + \frac{1}{10}) \) using the distributive property, and then verify your answer using a calculator.

IApply the distributive property: \( \frac{1}{2} \times (\frac{4}{5} + \frac{1}{10}) = (\frac{1}{2} \times \frac{4}{5}) + (\frac{1}{2} \times \frac{1}{10}) \)

IIPerform the multiplications:\n\( \frac{1}{2} \times \frac{4}{5} = \frac{1 \times 4}{2 \times 5} = \frac{4}{10} \)\n\( \frac{1}{2} \times \frac{1}{10} = \frac{1 \times 1}{2 \times 10} = \frac{1}{20} \)

IIIAdd the resulting fractions: \( \frac{4}{10} + \frac{1}{20} \)

IVFind the LCD of 10 and 20, which is 20.\n\( \frac{4}{10} = \frac{4 \times 2}{10 \times 2} = \frac{8}{20} \)

VAdd: \( \frac{8}{20} + \frac{1}{20} = \frac{9}{20} \)

VIVerify with a calculator:\nInput \( \frac{1}{2} \times (\frac{4}{5} + \frac{1}{10}) \) into the calculator using the fraction button.\nFirst, calculate the sum in the bracket: \( \frac{4}{5} + \frac{1}{10} = \frac{8}{10} + \frac{1}{10} = \frac{9}{10} \).\nThen, multiply: \( \frac{1}{2} \times \frac{9}{10} = \frac{9}{20} \).

Answer

\( \frac{9}{20} \)

The distributive property allows you to break down complex problems into simpler parts. Always use the fraction button (often labelled a b/c or similar) on your calculator for accuracy.

Common mistakes

✗Adding or subtracting the denominators when adding or subtracting fractions (e.g., \( \frac{1}{2} + \frac{1}{3} \neq \frac{2}{5} \)).
✗Not finding a common denominator before adding or subtracting fractions.
✗Forgetting to simplify fractions to their lowest terms at the end of a calculation.
✗Confusing the rules for multiplying fractions with the rules for adding/subtracting them.
✗Incorrectly entering fractions into a calculator, especially mixed numbers or complex expressions.

Exam tips

★Always show all steps in your working, even if you use a calculator for intermediate steps. This helps you get partial marks if your final answer is incorrect.
★Practise simplifying fractions regularly; it's a fundamental skill.
★Be familiar with the fraction button on your scientific calculator and how to use it for operations and conversions.
★When working with mixed operations, remember the order of operations (BODMAS/PEMDAS).
★Double-check your answers, especially for signs (positive/negative) and simplification.

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