Fractions.pdf

Head to www.savemyexams.com for more awesome resources AQA GCSE Maths Your notes Fractions Contents Basic Fractions Operations with Fractions...

Head to www.savemyexams.com for more awesome resources AQA GCSE Maths Your notes Fractions Contents Basic Fractions Operations with Fractions Page 1 of 12 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  Head to www.savemyexams.com for more awesome resources Basic Fractions Your notes Basic Fractions What is a fraction? A fraction is part of a whole a A fraction is written aswhere a and b are whole numbers (integers) b The number on the top, a, is called the numerator The number on the bottom, b, is called the denominator a means split something into b parts and take a of them b 1 means split something into 2 parts and take 1 of these parts 2 2 means split the something into 3 parts and take 2 of these parts 3 4 means split something into 5 parts and take 4 of these parts 5 How do I find equivalent fractions? “Splitting something into 2 parts and taking 1 of these parts” gives the same overall amount as “splitting something into 4 parts and taking 2 of these parts” 1 2 is the same as 2 4 2 4 8 3 Using the same idea, is the same as , or , or etc 4 8 16 6 Equivalent fractions are two fractions that represent the same amount (they are different ways of writing the same thing) To make equivalent fractions, multiply the top and bottom of a fraction by the same amount 5 5 × 2 10 5 × 3 15 5 × 4 20 is equivalent to = and = and = …etc 6 6 × 2 12 6 × 3 18 6 × 4 24 For each fraction, there are an infinite number of equivalent fractions To cancel a fraction down to its “simplest” equivalent form, divide the top and bottom of a fraction by the biggest whole number that goes into both (“common factor”) Page 2 of 12 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  Head to www.savemyexams.com for more awesome resources 12 12 ÷ 6 2 = = 18 18 ÷ 6 3 Your notes 25 25 ÷ 5 5 = = 45 45 ÷ 5 9 How do I find a fraction of an amount? Method 1: divide by the denominator and multiply by the numerator 1 To find of an amount, divide it by 2 2 1 To find of an amount, divide it by 3 3 2 To find of an amount, divide it by 5 then multiply it by 2 5 2 of 60: do 60 ÷ 5 = 12, then 12 × 2 = 24 5 Method 2 (if you know how to change fractions into decimals): change the fraction into a decimal, then multiply 1 To find of an amount, multiply the amount by 0.25 4 9 To find of an amount, multiply the amount by 0.9 10 Method 3 (if you know how to multiply fractions): write both numbers as fractions and multiply two fractions together 2 2 60 To find of 60, work out × 3 3 1 5 8 5 8 To find of , work out × 6 3 6 3 Page 3 of 12 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  Head to www.savemyexams.com for more awesome resources Mixed Numbers & Top Heavy Fractions What are mixed numbers & top heavy fractions? Your notes A mixed number has a whole number (integer) part and a fraction part 3 3 eg. 3 has the whole number 3 and the fraction , meaning “three and three quarters” 4 4 A top heavy fraction – also called an improper fraction – is one with the top (numerator) bigger than the bottom (denominator) 15 eg. means “fifteen quarters” 4 Turning mixed numbers into top heavy fractions 1. Multiply the whole number by the denominator (big × bottom) 2. Add that value to the numerator 3. Write the "new" numerator over the same denominator as before Turning top heavy fractions into mixed numbers Divide the top by the bottom (to get a whole number and a remainder) The whole number is the big number The fraction part is the remainder over the denominator Exam Tip Top heavy fractions are also called "improper" fractions Page 4 of 12 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  Head to www.savemyexams.com for more awesome resources Worked example Your notes 3 (a) Write 5 as an improper fraction 4 Multiply the whole number by the denominator, and add to the numerator. Keep the denominator the same. Simplify 17 (b) Write as a mixed number 5 Divide the top by the bottom 17 ÷ 5 = 3 remainder 2 The final answer is 3, with 2 parts still left over to be divided by 5, which can be written as a mixed number Page 5 of 12 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  Head to www.savemyexams.com for more awesome resources Operations with Fractions Your notes Adding & Subtracting Fractions Dealing with mixed numbers Always turn mixed numbers into top heavy fractions before adding or subtracting Adding & subtracting Adding and subtracting are treated in exactly the same way: Find the lowest common denominator (the smallest whole number that each denominator divides) Write each fraction as an equivalent fraction over this denominator (by multiplying top-and- bottom by the same amount) Add (or subtract) the numerators and write this over a single lowest common denominator do not add the denominators Check for any cancellation (or if asked to turn top heavy fractions back into mixed numbers) Page 6 of 12 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  Head to www.savemyexams.com for more awesome resources Worked example Your notes 2 1 (a) Find + 3 5 Find the lowest common denominator of 3 and 5 15 is the smallest number that divides both 3 and 5 the lowest common denominator is 15 Write both fractions as equivalent fractions over 15 (by multiplying top and bottom by the same amount) Add the numerators and write over a single denominator There is no cancellation 3 5 (b) Find 3 − giving your answer as a mixed number 4 8 Page 7 of 12 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  Head to www.savemyexams.com for more awesome resources Change the mixed number into a top heavy fraction (by multiplying the denominator, 4, by the whole number, 3, then adding the numerator, 3) Your notes To find first find the lowest common denominator of 4 and 8 8 is the smallest number that divides both 4 and 8 the lowest common denominator is 8 Write both fractions as equivalent fractions over 8 (by multiplying top and bottom by the same amount) Subtract the numerators and write over a single denominator Change into a mixed number (by dividing 25 by 8 to get 3 remainder 1) Page 8 of 12 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  Head to www.savemyexams.com for more awesome resources Your notes There is no more cancellation Page 9 of 12 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  Head to www.savemyexams.com for more awesome resources Multiplying Fractions Dealing with mixed numbers Your notes Always turn mixed numbers into top heavy fractions before multiplying Multiplying fractions Cancel any numbers on the tops of the fractions with numbers on the bottoms of the fractions (either fraction) Multiply the tops Multiply the bottoms Cancel again if possible Turn top heavy fractions back into mixed numbers (if necessary / asked for) Worked example 4 25 Find × 15 11 The 15 and 25 can be cancelled before multiplying (to make the next step easier) Multiply the numerators together and the denominators together There is no further cancelling that can be done Page 10 of 12 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  Head to www.savemyexams.com for more awesome resources Dividing Fractions Dealing with mixed numbers Your notes Always turn mixed numbers into top heavy fractions before dividing Dividing fractions Never try to divide fractions Instead “flip’n’times” (flip the second fraction and change ÷ into ×) a b So ÷ becomes × b a Then multiply the fractions (multiply tops and multiply bottoms) Cancel the final answer (if possible) Page 11 of 12 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  Head to www.savemyexams.com for more awesome resources Worked example Your notes 1 3 Divide 3 by , giving your answer as a mixed number 4 8 Rewrite as an improper fraction Turn the division into a multiplication, using the fact that dividing by a fraction is the same as multiplying by its reciprocal Multiply the fractions Simplify the fraction, by dividing the numerator and denominator by 4 Rewrite as a mixed number Page 12 of 12 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers