All the examples in this worksheet and all the answers to questions are available as answer sheets or videos.

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "All the examples in this worksheet and all the answers to questions are available as answer sheets or videos."

Transcription

1 BIRKBECK MATHS SUPPORT Numbers 3 In this section we will look at - improper fractions and mixed fractions - multiplying and dividing fractions - what decimals mean and exponents or indices can be used - how ratios and percentages work - exponents or indices with an introduction to algebra - some examples of calculations involving all numbers - some examples of word questions involving numbers Helping you practice At the end of the sheet there are some questions for you to practice. Don t worry if you can t do these but do try to think about them. This practice should help you improve. I find I often make mistakes the first few times I practice, but after a while I understand better. Videos All the examples in this worksheet and all the answers to questions are available as answer sheets or videos. Good luck and enjoy! Videos and more worksheets are available in other formats from

2 1. Improper and Mixed Fractions Here we will look at different ways that fractions can be written but mean the same thing. So far we have considered equivalent fractions, e.g. Now we will introduce Improper Fractions and Mixed Fractions. PROPER FRACTIONS Proper fractions are fractions where the numerator (number on the top) is always smaller than the denominator (number on the bottom), e.g. Any proper fraction is always less than one, and we can see that if we draw 2/3 as a diagram we don t quite have one whole, e.g. So all proper fractions are less than one. IMPROPER FRACTIONS Improper fractions are fractions where the numerator (number on the top) is always larger than the denominator (number at the bottom) e.g. Any improper fraction is always bigger than one whole and we can see this if we draw a diagram of four thirds.

3 So all improper fractions are bigger than the number one. MIXED FRACTIONS A mixed fraction is a fraction where we mix whole numbers and fractions. For example one and a third is a mixed fraction and we could write this as but by comparing the diagram above we can see that four thirds is the same as one whole and a third. We will now show this using maths Where we have used the fact that any number divided by itself is one, e.g. CONVERTING MIXED FRACTIONS AND IMPROPER FRACTIONS Here are some examples of converting between different types of fractions. To see videos explaining these calculations visit Example 1: is an improper fraction and can be written as a mixed fraction. First we need to know how many times the denominator of the fraction goes into the numerator of the fraction. Well 2 is the denominator and I can write the numerator 5 as So I can write So five out of two is the same as two and a half.

4 Example 2: One and a quarter is a mixed fraction and is written write it as an improper fraction in the following way. but we can Example 3: It also doesn t matter if the fraction is negative. If we have minus two and a fifth (a mixed fraction) we can change this to an improper fraction Notice that the whole expression is negative. This is because the whole of the mixed fraction is negative. 2. Multiplying and Dividing Fractions After the hard work on adding fractions and converting between improper and proper fractions you ll probably be pleased to know that multiplying and dividing fractions is much easy. We ll start with a simple example. MULTIPLYING FRACTIONS If I see a cake and eat a quarter of it and I do this three times in total, then I will have eaten three quarters of a cake. And another way to write this is as And here we can see that when we multiply fractions we just multiply all the numbers on the top (3 x 1) to get the total numerator (3), and all the numbers on the bottom (1 x 4) to get the total denominator (4).

5 Here is another example But notice that we can simplify the answer, since the numerator and the denominator can be written as multiples of 2: Key Points: When multiplying fractions make sure all the fractions are in the form of proper or improper fractions. If any fractions are written as mixed fractions you will need to change them to improper fractions. When multiplying fractions just multiply the numerators together (numbers on top) and the denominators together (numbers on bottom). Remember you may be able to simplify your answer. Example with mixed fractions 1) Calculate Well we can t multiply these at the moment as the first fraction is a mixed fraction. So first we have to change the mixed fraction to an improper fraction Now we can re-write the original equation and calculate the answer More videos of fractions are available at

6 DIVIDING FRACTIONS Dividing one fraction by another is not so difficult. To help us think about this, consider an example. If I have half of a cake and I divide this between 2 people what do I get? To see the answer we can draw a picture. Here is my half a cake: I now divide this between 2 people; and we can see that they each get a quarter of the original cake. Here is one of those pieces So this means one half divided by two is a quarter half divided by two equals a quarter Where each of these statements mean the same thing. Notice that we get the answer by turning the dividing fraction upside down, so To see examples of dividing fractions visit

7 4. Decimals Once we have started to understand fractions, then decimals are easier to get to grips with. We start with the simplest decimals Where 0.1 is pronounced as zero point one or nought point one. If we now consider some more general decimals So generally we can write the following eight fractions as decimals Any decimal that can be expressed exactly as a fraction is called a rational number. If a decimal can t be written exactly as a fraction it is called an irrational number. Decimals can be converted to fractions as follows: For more examples see the videos on

8 5. Percentages and Ratios If you can get confident with fractions then you will be much happier with ratios and percentages, as calculating ratios or percentages involve the same ideas. PERCENTAGES The word percentage means out of 100 and is written as %. So 50% means 50 out of 100, which we can write as a fraction then as a decimal But if we look at the fraction now we see that we can simplify the fraction So 50% is the same as 0.50 or the same as half. Here is another example. Seven percent is 7% which means 7 out of 100, and we can write it as a fraction and as a decimal We can t simplify this fraction though as no number goes into both 7 and 100. Here are some examples of questions that can be asked about percentage. We try to answer these questions in more than one way as this helps to check we have the right answer. Example 1: A dress is in the sale. The price says 25, but there is a 20% reduction. What is the actual price of the dress? Answer 1: first we need to find out what 20% of 25 is, and we can write this as 20% of 25 Therefore if the dress is reduced by 5, it means it is 25-5 = 20

9 Answer 2: Now we ll do the same question but a different way. If the dress is reduced by 20%, then 20% has been removed from the original price, since 100% - 20% = 80%. The new price is 80% of 25. So the new price of the jumper is 20, which is the same as the answer above. Example 2: A roofer quotes 400 to replace a roof. He adds VAT of 20%. What is the full cost to replace the roof? Answer 1: The total cost is 400 plus the extra 20%. So first we find out what 20% of 400 is Then the total cost is 400 plus the extra 80 = 480 Answer 2: The total cost is 20% on top of the quote for 400. The 400 is 100% and then there is an extra 20%. This makes 120% in total. So the full cost is 120% of 400 So 120% of 400 gives the total amount of 480. RATIOS Ratios are a simple way to compare amounts, usually when mixing things together. For example to make orange juice you need 1 part orange juice to 3 parts water. This is a ratio of 1 part 3 parts, so the glass would look like this. This glass has 3 parts water and 1 part orange juice way. It is in the ratio 3 to 1. Written as 3:1

10 We can now see that because there is 1 part of juice and 3 parts water, there are four parts in total. So orange juice is one part out of four, and the juice is three parts out of the four. We find we are back to fractions and percentages. So if we want to make 6 litres in total, we need 1/4 (or 25%) orange juice and 3/4 (or 75%) water, which can be calculated as Or we can carry out the same calculation using percentages 6. Exponents, Indices or Powers Lots of jobs, such as nursing, business or engineering involve large and small numbers and decimals. A very useful way to write very small and very large numbers is to use exponents, indices or powers. For example we could write 10x10 as 10 to the power 2, which means multiply 10 by its self two times, so In the case of 10 3, the number 10 is the base and the 3 is the exponent, index or power. We can use the same idea for any number, so

11 This becomes useful though for writing very large numbers. For example This way of writing numbers is called Scientific Notation or Standard Form, and we will look at this more in the worksheet on Measurements and Units But before we go any further we need to think about indices a bit more. RULES OF INDICES To make calculations and simplifications with numbers written with powers or indices we need to learn and use the Rules of Indices. Below we show how they are true, give examples and the general rule. We have used a small bit of algebra in this section and if you feel lost try reading the next section on algebra or watching the video on indices. Notice that This means that Now notice how the indices on the left-hand side (which are 2 and 3) add up to give the index (this is singular for indices) of the answer (which is 5). We can write is as a general rule For example or we could also write using symbols, where x could be any number, for example in this case 2 and y is any other number, in this example 3

12 But x could be any number and y could any other number and the rule would still be true. Let us look at We can see that the rule still works, and further that if we change 4 for 6, So if we use the symbol a to mean any number we can write the First Rule of Indices is Where n, x and y represents any numbers. Notice now what happens if we write the following fraction Well we know that if something multiplies the top of a fraction and the bottom of a fraction then we can cancel, so this fraction can be simplified to But if we re-write the top and bottom in terms of indices we have We can see that the index on the right hand side can be got from the index of the numerator minus the index of the denominator: 5-3 = 2. The second rule of indices is

13 We now consider what happens when we have 2x2x2 and multiply this by itself four times, well lets write it out (we have left extra spaces in between each group of 2x2x2) But we can see on the left hand side that the total number of 2s multiplying each other is twelve which is the same as 4x3, so we can write (If you are not completely comfortable with using brackets there are lots of examples explaining brackets in the first worksheet in the Calculations section.) Writing this as a general rule for any number we have The third rule of indices is Finally: we notice three other things about indices: Often it is only through practicing examples that things become clear. So try these examples then read the theory again and it will become clearer. Examples: simplifying using indices: 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) It is possible to have fractional indices and we will look at the meaning of these in the Calculations section.

14 6. Writing big and small numbers with indices We already mentioned that it is possible to write large numbers using index notation. For example. But we will see now that it is also possible to write small numbers in index form: Consider the following decimals: Continuing to follow this pattern we see that we could write So for any small number, for example we can write If your job is writing small numbers regularly then it is easy to miss out the zeros and so using indices is a much safer and easier way to write the numbers and the index tells you how many zeros, including the one before the decimal point. Here we will look at more examples: 1) 2) 3) 5) 6) 7) 8) 9)

15 7. So what is Algebra? Algebra uses the rules of numbers, such as addition, division, but instead of using numbers it use variables. For example if I buy two arm chairs and a sofa, then I can use the variable A to represent the armchairs and the variable S to represent the sofa, then I have If I now decide to buy another arm chair I have But this is the same as three armchairs and a sofa, so we can simplify SUBSTITUTION Often the letters x and y are used to represent variables. For example we can let x represent one burger and y represent one portion of fries. So if the first customer orders two burgers and three portions of fries I can write this as But now the second customer orders five burgers and two fries we write Suppose we now find out that one place sells burgers for 1.50 and fries for 1.00, we can substitute these in the following way But if the burgers are 1.00 and the fries are 1.20, substitution gives So we can see that it is cheaper for the first customer to go the first cafe but it is cheaper for the second customer to go to the second cafe.

16 8. Now your turn Generally the more maths you practice the easier it gets. If you make mistakes don t worry. I generally find that if I make lots of mistakes I understand the subject better when I have finished. If you want to see videos explaining these ideas and showing the answers visit A) Convert the following improper factions into mixed fractions 1) 2) 3) 4) 5) 6) B) Multiplying and Dividing Fractions, try also to simplify your answer 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) C) Decimals, Percentages and Ratios 1) 2) 3) 4) 5) 6) 7) Water is mixed with glue in the ratio 3:1. If I use 1 litre of glue, how much water do I need? 8) Orange juice is mixed in the ratio 1 parts juice to 4 parts water. How much water is in 100ml of mixed juice? 9) The ratio of milk to dark chocolates is 3:4. If there are 140 chocolates in a bag, how many are milk chocolates?

17 D) Working with indices, simplify the following 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) E) Word Questions 1) In a restaurant each slice of cake sold is one eight of a total cake. If there are three and a half cakes left how can you show mathematically how many slices are left for sale? 2) A suit has a price tag 100 and a shirt has a price tag 40. But there is a special offer and everything is reduced by 15%. How much would the suit and shirt cost with the reduction? 3) Humans produce 2 million red blood cells each second. If each blood cell has mass of. Show using indices the mass of blood cells produced each second. 4) Two brothers decide to share the cost of a car. One of them pays 800 and the other one 200. Write these amounts as the simplest ratio possible. 5) Jeans are sold for 30 each. The selling price is 50% more than the cost price. What is the cost price of each pair of jeans? 6) The ratio of men to women working in a factory is 5:4. There are a total of 20 women. How many people work at the factory in total? 7) A survey reports that one half of families living in a particular block of flats have a pet. Of these people half had a fish, a quarter had a hamster and a quarter had a dog. What fraction of families had a fish? 8) A television has a mark up on the cost price of 40% and is then sold in the sale for 10% discount. If the TV originally cost 100, how much was it sold for?

Maths Workshop for Parents 2. Fractions and Algebra

Maths Workshop for Parents 2. Fractions and Algebra Maths Workshop for Parents 2 Fractions and Algebra What is a fraction? A fraction is a part of a whole. There are two numbers to every fraction: 2 7 Numerator Denominator 2 7 This is a proper (or common)

More information

Measurements 1. BIRKBECK MATHS SUPPORT www.mathsupport.wordpress.com. In this section we will look at. Helping you practice. Online Quizzes and Videos

Measurements 1. BIRKBECK MATHS SUPPORT www.mathsupport.wordpress.com. In this section we will look at. Helping you practice. Online Quizzes and Videos BIRKBECK MATHS SUPPORT www.mathsupport.wordpress.com Measurements 1 In this section we will look at - Examples of everyday measurement - Some units we use to take measurements - Symbols for units and converting

More information

Decimals and other fractions

Decimals and other fractions Chapter 2 Decimals and other fractions How to deal with the bits and pieces When drugs come from the manufacturer they are in doses to suit most adult patients. However, many of your patients will be very

More information

Fractions. Cavendish Community Primary School

Fractions. Cavendish Community Primary School Fractions Children in the Foundation Stage should be introduced to the concept of halves and quarters through play and practical activities in preparation for calculation at Key Stage One. Y Understand

More information

Different types of fraction

Different types of fraction N/E. Different types of fraction There are different types of fraction. Two types are unit fractions and non-unit fractions. Unit fractions Unit means one. Here are some examples of unit fractions: Can

More information

Fractions. If the top and bottom numbers of a fraction are the same then you have a whole one.

Fractions. If the top and bottom numbers of a fraction are the same then you have a whole one. What do fractions mean? Fractions Academic Skills Advice Look at the bottom of the fraction first this tells you how many pieces the shape (or number) has been cut into. Then look at the top of the fraction

More information

Fractions. Chapter 3. 3.1 Understanding fractions

Fractions. Chapter 3. 3.1 Understanding fractions Chapter Fractions This chapter will show you how to find equivalent fractions and write a fraction in its simplest form put fractions in order of size find a fraction of a quantity use improper fractions

More information

CONNECT: Powers and logs POWERS, INDICES, EXPONENTS, LOGARITHMS THEY ARE ALL THE SAME!

CONNECT: Powers and logs POWERS, INDICES, EXPONENTS, LOGARITHMS THEY ARE ALL THE SAME! CONNECT: Powers and logs POWERS, INDICES, EXPONENTS, LOGARITHMS THEY ARE ALL THE SAME! You may have come across the terms powers, indices, exponents and logarithms. But what do they mean? The terms power(s),

More information

Welcome to Basic Math Skills!

Welcome to Basic Math Skills! Basic Math Skills Welcome to Basic Math Skills! Most students find the math sections to be the most difficult. Basic Math Skills was designed to give you a refresher on the basics of math. There are lots

More information

ACCUPLACER MATH TEST REVIEW

ACCUPLACER MATH TEST REVIEW ACCUPLACER MATH TEST REVIEW ARITHMETIC ELEMENTARY ALGEBRA COLLEGE ALGEBRA The following pages are a comprehensive tool used to maneuver the ACCUPLACER UAS Math portion. This tests your mathematical capabilities

More information

Integers, I, is a set of numbers that include positive and negative numbers and zero.

Integers, I, is a set of numbers that include positive and negative numbers and zero. Grade 9 Math Unit 3: Rational Numbers Section 3.1: What is a Rational Number? Integers, I, is a set of numbers that include positive and negative numbers and zero. Imagine a number line These numbers are

More information

Accuplacer Arithmetic Study Guide

Accuplacer Arithmetic Study Guide Testing Center Student Success Center Accuplacer Arithmetic Study Guide I. Terms Numerator: which tells how many parts you have (the number on top) Denominator: which tells how many parts in the whole

More information

COMPASS Numerical Skills/Pre-Algebra Preparation Guide. Introduction Operations with Integers Absolute Value of Numbers 13

COMPASS Numerical Skills/Pre-Algebra Preparation Guide. Introduction Operations with Integers Absolute Value of Numbers 13 COMPASS Numerical Skills/Pre-Algebra Preparation Guide Please note that the guide is for reference only and that it does not represent an exact match with the assessment content. The Assessment Centre

More information

HOSPITALITY Math Assessment Preparation Guide. Introduction Operations with Whole Numbers Operations with Integers 9

HOSPITALITY Math Assessment Preparation Guide. Introduction Operations with Whole Numbers Operations with Integers 9 HOSPITALITY Math Assessment Preparation Guide Please note that the guide is for reference only and that it does not represent an exact match with the assessment content. The Assessment Centre at George

More information

Accuplacer Elementary Algebra Study Guide for Screen Readers

Accuplacer Elementary Algebra Study Guide for Screen Readers Accuplacer Elementary Algebra Study Guide for Screen Readers The following sample questions are similar to the format and content of questions on the Accuplacer Elementary Algebra test. Reviewing these

More information

Introduction to Fractions

Introduction to Fractions Introduction to Fractions Fractions represent parts of a whole. The top part of a fraction is called the numerator, while the bottom part of a fraction is called the denominator. The denominator states

More information

Key. Introduction. What is a Fraction. Better Math Numeracy Basics Fractions. On screen content. Narration voice-over

Key. Introduction. What is a Fraction. Better Math Numeracy Basics Fractions. On screen content. Narration voice-over Key On screen content Narration voice-over Activity Under the Activities heading of the online program Introduction This topic will cover how to: identify and distinguish between proper fractions, improper

More information

FRACTIONS 1 MANIPULATING FRACTIONS. the denominator represents the kind of pieces the whole has been divided into

FRACTIONS 1 MANIPULATING FRACTIONS. the denominator represents the kind of pieces the whole has been divided into CONNECT: Fractions FRACTIONS 1 MANIPULATING FRACTIONS Firstly, let s think about what a fraction is. 1. One way to look at a fraction is as part of a whole. Fractions consist of a numerator and a denominator:

More information

Pre-Algebra Lecture 6

Pre-Algebra Lecture 6 Pre-Algebra Lecture 6 Today we will discuss Decimals and Percentages. Outline: 1. Decimals 2. Ordering Decimals 3. Rounding Decimals 4. Adding and subtracting Decimals 5. Multiplying and Dividing Decimals

More information

Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving

Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Section 7 Algebraic Manipulations and Solving Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Before launching into the mathematics, let s take a moment to talk about the words

More information

All the examples in this worksheet and all the answers to questions are available as answer sheets or videos.

All the examples in this worksheet and all the answers to questions are available as answer sheets or videos. BIRKBECK MATHS SUPPORT www.mathsupport.wordpress.com Calculations 2 In this section we will look at - rounding numbers up or down - decimal places and significant figures - scientific notation - using

More information

eday Lessons Mathematics Grade 8 Student Name:

eday Lessons Mathematics Grade 8 Student Name: eday Lessons Mathematics Grade 8 Student Name: Common Core State Standards- Expressions and Equations Work with radicals and integer exponents. 3. Use numbers expressed in the form of a single digit times

More information

+ = has become. has become. Maths in School. Fraction Calculations in School. by Kate Robinson

+ = has become. has become. Maths in School. Fraction Calculations in School. by Kate Robinson + has become 0 Maths in School has become 0 Fraction Calculations in School by Kate Robinson Fractions Calculations in School Contents Introduction p. Simplifying fractions (cancelling down) p. Adding

More information

Simply Math. Everyday Math Skills NWT Literacy Council

Simply Math. Everyday Math Skills NWT Literacy Council Simply Math Everyday Math Skills 2009 NWT Literacy Council Acknowledgement The NWT Literacy Council gratefully acknowledges the financial assistance for this project from the Department of Education, Culture

More information

2. Simplify. College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses

2. Simplify. College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key 1. Multiply 2 3 5 1 Use the distributive property to remove the parentheses 2 3 5 1 2 25 21 3 35 31 2 10 2 3 15 3 2 13 2 15 3 2

More information

Learning new things and building basic skills

Learning new things and building basic skills Math Review TABE Answer Key 2 Learning new things and building basic skills may be challenging for you, but they also can be very exciting. When you follow the guidelines for learning basic skills, you

More information

47 Numerator Denominator

47 Numerator Denominator JH WEEKLIES ISSUE #22 2012-2013 Mathematics Fractions Mathematicians often have to deal with numbers that are not whole numbers (1, 2, 3 etc.). The preferred way to represent these partial numbers (rational

More information

Preliminary Mathematics

Preliminary Mathematics Preliminary Mathematics The purpose of this document is to provide you with a refresher over some topics that will be essential for what we do in this class. We will begin with fractions, decimals, and

More information

Numeracy Preparation Guide. for the. VETASSESS Test for Certificate IV in Nursing (Enrolled / Division 2 Nursing) course

Numeracy Preparation Guide. for the. VETASSESS Test for Certificate IV in Nursing (Enrolled / Division 2 Nursing) course Numeracy Preparation Guide for the VETASSESS Test for Certificate IV in Nursing (Enrolled / Division Nursing) course Introduction The Nursing course selection (or entrance) test used by various Registered

More information

Chapter 1: Order of Operations, Fractions & Percents

Chapter 1: Order of Operations, Fractions & Percents HOSP 1107 (Business Math) Learning Centre Chapter 1: Order of Operations, Fractions & Percents ORDER OF OPERATIONS When finding the value of an expression, the operations must be carried out in a certain

More information

Solution Guide Chapter 14 Mixing Fractions, Decimals, and Percents Together

Solution Guide Chapter 14 Mixing Fractions, Decimals, and Percents Together Solution Guide Chapter 4 Mixing Fractions, Decimals, and Percents Together Doing the Math from p. 80 2. 0.72 9 =? 0.08 To change it to decimal, we can tip it over and divide: 9 0.72 To make 0.72 into a

More information

Math 016. Materials With Exercises

Math 016. Materials With Exercises Math 06 Materials With Exercises June 00, nd version TABLE OF CONTENTS Lesson Natural numbers; Operations on natural numbers: Multiplication by powers of 0; Opposite operations; Commutative Property of

More information

PREPARATION FOR MATH TESTING at CityLab Academy

PREPARATION FOR MATH TESTING at CityLab Academy PREPARATION FOR MATH TESTING at CityLab Academy compiled by Gloria Vachino, M.S. Refresh your math skills with a MATH REVIEW and find out if you are ready for the math entrance test by taking a PRE-TEST

More information

Exponent Properties Involving Products

Exponent Properties Involving Products Exponent Properties Involving Products Learning Objectives Use the product of a power property. Use the power of a product property. Simplify expressions involving product properties of exponents. Introduction

More information

Click on the links below to jump directly to the relevant section

Click on the links below to jump directly to the relevant section Click on the links below to jump directly to the relevant section Basic review Writing fractions in simplest form Comparing fractions Converting between Improper fractions and whole/mixed numbers Operations

More information

EQUATIONS. Main Overarching Questions: 1. What is a variable and what does it represent?

EQUATIONS. Main Overarching Questions: 1. What is a variable and what does it represent? EQUATIONS Introduction to Variables, Algebraic Expressions, and Equations (2 days) Overview of Objectives, students should be able to: Main Overarching Questions: 1. Evaluate algebraic expressions given

More information

RATIONAL NUMBERS CHAPTER

RATIONAL NUMBERS CHAPTER RATIONAL NUMBERS CHAPTER 70 CHAPTER RATIONAL NUMBERS Section. Recognizing, Reading, Writing and Simplifying Fractions What is a fraction? You have a circle. Cut it into two equal parts. Each part is called

More information

Section 1.5 Exponents, Square Roots, and the Order of Operations

Section 1.5 Exponents, Square Roots, and the Order of Operations Section 1.5 Exponents, Square Roots, and the Order of Operations Objectives In this section, you will learn to: To successfully complete this section, you need to understand: Identify perfect squares.

More information

1.3 Order of Operations

1.3 Order of Operations 1.3 Order of Operations As it turns out, there are more than just 4 basic operations. There are five. The fifth basic operation is that of repeated multiplication. We call these exponents. There is a bit

More information

No Solution Equations Let s look at the following equation: 2 +3=2 +7

No Solution Equations Let s look at the following equation: 2 +3=2 +7 5.4 Solving Equations with Infinite or No Solutions So far we have looked at equations where there is exactly one solution. It is possible to have more than solution in other types of equations that are

More information

Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 1 Section 8 Powers and Exponents

Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 1 Section 8 Powers and Exponents Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 1 Please watch Section 8 of this DVD before working these problems. The DVD is located at: http://www.mathtutordvd.com/products/item66.cfm

More information

Unit 3: Algebra. Date Topic Page (s) Algebra Terminology 2. Variables and Algebra Tiles 3 5. Like Terms 6 8. Adding/Subtracting Polynomials 9 12

Unit 3: Algebra. Date Topic Page (s) Algebra Terminology 2. Variables and Algebra Tiles 3 5. Like Terms 6 8. Adding/Subtracting Polynomials 9 12 Unit 3: Algebra Date Topic Page (s) Algebra Terminology Variables and Algebra Tiles 3 5 Like Terms 6 8 Adding/Subtracting Polynomials 9 1 Expanding Polynomials 13 15 Introduction to Equations 16 17 One

More information

5.4 The Quadratic Formula

5.4 The Quadratic Formula Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function

More information

Fractions, Ratios, and Proportions Work Sheets. Contents

Fractions, Ratios, and Proportions Work Sheets. Contents Fractions, Ratios, and Proportions Work Sheets The work sheets are grouped according to math skill. Each skill is then arranged in a sequence of work sheets that build from simple to complex. Choose the

More information

Using Proportions to Solve Percent Problems I

Using Proportions to Solve Percent Problems I RP7-1 Using Proportions to Solve Percent Problems I Pages 46 48 Standards: 7.RP.A. Goals: Students will write equivalent statements for proportions by keeping track of the part and the whole, and by solving

More information

0.8 Rational Expressions and Equations

0.8 Rational Expressions and Equations 96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to

More information

Order of Operations - PEMDAS. Rules for Multiplying or Dividing Positive/Negative Numbers

Order of Operations - PEMDAS. Rules for Multiplying or Dividing Positive/Negative Numbers Order of Operations - PEMDAS *When evaluating an expression, follow this order to complete the simplification: Parenthesis ( ) EX. (5-2)+3=6 (5 minus 2 must be done before adding 3 because it is in parenthesis.)

More information

Algebra 1A and 1B Summer Packet

Algebra 1A and 1B Summer Packet Algebra 1A and 1B Summer Packet Name: Calculators are not allowed on the summer math packet. This packet is due the first week of school and will be counted as a grade. You will also be tested over the

More information

4 Percentages Chapter notes

4 Percentages Chapter notes 4 Percentages Chapter notes GCSE Specification concepts and skills Find a percentage of a quantity (N o): 4. Use percentages to solve problems (N m): 4., 4.2, 4., 4.4 Use percentages in real-life situations:

More information

(- 7) + 4 = (-9) = - 3 (- 3) + 7 = ( -3) = 2

(- 7) + 4 = (-9) = - 3 (- 3) + 7 = ( -3) = 2 WORKING WITH INTEGERS: 1. Adding Rules: Positive + Positive = Positive: 5 + 4 = 9 Negative + Negative = Negative: (- 7) + (- 2) = - 9 The sum of a negative and a positive number: First subtract: The answer

More information

This assignment will help you to prepare for Algebra 1 by reviewing some of the things you learned in Middle School. If you cannot remember how to complete a specific problem, there is an example at the

More information

Pre-Algebra Class 3 - Fractions I

Pre-Algebra Class 3 - Fractions I Pre-Algebra Class 3 - Fractions I Contents 1 What is a fraction? 1 1.1 Fractions as division............................... 2 2 Representations of fractions 3 2.1 Improper fractions................................

More information

Fractions, decimals and percentages

Fractions, decimals and percentages Fractions, decimals and percentages Some notes for the lesson. Extra practice questions available. A. Quick quiz on units Some of the exam questions will have units in them, and you may have to convert

More information

Chapter 1. Real Numbers Operations

Chapter 1. Real Numbers Operations www.ck1.org Chapter 1. Real Numbers Operations Review Answers 1 1. (a) 101 (b) 8 (c) 1 1 (d) 1 7 (e) xy z. (a) 10 (b) 14 (c) 5 66 (d) 1 (e) 7x 10 (f) y x (g) 5 (h) (i) 44 x. At 48 square feet per pint

More information

Math Refresher. Book #2. Workers Opportunities Resources Knowledge

Math Refresher. Book #2. Workers Opportunities Resources Knowledge Math Refresher Book #2 Workers Opportunities Resources Knowledge Contents Introduction...1 Basic Math Concepts...2 1. Fractions...2 2. Decimals...11 3. Percentages...15 4. Ratios...17 Sample Questions...18

More information

3.1. RATIONAL EXPRESSIONS

3.1. RATIONAL EXPRESSIONS 3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers

More information

Sometimes it is easier to leave a number written as an exponent. For example, it is much easier to write

Sometimes it is easier to leave a number written as an exponent. For example, it is much easier to write 4.0 Exponent Property Review First let s start with a review of what exponents are. Recall that 3 means taking four 3 s and multiplying them together. So we know that 3 3 3 3 381. You might also recall

More information

Grade 9 Mathematics Unit #1 Number Sense Sub-Unit #1 Rational Numbers. with Integers Divide Integers

Grade 9 Mathematics Unit #1 Number Sense Sub-Unit #1 Rational Numbers. with Integers Divide Integers Page1 Grade 9 Mathematics Unit #1 Number Sense Sub-Unit #1 Rational Numbers Lesson Topic I Can 1 Ordering & Adding Create a number line to order integers Integers Identify integers Add integers 2 Subtracting

More information

10-4-10 Year 9 mathematics: holiday revision. 2 How many nines are there in fifty-four?

10-4-10 Year 9 mathematics: holiday revision. 2 How many nines are there in fifty-four? DAY 1 Mental questions 1 Multiply seven by seven. 49 2 How many nines are there in fifty-four? 54 9 = 6 6 3 What number should you add to negative three to get the answer five? 8 4 Add two point five to

More information

Ratios (pages 288 291)

Ratios (pages 288 291) A Ratios (pages 2 29) A ratio is a comparison of two numbers by division. Ratio Arithmetic: to : Algebra: a to b a:b a b When you write a ratio as a fraction, write it in simplest form. Two ratios that

More information

Exponents. Exponents tell us how many times to multiply a base number by itself.

Exponents. Exponents tell us how many times to multiply a base number by itself. Exponents Exponents tell us how many times to multiply a base number by itself. Exponential form: 5 4 exponent base number Expanded form: 5 5 5 5 25 5 5 125 5 625 To use a calculator: put in the base number,

More information

Exponents, Factors, and Fractions. Chapter 3

Exponents, Factors, and Fractions. Chapter 3 Exponents, Factors, and Fractions Chapter 3 Exponents and Order of Operations Lesson 3-1 Terms An exponent tells you how many times a number is used as a factor A base is the number that is multiplied

More information

PERCENTS. Percent means per hundred. Writing a number as a percent is a way of comparing the number with 100. For example: 42% =

PERCENTS. Percent means per hundred. Writing a number as a percent is a way of comparing the number with 100. For example: 42% = PERCENTS Percent means per hundred. Writing a number as a percent is a way of comparing the number with 100. For example: 42% = Percents are really fractions (or ratios) with a denominator of 100. Any

More information

Mathematics Higher Tier, Algebraic Fractions

Mathematics Higher Tier, Algebraic Fractions These solutions are for your personal use only. DO NOT photocopy or pass on to third parties. If you are a school or an organisation and would like to purchase these solutions please contact Chatterton

More information

Order of Operations. 2 1 r + 1 s. average speed = where r is the average speed from A to B and s is the average speed from B to A.

Order of Operations. 2 1 r + 1 s. average speed = where r is the average speed from A to B and s is the average speed from B to A. Order of Operations Section 1: Introduction You know from previous courses that if two quantities are added, it does not make a difference which quantity is added to which. For example, 5 + 6 = 6 + 5.

More information

Self-Directed Course: Transitional Math Module 2: Fractions

Self-Directed Course: Transitional Math Module 2: Fractions Lesson #1: Comparing Fractions Comparing fractions means finding out which fraction is larger or smaller than the other. To compare fractions, use the following inequality and equal signs: - greater than

More information

Rule 2: If the decimal point is moved to the left, the exponent is positive.

Rule 2: If the decimal point is moved to the left, the exponent is positive. Scientific Notation Any quantity can be expressed using a power of ten. As you move the decimal point, you multiply by 10 as many times as necessary to make the numbers equal. Consider the following examples:

More information

SAT Math Facts & Formulas Review Quiz

SAT Math Facts & Formulas Review Quiz Test your knowledge of SAT math facts, formulas, and vocabulary with the following quiz. Some questions are more challenging, just like a few of the questions that you ll encounter on the SAT; these questions

More information

23. RATIONAL EXPONENTS

23. RATIONAL EXPONENTS 23. RATIONAL EXPONENTS renaming radicals rational numbers writing radicals with rational exponents When serious work needs to be done with radicals, they are usually changed to a name that uses exponents,

More information

Introduction to Fractions

Introduction to Fractions Section 0.6 Contents: Vocabulary of Fractions A Fraction as division Undefined Values First Rules of Fractions Equivalent Fractions Building Up Fractions VOCABULARY OF FRACTIONS Simplifying Fractions Multiplying

More information

Mathematics Practice for Nursing and Midwifery Ratio Percentage. 3:2 means that for every 3 items of the first type we have 2 items of the second.

Mathematics Practice for Nursing and Midwifery Ratio Percentage. 3:2 means that for every 3 items of the first type we have 2 items of the second. Study Advice Service Student Support Services Author: Lynn Ireland, revised by Dave Longstaff Mathematics Practice for Nursing and Midwifery Ratio Percentage Ratio Ratio describes the relationship between

More information

Fractions Packet. Contents

Fractions Packet. Contents Fractions Packet Contents Intro to Fractions.. page Reducing Fractions.. page Ordering Fractions page Multiplication and Division of Fractions page Addition and Subtraction of Fractions.. page Answer Keys..

More information

Year 9 set 1 Mathematics notes, to accompany the 9H book.

Year 9 set 1 Mathematics notes, to accompany the 9H book. Part 1: Year 9 set 1 Mathematics notes, to accompany the 9H book. equations 1. (p.1), 1.6 (p. 44), 4.6 (p.196) sequences 3. (p.115) Pupils use the Elmwood Press Essential Maths book by David Raymer (9H

More information

Years after 2000. US Student to Teacher Ratio 0 16.048 1 15.893 2 15.900 3 15.900 4 15.800 5 15.657 6 15.540

Years after 2000. US Student to Teacher Ratio 0 16.048 1 15.893 2 15.900 3 15.900 4 15.800 5 15.657 6 15.540 To complete this technology assignment, you should already have created a scatter plot for your data on your calculator and/or in Excel. You could do this with any two columns of data, but for demonstration

More information

MATH 65 NOTEBOOK CERTIFICATIONS

MATH 65 NOTEBOOK CERTIFICATIONS MATH 65 NOTEBOOK CERTIFICATIONS Review Material from Math 60 2.5 4.3 4.4a Chapter #8: Systems of Linear Equations 8.1 8.2 8.3 Chapter #5: Exponents and Polynomials 5.1 5.2a 5.2b 5.3 5.4 5.5 5.6a 5.7a 1

More information

Order of Operations More Essential Practice

Order of Operations More Essential Practice Order of Operations More Essential Practice We will be simplifying expressions using the order of operations in this section. Automatic Skill: Order of operations needs to become an automatic skill. Failure

More information

Fraction Problems. Figure 1: Five Rectangular Plots of Land

Fraction Problems. Figure 1: Five Rectangular Plots of Land Fraction Problems 1. Anna says that the dark blocks pictured below can t represent 1 because there are 6 dark blocks and 6 is more than 1 but 1 is supposed to be less than 1. What must Anna learn about

More information

Translating Mathematical Formulas Into Excel s Language

Translating Mathematical Formulas Into Excel s Language Translating Mathematical Formulas Into Excel s Language Introduction Microsoft Excel is a very powerful calculator; you can use it to compute a wide variety of mathematical expressions. Before exploring

More information

Multiplication and Division with Rational Numbers

Multiplication and Division with Rational Numbers Multiplication and Division with Rational Numbers Kitty Hawk, North Carolina, is famous for being the place where the first airplane flight took place. The brothers who flew these first flights grew up

More information

Linear Programming Notes VII Sensitivity Analysis

Linear Programming Notes VII Sensitivity Analysis Linear Programming Notes VII Sensitivity Analysis 1 Introduction When you use a mathematical model to describe reality you must make approximations. The world is more complicated than the kinds of optimization

More information

Lesson 4: Convert Fractions, Review Order of Operations

Lesson 4: Convert Fractions, Review Order of Operations Lesson 4: Convert Fractions, Review Order of Operations LESSON 4: Convert Fractions, Do Order of Operations Weekly Focus: fractions, decimals, percent, order of operations Weekly Skill: convert, compute

More information

5 Decimal numbers. 1 Decimal numbers. 2 How to read decimal numbers.

5 Decimal numbers. 1 Decimal numbers. 2 How to read decimal numbers. 5 Decimal numbers 1 Decimal numbers Decimal numbers such as 3.762 are used in situations in which we look for more precision than whole numbers provide. As with whole numbers, a digit in a decimal number

More information

Multiplying Fractions

Multiplying Fractions . Multiplying Fractions. OBJECTIVES 1. Multiply two fractions. Multiply two mixed numbers. Simplify before multiplying fractions 4. Estimate products by rounding Multiplication is the easiest of the four

More information

Unit 1 Number Sense. In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions.

Unit 1 Number Sense. In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions. Unit 1 Number Sense In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions. BLM Three Types of Percent Problems (p L-34) is a summary BLM for the material

More information

Repton Manor Primary School. Maths Targets

Repton Manor Primary School. Maths Targets Repton Manor Primary School Maths Targets Which target is for my child? Every child at Repton Manor Primary School will have a Maths Target, which they will keep in their Maths Book. The teachers work

More information

Add Decimal Numbers LESSON 4

Add Decimal Numbers LESSON 4 LESSON 4 Add Decimal Numbers In this lesson, we will begin using the algebra-decimal inserts to represent decimals. Turn a red hundred square upside down so the hollow side is showing, and snap the flat

More information

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers. Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

More information

Exponents and Radicals

Exponents and Radicals Exponents and Radicals (a + b) 10 Exponents are a very important part of algebra. An exponent is just a convenient way of writing repeated multiplications of the same number. Radicals involve the use of

More information

c sigma & CEMTL

c sigma & CEMTL c sigma & CEMTL Foreword The Regional Centre for Excellence in Mathematics Teaching and Learning (CEMTL) is collaboration between the Shannon Consortium Partners: University of Limerick, Institute of Technology,

More information

1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N.

1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N. CHAPTER 3: EXPONENTS AND POWER FUNCTIONS 1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N. For example: In general, if

More information

Round decimals to the nearest whole number

Round decimals to the nearest whole number Round decimals to the nearest whole number Learning Objective Simplifying Fractions Simplified Fractions To simplify a fraction, we find an equivalent fraction which uses the smallest numbers possible.

More information

Maths Module 2. Working with Fractions. This module covers fraction concepts such as:

Maths Module 2. Working with Fractions. This module covers fraction concepts such as: Maths Module Working with Fractions This module covers fraction concepts such as: identifying different types of fractions converting fractions addition and subtraction of fractions multiplication and

More information

Exponents, Radicals, and Scientific Notation

Exponents, Radicals, and Scientific Notation General Exponent Rules: Exponents, Radicals, and Scientific Notation x m x n = x m+n Example 1: x 5 x = x 5+ = x 7 (x m ) n = x mn Example : (x 5 ) = x 5 = x 10 (x m y n ) p = x mp y np Example : (x) =

More information

MATH-0910 Review Concepts (Haugen)

MATH-0910 Review Concepts (Haugen) Unit 1 Whole Numbers and Fractions MATH-0910 Review Concepts (Haugen) Exam 1 Sections 1.5, 1.6, 1.7, 1.8, 2.1, 2.2, 2.3, 2.4, and 2.5 Dividing Whole Numbers Equivalent ways of expressing division: a b,

More information

Math Matters: Why Do I Need To Know This? 1 Probability and counting Lottery likelihoods

Math Matters: Why Do I Need To Know This? 1 Probability and counting Lottery likelihoods Math Matters: Why Do I Need To Know This? Bruce Kessler, Department of Mathematics Western Kentucky University Episode Four 1 Probability and counting Lottery likelihoods Objective: To demonstrate the

More information

Use your TI-84 graphing calculator to check as many problems as possible.

Use your TI-84 graphing calculator to check as many problems as possible. Name: Date: Period: Dear Future Algebra Honors student, We hope that you enjoy your summer vacation to the fullest. We look forward to working with you next year. As you enter your new math class, you

More information

Amount, Base, and Rate

Amount, Base, and Rate The Mathematics 11 Competency Test, Base, and Rate Sorting out a percent problem always involves correctly attaching numbers to three fundamental quantities: the percent (rate), the value to which the

More information

Trigonometry. Week 1 Right Triangle Trigonometry

Trigonometry. Week 1 Right Triangle Trigonometry Trigonometry Introduction Trigonometry is the study of triangle measurement, but it has expanded far beyond that. It is not an independent subject of mathematics. In fact, it depends on your knowledge

More information

Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 1 Section 9 Order of Operations

Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 1 Section 9 Order of Operations Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 1 Please watch Section 9 of this DVD before working these problems. The DVD is located at: http://www.mathtutordvd.com/products/item66.cfm

More information

T. H. Rogers School Summer Math Assignment

T. H. Rogers School Summer Math Assignment T. H. Rogers School Summer Math Assignment Mastery of all these skills is extremely important in order to develop a solid math foundation. I believe each year builds upon the previous year s skills in

More information