The main focus of this unit is exploring how numbers can be expressed in different ways, and how the different forms of expression can make calculations or problems easier to tackle. By the end of the unit, you should feel more confident in dealing with fractions and decimals in everyday life.
Before you start work on the activities in this unit, skim through it to see what it contains. Make a note of the subheadings and activity headings that you recognize, as well as those that appear new to you. Then, based on your experience of the previous five units, try to estimate how long this unit is going to take you. Make a note of your estimate in your math notebook. The time you require will depend on your previous experience and your mathematical knowledge; everyone is different. However, this unit does contain some essential ideas that will be needed later in the course, so it is important to work through the activities thoroughly on the topics that are new to you, or perhaps long forgotten!
This unit should take around 11 hours to complete. In this unit you will learn about:
Compared to the previous five units, Unit 6 will most likely take you more time to work through. This unit, which focuses on fractions, is a longer unit because we all need extra practice working with fractions. In this unit, you will learn how to use the calculator to perform operations with fractions, as well as other practical skills.
As you work through this unit, take breaks and be sure to talk to a friend or family member if you need extra help. The Internet can also provide you with additional guidance. Take advantage of all of the resources at your disposal. Finally, keep in mind that we all get stuck at some point on our math journey, and getting past those rough patches often requires a little help.
Additionally, this unit will continue to show how to write good, clear mathematics, so that other people can easily follow your ideas and mathematical solutions. As you work, be sure to label and neatly write out your work in your math notebook, especially for activities or concepts that are unfamiliar to you. This unit contains essential ideas that will be needed later in the course, so you will need to work through the activities thoroughly on the topics that are new to you or perhaps long forgotten.
Use the various strategies discussed in Unit 1, and don’t give up. You can do this!
If you would like to check your understanding of some basic techniques of expressing numbers in different ways before you start, give the Unit 6 pre quiz a try, then use the feedback to help you plan your study.
The quiz does not check all the topics in the unit, but it should give you some idea of the areas you may need to spend most time on. Remember, it doesn’t matter if you get some or even all of the questions wrong—it just indicates how much time you may need for this unit!
As you know, the upcoming section is about fractions.
Before you start working on the activities in this unit, take a look at this video.
You can see that fractions are everywhere! Can you think of other examples where you see or use fractions?
Try asking your friends and family, too.
You use fractions in the kitchen when baking an apple pie.
You use fractions when building steps for your front porch.
You use fractions when you pay for your gas.
You use fractions when you are tuning up your bicycle.
Of course, you might have come up with some other examples.
Most people use fractions in their everyday life when they talk about time (quarter after ten), parts of pizzas and cakes (halves and quarters) or when shopping (two-thirds off marked prices). You may also see fractions in the headlines. The following headline appeared in a press release from the British Wind Energy Association in June 2005.
“Three-quarters of people in Wales believe wind farms are necessary, says new poll.”
What did you think when you read the headline? Do a majority of the Welsh support this view? Do you know how many people actually think in this way?
No, you don’t actually know the number of people who think this way; the headline just tells you the proportion of those who support wind farms out of the group of people who were interviewed. If you gathered together all the people who were polled, you could arrange them into four equal groups, so that the people in three of the groups would have supported this view, and those in the fourth would not.
If only four people had been interviewed, three would have agreed that wind farms are necessary. If 4,000 people were interviewed, then 3,000 would have agreed, and so on. So, how much notice you should take of the headline would probably depend on both the number of people who were surveyed and how they were selected.
Interviewing a lot of people who had been selected at random may give a better indication of the views of the general population than polling just a few people who lived a long way from any wind farm would.
Reference
BWEA (2005) BWEA Press Release (accessed August 26, 2011).
Fortunately, some information on who was interviewed was provided in the press release: 500 Welsh people were surveyed.
By dividing this group into quarters, work out how many of the people agreed with the view in the headline. How many did not agree?
Did you break 500 into four equal parts? Out of those four parts, how many parts agree that wind farms are necessary? What do you need to do with these pieces of information, then, to determine how many survey participants did and did not say that wind farms were necessary?
Three-quarters of the 500 people agreed.
First, split the group into quarters by dividing 500 by 4: people—one-quarter of the people surveyed.
Three-quarters of the group means we need three sets of 125 people.
Since , 375 people agreed with the statement and 125 did not. (You can check your arithmetic by noting that .)
In a similar way, you can often make sense of most everyday fractions by:
There will be occasions when you will need to carry out more involved calculations involving fractions. The next part of this section introduces fractions, and then considers some calculations that you can work out using your calculator.
Take a piece of blank notebook paper and fold it in half, creasing along the fold. Open it up and shade the left-hand side.
Since the paper has been divided into two equal parts, each piece is half of the original. This fraction is written as and read as “one-half.”
Now fold the paper back along the original crease and then in half again along the long side. If you open up the paper, you should see four pieces of the same size, with two of them shaded. The paper is now divided into quarters (or fourths), and the fraction of the paper shaded is .
Since you haven’t altered the shading in any way, this demonstration shows that one half is equal to two quarters: . Now, fold the paper back into quarters along the crease lines and then fold into three equal pieces or thirds along the long side. If you now open the paper up, you can see that there are 12 equal pieces. These pieces are “twelfths,” of which six are shaded, so of the paper is shaded. This fraction also represents the same amount as .
You can continue to fold the paper into smaller and smaller pieces. Each time you open up the paper, it will be divided into smaller fractions, but half of it will still be shaded. The fractions which represent the shaded part are all equivalent to each other.
[ Why doesn’t this work with zero? You thought about that in Unit 2. ] If you multiply the numerator (the number on top) and the denominator (the number underneath) of any fraction by the same number (except zero), you will get a fraction that is equivalent to the original one:
Note: You must multiply the numerator and denominator by the same number because it is the same as multiplying by one, so it doesn’t change the value of the fraction.
You can also generate equivalent fractions by dividing the numerator (top number) and the denominator (bottom number) of the fraction by the same number (again, not zero).
Would you have reduced this fraction using different steps? This process of dividing the numerator (the number above the line) and the denominator (the value below the line) by the same number is known as “cancelling,” “reducing,” or “simplifying.” If there is no whole number that can be divided into both the numerator and the denominator, the fraction is said to be in lowest terms, or fully reduced. Answers are usually left in this form, as they are easier to visualize and understand (click on “View document”).
[ Want to play more with fractions? Check out this interactive website. ]
Imagine these paper circles as pizzas. You can see that of a pizza is the same as of a pizza, and also the same as of a pizza.
This activity may require you to work out some of the solutions on paper. It is recommended that you use your math notebook to keep track of your work.
(a) Use your knowledge of equivalent fractions to fill in the missing numbers.
The denominator of the fraction on the left is 3. What do you have to multiply it by to get 18? Remember that if you multiply the denominator by a particular number, you must do the same to the numerator.
(a)
(i) The 3 is multiplied by 6 to reach 18, so this is the number that is used to multiply the numerator and the denominator. The missing number is 12.
(ii) Similarly, since , multiply the numerator and the denominator by four.
The missing number is 16.
(b) Which of the following three fractions are equivalent to each other?
Did you check to see if multiplying the numerator, 7, by any numbers gave the numerator of another of the given fractions? If so, try multiplying the denominator, 8, by the same number and see what denominator is yielded.
(b) Multiplying the numerator and the denominator of the first fraction by five gives
Only the two fractions labelled (i) and (ii) are equivalent.
You cannot create from by multiplying numerator and denominator by the same number.
If you multiply by 7 top and bottom, which is to multiply it by , you get , which isn’t , and if you multiply by 8 top and bottom (that is, multiply it by ) you get , which isn’t , either.
(c) Reduce (cancel) the following fractions into their lowest terms.
Look at each fraction individually. What number can divide evenly into both the numerator and the denominator of each? Remember, there could be more than one option.
(c)
(i) Dividing the top and the bottom by 2 gives .
(ii) Dividing the top and the bottom by 10 (or by 5 and then by 2) gives .
(iii) Dividing the top and the bottom by 9 and then by 3 gives .
Dividing the top and the bottom by 4 and then 3 gives: .
(Of course it does not matter which number you divide by first, and there are even more choices than the ones shown. Reducing is a process.)
If you find it difficult to spot the numbers to divide by, try to work systematically by trying 2, 3, 5, … in turn. You can check these results by using your calculator, as you will see shortly.
Does the notation of reducing a fraction used in part (c) of this activity look new to you? Click the “View document” link for an explanation.
From a survey of a group of 560 people, 245 say that they have taken a college course during the last year, and 140 say that they have taken a math course.
(a) What fraction of the total group has taken a college course during the last year?
What is the total number of people surveyed? How many of those reported taking a college course during the last year? Remember to reduce your fraction!
(a) The fraction who have taken a college course during the last year is . Now, both 245 and 560 look as though they can be divided by 5. Let’s try it. Yes! That gives us . Now, 7 is one of the few numbers that can divide evenly into 49. Does it also divide 112? Yes again! This fraction can be simplified by dividing the top and bottom by 5 and then by 7:
(b) What fraction of the total group has taken a math course?
How many out of the total number of people interviewed took a math course? Remember to reduce your fraction! |
(b) The fraction who have taken a math course is .
Cancel by dividing the top and bottom by 10, then by 7, and finally by 2. (A different order, or even a different choice of numbers to reduce by can be used, but many people find it easy to divide by 10 first.)
Again, there are no more numbers that will divide evenly into both the numerator and the denominator, so we know we have our answer. The fraction who have taken a math course is .
This activity is different from the numerical activities you’ve seen so far: it asks you to consider some general statements about fractions. |
Read the following statements and decide whether they are true or false.
(a) “Only a fraction of the group were on time” always means that less than half the people were on time.
Can you think of a fraction that is larger than one half?
a.
True
b.
False
The correct answer is b.
(a) False. The fraction could be or , which are both bigger than one half. Be careful! The use of the word “only” may suggest to you that it is a small fraction, perhaps less than one half, but this could be a wrong interpretation.
(b) You can write any fraction in a decimal form.
What operation does a fraction bar represent?
a.
True
b.
False
The correct answer is a.
(b) True. A fraction can be thought of as a division problem. For example, is the same as . However, some decimal fractions do not stop; instead they have a repeating set of digits, such as .
These are known as repeating decimals. They are accurately represented by placing a bar over the repeating set, like this: .
An alternative representation for a repeating decimal is to place a dot over the first and last digit of the repeating set.
Often, these decimal numbers are rounded, so you might see rounded off to 0.29 or 0.286.
Keep in mind that rounded values, while useful for some purposes, are not the same as accurate representations.
(c) Fractions always have a value of less than one.
Could the numerator (top number) of a fraction be larger than its denominator?
a.
True
b.
False
The correct answer is b.
(c) False. A positive fraction in which the numerator is greater than the denominator has a value greater than 1. For example, (seven-thirds) has a value greater than 1, because it means , which is greater than 2.
(d) A number can always be written as a fraction.
Can you write a whole number as a fraction? Can you write a decimal number such as 0.375 as a fraction?
a.
True
b.
False
The correct answer is b.
(d) False. Some numbers, but not all, can be written as fractions. Whole numbers, such as 8, can be written with a denominator of 1, like this: . So, if you have a finite number that stops after a certain number of decimal places, then it can be written as a fraction. For example: . These are not approximations or repeating fractions: 0.375 is exactly equal to .
Even decimals that have a repeating sequence can be written as fractions. For example: and . However, it can be proven that there are some numbers, such as (pi) and that cannot be written as fractions. These are known as irrational numbers. In math, the word irrational means it cannot be written as a ratio of two integers.
In most of the examples considered so far, you have been using fractions to describe part of a whole. The result is a positive number whose value is less than 1. Fractions in which the numerator (top number) is less than the denominator (bottom number), such as and , are known as proper fractions.
A mixed number consists of a whole number and a proper fraction. Consider .
Mixed numbers can be represented on a number line by dividing each unit interval into parts. For example, to mark on the number line, the interval from 2 to 3 can be divided into thirds.
Then, can be marked at the point one third of the way along the interval from 2 to 3.
Draw a number line from ^{−}2 to 5 in your math notebook. Plot the following fractions on the number line:
, , , and .
Which two whole numbers does lie between? Into how many parts do you want to break the distance between these whole numbers if you have fourths?
You can determine the position of a number by dividing the length between the two appropriate whole numbers on the scale into 2, 4, 8 or 5 equal parts (or pieces) as appropriate, and then plotting the fractions on it as shown below.
A fraction in which the numerator is greater than or equal to the denominator is known as an improper fraction. You can think of these as “top-heavy” fractions: , , and the like.
Mixed numbers, such as can be rewritten as improper fractions. You can imagine this in terms of pizzas. If you have two and one third pizzas, and cut each of the whole pizzas into thirds, then how many thirds of a pizza will you have?
In this case, the total number of thirds will be , so therefore .
You don’t want to have to draw out a picture each time you need to convert a mixed number into an improper fraction. Think about how we found the seven thirds. Because there were two whole parts broken into thirds, plus 1 extra third, we multiplied 2 by 3, then added 1. Since these are all thirds, we placed the 7 over 3 in fractional notation. We can work it out like this: .
This works for any mixed number that we need to convert into an improper fraction. So, the rule is:
Want to see another example before you jump in? Click on the “View document” link below.
You can also change improper fractions back into mixed numbers. For example, for , imagine you have some pizzas cut into eight equal slices (eighths), and that you have 17 slices. You know that eight slices make one whole pizza, and that two pizzas would be 16 slices. There would be one-eighth (one slice) left over. So, . You can also carry out the division:
Since eight goes into 17 at most two whole times, and there is one out of eight parts left over. This tell us .
Need a little refresher on long division? (Click on “View document”).
(a) Change the following mixed numbers into improper fractions.
If you are having trouble with this, did you try using a picture? To get started, look at the extra example offered in the example at the end of the last screen, or watch the pencast that is located in the same place.
(a)
(i) Using our “pizza math,” the calculation would look like this:
So, as there are four quarters in each whole, three wholes will give quarters. The one extra quarter makes 13 overall.
Thus, .
(ii) We could use the following shortcut to solve this one:
(iii) There are eight eighths in each whole, so seven wholes will give 56 eighths. The extra three eighths makes 59 overall, so the fraction is . Hence, .
(b) Change the following improper fractions into mixed numbers.
Have you thought about drawing pizzas that contain five slices each? What about performing long division?
(b)
(i) Since , 20 fifths will make up 4 wholes. This leaves an extra 3 fifths, so the fraction is . Thus, .
(ii) Let’s try using long division:
Since , 14 sevenths will make up two wholes. This leaves one seventh over, so the fraction is . Hence, .
(iii) Since , 16 quarters will make up four wholes. This leaves one quarter over, so the fraction is . Consequently, .
You can use the calculator to carry out calculations involving fractions. In the following exploration, you will learn how to use the calculator:
The calculator can be accessed on the left-hand side bar under Toolkit.
A fraction is one number divided by another, so to input a fraction on the calculator, use the division key: .
To enter the fraction one-half, the calculator sequence is .
Or, you can type on your keyboard, if you prefer.
Enter one-half, then click on the equals sign (or Enter on your keyboard). In the top window, you should see the fraction, followed by the decimal equivalent.
Try entering the following fractions and see what happens. Remember to clear each one by clicking the button before you enter the next fraction.
(a)
Enter the numerator (top), then the division sign, then the denominator (bottom), and finally the equals sign. When you reach the mixed number in part (e), enter it as 2 whole units plus the fraction part.
(b)
We say that is a repeating decimal, but it is shown on the calculator rounded to 14 decimal places. Why is the final digit 7 rather than 6? If you were asked to round this to 2 decimal places, what would your answer be?
Think before you read on.
Let’s round the displayed number. Remember from Unit 2, that when you are rounding a number, you look at the first digit that you want to cut off. If this digit is 5 or more, then round up the previous digit, which becomes the final digit in your answer. So, , when written as a decimal rounded to 2 decimal places, is 0.67.
(c)
can be reduced, so the calculator shows first, then the reduced form, , and finally, the decimal equivalent.
(d)
is an improper fraction. The calculator shows it as a mixed number before converting it to a rounded decimal.
(e)
means 2 with an extra added on, so enter it like this on the calculator:
The calculator can help you write a terminating decimal as a fraction too. Enter the decimal 0.4375 into the calculator, then click on . In the top screen you should see:
Use the calculator to convert the following decimal numbers to fractions:
(a) 0.65
Clear the previous calculation from the calculator, enter the decimal number and click on the equals sign.
(a)
(b) 7.36
(b)
The number is greater than 1, so the calculator displays it first as an improper fraction, then as a mixed number.
(c) 0.986
(c)
The calculator has its limitations. When the fractional equivalent of a decimal has a large denominator, the calculator gives an approximation. The sign ≈ means “is approximately equal to.”
rounded to 6 decimal places, so the fraction approximation for 0.986 given by the calculator is quite close to the exact answer. If you need an exact answer, you will need to work it out for yourself by remembering that 0.986 is the same as , and then reducing that fraction as far as possible.
If you feel that you need extra practice in converting fractions from one form to another, try making up some of your own examples and then check your answers on the calculator.
In this section, we are going to look at how we add and subtract numbers that include fractions.
The chocolate bar pictured above has 24 pieces. Each piece is of the bar, so the whole bar is , or 1. There are four identical rows, so each row is of the bar. There are also six identical columns, so each column is of the bar.
Now, imagine that you eat one third of the chocolate bar. That is eight out of the 24 pieces (because ). You are still feeling hungry five minutes later, so you decide to eat another five pieces. What fraction of the bar will you have eaten in total? What fraction of the bar is left?
The fraction eaten will be , but it’s not easy to quickly determine what fraction this is because the two fractions are out of different numbers of parts. However, if you recall that eating of the bar is the same as eating , the fraction eaten is . You can check this answer on the bar of chocolate: just count the number of squares that have been eaten!
Would you like to look at another example before you go on? (Click on “View document” below).
The fraction that is left will be . So, thinking of the whole bar as , the calculation becomes . You can check this by counting the number of squares that are left.
This method of changing the fractions into the same kind using equivalent fractions can be used for adding and subtracting fractions in general.
If you need to add or subtract fractions, the first question you need to ask yourself is “Are these the same kind of fractions?” That is, are they divided into the same number of parts? If the answer is no, then you need to convert the fractions into equivalent fractions with the same denominator in order to add or subtract them.
In the following exploration you’ll learn how to use the calculator to add and subtract fractions.
The calculator can be accessed on the left-hand side bar under the Toolkit.
Let’s try adding some fractions on the calculator.
Enter , then click , or press Enter on your keyboard. You should see the following:
So, adding one-fifth and two-fifths gives three-fifths, which should make sense.
Try the following additions and subtractions on the calculator.
(a)
Remember to clear the previous calculation before you start by clicking . Then, enter each calculation exactly as it is written. With complicated calculations like these, it is important to watch the top screen to make sure that the calculator is doing what you want it to do.
(a) Both fractions in this calculation are eighths, so you can add them directly, giving . Because the result is bigger than 1, it is given as a mixed number and then as a decimal number.
(b)
(b) You can subtract fractions, too. Both fractions are tenths, so subtracting
from gives , which reduces to by dividing both the numerator and the denominator by 2.
(c)
(c) This time the fractions have different denominators. The calculator has converted to twenty-fourths by multiplying top and bottom by 8.
This gives
The decimal form has been rounded here, so is not exact. For this reason, mathematicians often prefer to leave numbers as exact fractions rather than expressing them as rounded decimals.
(d)
(d) The calculator has converted all the fractions to twelfths, because twelve is the smallest number that 2, 3, and 4 all divide into evenly. The sum is written as . Now the fractions can be added, because they are all have the same denominator, giving the answer , or .
(e)
(e) This time, all the fractions are written as eighteenths, because 18 is the smallest number that 9, 3, and 2 divide evenly into. So, . The calculation in the numerator is . (You learned how to work with negative numbers in Unit 2.)
So, the answer is
Notice here that there is only room on the top screen for the calculator to display the first 8 decimal places of the decimal answer, but the answer correct to 14 decimal places is shown in the screen below.
The calculator can be accessed on the left-hand side bar under the Toolkit.
Try adding these mixed numbers on the calculator. Can you work out how to do this?
(a)
Enter each number as the integer (whole number) part plus the fraction part.
(a) You can rewrite the sum as . You may find it easy to add the whole numbers, then add the fraction parts separately, and then combine the results. However,there is no need to reorder.
Let’s enter the same numbers in the calculator without rearranging. You should see the following:
A final word of warning—the calculator does not always give the answer as a fraction.
(b) Try adding on the calculator.
(b) The calculator goes directly to the decimal form of the answer: .
But, remember from our earlier explorations that you can convert a decimal to a fraction by clicking on the equals sign or hitting Enter. Try this, and you should find that the calculator changes 0.5 back into a fraction:
Here are another few examples to make sure that adding and subtracting fractions works for you.
(c)
Give your answer as a mixed number.
(c) The calculator goes straight to the decimal form:
Click on the equals sign to convert back to a fraction.
To check the calculation without a calculator, write all the fractions as twenty-fourths, because this is the smallest number that 24, 6 and 3 all divide into. This gives , which can be reduced to .
(d)
Think hard about how you enter the second mixed number, ; remember that you want to subtract all of , that is both the 3 and the .
(b) When you enter the second mixed number, you need to subtract both parts. One way to do this is to put parentheses around the , to show that you are subtracting both parts.
The key sequence to achieve this would be:
The calculator shows:
Notice that the calculator has interpreted your entry as “subtract 3 and also subtract one-half.” This gives another way of entering the calculation.
You can choose the method that makes most sense to you but think carefully about what you subtract.
The calculator can be accessed on the left-hand side bar under the Toolkit.
In your math notebook, work out the following by hand. Then, check your results using the calculator.
(a)
Are both fractions out of the same number of parts? Can you relate this to the chocolate bar example?
(a) Both the given fractions are eighteenths, so they can be added together directly: . Don’t forget to reduce to , by dividing the numerator (top) and the denominator (bottom) by 6.
(b)
Can you find equivalent fractions for each given fraction that all share the same denominator? What number can be divided by both 6 and 7?
(b) This sum involves sixths and sevenths, which are different types of fraction. However, you can change both into forty-seconds, since both 6 and 7 evenly divide into 42. So, and . Thus, the sum is .
(c)
Try adding the whole numbers first, then add the fractional parts together.
(c) Here, you can add the whole numbers first, , and then add the fractions, but first you must convert each fraction into twenty-fourths. So, the sum is .
(d)
Imagine that you have an extra large pizza with 16 slices. After you eat one slice, what fraction of the pizza would be left? How much would be left after you eat 5 more slices?
(d) Both the fractions are sixteenths. You can imagine subtracting five slices of pizza from 15 slices will leave 10 slices out of the original 16: .
Think of our paper pizzas again:
(e)
This problem is very similar to the chocolate bar example. Try drawing a picture to help.
(e) This calculation is similar to the chocolate bar example. The problem is that we need both fractions to be out of the same number of parts—remember the 24 pieces of chocolate? So, to solve it the same way, the calculation becomes . Don’t forget to reduce your answer!
Now that you’ve warmed up, you must be ready for a challenge!
Whether you choose to be the bike or airplane, you will need to use your new skills to win this game. Good luck!
If you’d like another challenge after that, then here’s a very old one. What would happen if you added the following pattern of fractions together?
and so on?
Well, the answer is that the sum would get closer and closer to 1, but you would never get quite there! This is an example of an old mathematical puzzle called Zeno’s Paradox, about a race between the Greek warrior Achilles and a tortoise. It dates from around 450 bce.
Watch a short video clip and read a brief article on the paradox.
If you study more math, you’ll discover how to continue adding these fractions forever, and then their sum is 1!
[ Why do you think the authors used fractions of the group (rather than the numbers of people) to report their results? ] The next activity illustrates how fractions are used to report results from a survey.
In 2001, the New Policy Institute (whose mission is to advance social justice in a market economy) published a report titled “Young people, financial responsibilities and social exclusion.” The report summarized the results of a survey on 210 young people, two-thirds of whom lived in rooms at the YMCA. Of the young people who were surveyed, one-half were in debt and, of these, a one-third owed more than $1,600.
(a) How many of the people surveyed lived in rooms at the YMCA?
Two out of every three people surveyed lived in the YMCA. Into how many parts do you need to divide your group of people?
(a) One-third of the group indicates that the group is evenly divided into 3 parts, which is people. So, of the group will be two sets of 70 people each, which is 140 people.
(b) How many of the young people were in debt?
What’s the quickest way to find half of a given amount?
(b) If we want half of the group, that’s the same as dividing it by 2. So, , which means that 105 of the young people interviewed were in debt.
(c) Use your answer from part (b) to work out how many of these young people with debt owed more than $1,600.
If we want to find a third of a number, how many equal parts do we want to break it into? Notice the wording of the original survey results: One-third of the people who were in debt owed more than $1,600. So, it’s not one-third of all of the survey participants.
(c) One-third of the group of 105 people owed more than $1,600. Since , we know that 35 people owed more than $1,600.
(d) What fraction of the total number of young people surveyed owed more than $1,600?
How many people owe more than $1,600? You found this in part (c). How many people were interviewed in all?
(d) Since 210 people were surveyed, the fraction who owed more than $1,600 is , which reduces to . Hence one-third of one-half equals one-sixth.
Reference:
New Policy Unit (2001) Young People, Financial Responsibilities and Social Exclusion. Available online at http://www.npi.org.uk/ files/ New%20Policy%20Institute/ More%20than%20PIN.pdf [accessed 24 Sept 2011]
Part (d) of the activity on the previous page does not give a surprising result, mathematically speaking. One-third of one-half is indeed one-sixth. To convince yourself, take a piece of notebook paper. Fold it in half along the long side and shade one half of the paper. Then, fold it into thirds along the short side. The paper is now split into six equal pieces, or sixths. If you look at the shaded half, you can see that one out of three parts of this portion represents one out of six parts of the entire paper as shown below.
Mathematically, this is written as .
To multiply two fractions together, you must multiply the numerators together, then multiply the denominators together, and reduce the answer if necessary.
It is often easier to reduce before multiplying out. For example, let’s consider .
Multiplying the numerators together, then multiplying the denominators together gives , but we can cancel out the 2s using division: .
This calculation is equivalent to canceling at the start of the calculation:
The next page summarizes this procedure, followed by an exploration to help you learn how to use your calculator to aid in the multiplication of fractions.
Let's calculate .
First, multiply the numerators together, then multiply the denominators together: .
We write the answer as a mixed number, if appropriate. We usually do this if the original numbers were given as mixed numbers.
Here is a more detailed explanation (click on “View document”).
The calculator can be accessed on the left-hand side bar under the Toolkit.
In this exploration, you will learn how to multiply fractions on the calculator.
When you used the calculator to add fractions, you didn’t have to think too hard about entering them correctly, because the calculator follows the correct order of operations. It prioritizes division over addition, and the fractions are automatically entered as you intend.
More care is needed for multiplication of fractions, because division and multiplication have equal priority where order of operations is applied.
To be sure that the calculator is performing the calculation that you want it to, it is best to insert parentheses around each fraction.
So, to multiply , you must enter the calculation as on your calculator.
The proper key sequence is:
The calculator should show:
Note how the calculator automatically converts into before it displays the answer as a fraction and a decimal.
Try the next examples for yourself.
(a)
(a) Remember to put parentheses around the fractions.
By hand:
Using the calculator:
(b)
To do the calculation without the calculator, you need to change the mixed number into an improper fraction. You can enter the fraction as a mixed number on the calculator.
By hand:
Using the calculator, you can enter the mixed number as a sum. You need parentheses around the whole of to show that you are multiplying all of this by .
(Here, the top window of the calculator only displays the first eight of the 14 decimal places. The entire rounded output is shown in the window below.)
In the New Policy Institute survey introduced before, of the 210 young people had bank accounts. Of these, had overdraft protection and approximately had checkbooks.
(a) How many of the people surveyed had overdraft protection?
How many people from the group had a bank account? The word “of” often translates into multiplication and a whole number can be written as a fraction by placing it over 1. What fraction of those with bank accounts have overdraft protection?
(a) Let’s use our new knowledge of fraction multiplication. of 210 translates to .
So, we know that 126 people have bank accounts. One third of this group have overdraft protection, so .
Therefore, 42 of the people surveyed have overdraft protection.
(b) How many survey participants had checkbooks?
(b) We already know how many people have bank accounts: 126. We then need to find of this group.
We can calculate that people.
This fraction, it turns out, is an approximation. You need to interpret the results of your calculation carefully, particularly if it involves fractions of a person!
So, about 50 people have checkbooks.
In the previous activity, you might have approached the problem differently. Perhaps you found what one-fifth of the group was first by using division, and then used this portion to find three of those sets. Once you had this value, which indeed is 126 people, you could have then found the number of people with overdraft protection and checkbooks as done above.
Both approaches are valid and will give you the correct answers. Choose whichever method is easier for you.
Dividing fractions is a little more difficult, so we’ll try out two different approaches: one using your calculator so that you can get a feel for what’s happening, and another by considering a simple example first and seeing if we can translate that to a more complex question by applying a similar strategy.
In this exploration, you will learn how to use your calculator to divide fractions, and also discover a useful method for dividing fractions by hand.
The calculator can be accessed on the left-hand side bar under the Toolkit.
For this activity, suppose that you have three pizzas, and you divide each of them in half.
How many halves do you have? You can see that there are six halves in three pizzas. When you calculate , you can think of this as “How many halves are there in three?”
Try the calculation on the calculator. Remember to put in parentheses, because you are dividing three by the entire fraction . Do you get the answer six?
The key sequence is
Remember that a division sign is equivalent to a fraction bar, and this is how the calculator displays the calculation. The fraction bar under the three is larger than the one in the middle of the , indicating that you are dividing 3 by the fraction .
To show that the parentheses are not optional, enter the same calculation without parentheses.
The calculator follows the order of operations. There are two divisions, so the calculator works from the left and divides three by one, then the result by two.
Can you see how the length of the fraction bars shows this? Compare this result to your previous expression, where you put the into parentheses. Whenever you divide fractions, take great care to use parentheses around each fraction so that the calculator does as you intend.
Now try the following divisions. First, think about what the answer should be, and then try the calculation on the calculator.
(a)
Remember to clear the last calculation before you start, and don’t forget to put the fraction in parentheses.
(a) is asking how many thirds there are in three. Think about the pizzas; you should be able to see that there are nine thirds in three pizzas.
The calculator agrees.
(b)
Can you see a pattern in the calculations? Look at them again and see if you can describe what’s happening.
Look at the last three calculations. They asked you to divide 3 by , by , and 3 by .
How do you create the result from the numbers involved?
When you divide 3 by , you get 6, which is the same as multiplying 3 by 2.
When you divide 3 by , you get 9, which is the same as multiplying 3 by 3.
When you divide 3 by , you get 12, which is the same as multiplying 3 by 4.
Let’s investigate further. Try the following calculations, and make a note of the answers.
(a) and
(b) and
(c) and
In each case, you should obtain the same answer for the division as for the multiplication. For (a), you should get or for (b) or ; and for (c) the answer is 7.
Look again for a pattern. In each case, you start with 3. Look at the fraction that you divide by, and the fraction that you multiply by. What is the connection in each case?
In each case, the fraction that you multiply by is the fraction that you divide by turned upside down. In other words, the numerator and the denominator switch places.
This is the exciting part of mathematics, spotting patterns and then seeing whether they hold for other cases. We have a conjecture!
If you think you have spotted the pattern, then let's try to write it out. Completing the following sentence:
To divide by a fraction …
To divide by a fraction, switch the numerator and the denominator, then multiply.
There is another way to write this. When the numerator and the denominator trade places, the result is called the reciprocal of the original fraction. Dividing by a fraction is the same as multiplying by its reciprocal.
To convince yourself that this is true, try two additional examples on the calculator. Divide the first fraction by the second, and note your answer. Then, multiply the first fraction by the reciprocal of the second fraction.
(a)
Remember to put each fraction in parentheses. Watch the screen at the top as you input the calculation to make sure that you are entering it correctly.
(a) The calculator doesn’t enter the division very elegantly, but you can tell that it is correct because the middle fraction line is longer than the other two. This means that the whole of the first fraction is divided by the whole of the second fraction.
Multiplying by the reciprocal gives the same answer:
(b)
(b) The division results in:
Meanwhile, multiplication by the reciprocal gives:
From the calculator exercises on the previous page, you can see that to divide by a fraction, you convert the second fraction (the one you are dividing by) into its reciprocal, then multiply by that fraction instead. If you’re interested in finding out why this works, it will be covered later in a section titled “Division or multiplication? That is the question.”
Just once more, let’s consider . If you switch the numerator and the denominator of , you will have : this is the reciprocal of .
Thus, .
So, dividing a number by is the same as multiplying the number by .
This process of finding the reciprocal of the fraction that appears after the division sign, and then multiplying by that value, can be used in any division problem. If you’re interested in finding out why this works, it is covered in Unit 7.
The same principles apply to dividing mixed numbers as dividing proper fractions.
Let's try to calculate .
First, we know to rewrite the mixed numbers as improper fractions: .
Then, we find the reciprocal of the fraction after the ÷ sign, and change the division symbol to the multiplication symbol, so our new calculation is: .
Then, we cancel and multiply out: .
Because our original numbers were mixed numbers, it may be appropriate to rewrite our answer as a mixed number as well. In this case, our answer is equivalent to .
Do you see what happened there? We used an approach we already knew from before on a new type of problem, and arrived at the correct answer!
Here’s another example: .
Convert into an improper fraction: