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If you have not completed Unit 6, it is recommended that you go back and finish that unit first.

This unit should take around three hours to complete. In this unit you will learn about:

- How fractions can be used in real life.
- Breaking a problem into small parts.
- Using sketches to make sense of a problem.

You will have further practice in:

- Adding, subtracting, multiplying, and dividing various types of fractions.
- Using a calculator for problems involving fractions.

In this unit, you will be using the ideas and methods from Unit 6. It is a shorter unit than Unit 6, but it might take you about the same time to work through, as some of the calculations are longer. You will be following steps to solve one particular real-life problem.

You will need to have a good knowledge of calculating with fractions from Unit 6—you will be able to use your calculator for most of these, but you should do simple checks to ensure that you haven’t made an error in using the calculator. Take your time, doing the activities in the unit carefully, and take breaks frequently. You will find that drawing your own diagrams will help you to make sense of the calculations. Talking to a family member or friend will be useful. Keep in mind that we all learn more from making mistakes and devising or finding methods to correct those mistakes.

To check your understanding of the basic techniques of manipulating fractions before you start, try the Unit 7 pre quiz, and then use the feedback to help you work through the unit. 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!

Even though fractions are a part of our daily lives, you may have found some of the examples and activities in Unit 6 quite challenging. As you discovered in Unit 1, everyone gets stuck at some point when they are doing math, so it is important to have strategies for tackling problems and getting yourself going again. Remember that discussing the problem with someone else or taking a break can help. Often, drawing a diagram or thinking about the problem realistically might prove useful, too.

This section is designed to direct you through a challenging problem and illustrate strategies that can help make an exercise that contains fractions easier to tackle. You are not expected to complete this problem from scratch, but rather work through it as presented, in pieces, with guidance. Be sure to pay attention to the strategies used, as well as keep track of any difficulties you run into. Here’s the example of how fractions can be used in a practical problem.

Suppose you find an old photograph that you would like to hang in your house. The dimensions of the photograph are unique. You know custom framing and matting can be quite expensive, so you plan to do the work yourself. The photo will be framed by “matting” a piece of rigid cardstock material with a rectangular window cut out of it for the photo to show through. Then, you want to place the matted photo into a frame.

Note: The dimensions of this picture **do not** match those of the
photograph you want to frame.

Matting is sold in standard sizes, so you will need to purchase a big enough piece and trim it down to the desired size. You will also need to cut an opening of the right size and shape to correctly expose the photograph. Next, you will mount the photo to the matting so that it is exposed through the “window.” The matting (with the enclosed photo) will then be placed in the frame. (For the second part of the problem, you can think of the matting as the “new” photo.) In other words, once you have the matting properly sized, you will need to purchase a frame that fits the matting rather than the photo. The photograph above shows a photograph that has been matted and framed, but with very different dimensions and proportions than the one you will be calculating in this exercise.

How much of the matting will need to be cut off and discarded? What are the overall dimensions of the framed picture?

The first step in any mathematical situation is to check that you understand the problem. Can you explain it in your own words? Here, we need to work with the dimensions of the photograph to determine how to properly calculate the size of the matting and frame it.

It is often helpful to write down what you know and what you want to find. You should include anything that you think might be relevant. After measuring the picture, you do some research on photo framing and purchasing the matting, with the following results:

Do you need any further information? What is it?

The matting is almost like another frame that goes on top of the picture. You will need to cut a rectangle in the matting for the photograph to show through. What other dimensions would you need to know to determine how to reduce the matting? In order to determine the size of the frame you should buy, you will need to know the final overall dimensions of the mat.

You still need to know how much of the photo should be exposed through the window of the matting. Based on the composition of the picture, you decide that it would look best if inches by inches of the photo was exposed by the matting.

Additionally, you want the matting to create a -inch border around the entire picture.

You might realize later that you need more information. If this happens, then you just add it to your list from above. Problem solving is a work in progress.

A sketch (not to scale) containing the above measurements would be useful at this stage, particularly if you are a visual learner.

This is not drawn to scale, and this sketch is a printed one so that the measurements are clear—you just need to do a rough sketch—it's the measurements you write on the sketch that are important.

This is an involved problem because there are many fractions floating around, so a useful strategy here would be to break it down into more manageable chunks by concentrating on one piece at a time. There are two separate problems.

- How will you cut down the matting to achieve the appropriate border and expose the desired amount of the photo?
- What will the final dimensions of the framed picture be? How much space do you need to have available to hang it?

The first problem will require addition and subtraction of fractions, based on the size of the matting, as well as the amount of the photograph you plan to reveal. The second problem relies on the answer you get from the first question.

Let’s focus on each of these questions one at a time. Be sure to follow along. It may be helpful to include sketches (no artistic talent required) along with any of the math.

How will you cut down the matting to achieve the appropriate border and expose the desired portion of the photo?

We know that we want to cut a -inch by -inch window in the matting to reveal the picture. Since you also want a particular size border, you can start by measuring inches from the top, and inches from the left of the matting. At this point, another sketch would be beneficial:

Because you know how much of the picture you want revealed, you could start by measuring the top and left border, then measure inches down, and inches across, then add the right and bottom borders, making a rectangle which will provide the proper “window” for the photo.

It would be useful at this stage to calculate the actual width and length of the matting you need for the photo. This can then be used as a check before you cut the matting.

In order to figure out how much the matting was reduced by, the dimensions of the final matting must be determined.

The width would require the matting border on the top and bottom plus the height of the cut out window to be added. If you haven’t done so already, make a sketch of the piece of matting you are now working with, or use the blue shaded matting sketch on the previous page to follow along.

You can check by using the calculator.

The calculator can be accessed on the left-hand side bar under Toolkit.

The width of the photo plus the borders is inches:

After reducing it, the matting is inches in width.

Now that we have found the width of the matting, we still need to determine the length.

Use the same methods as the previous activity to find the length of the matting.

We will use the same method to find the length of the picture. Add the matting border on both sides to the length of the cut out window. Use a sketch to follow along, as you did before.

You can check by using the calculator.

The calculator can be accessed on the left-hand side bar under Toolkit.

The length of the photo plus the borders is inches:

After reducing it, the matting is inches in length. Therefore, the measurements are inches by inches.

It might be difficult to find a ruler that measures thirds, but because two thirds is close to seven tenths, you could use this as an approximation. There are rulers that measure tenths, and you can find printable versions on the Internet, such as the ones on this web page, where the ruler at the far left is marked in tenths.

The border on the right and bottom are too large. To finish cutting down the matting, you could measure inches down from the bottom and to the right of the inscribed rectangle you just cut out.

Before you do any cutting, you should check that the width of matting you propose to cut off is close to inches and the length is close to inches.

The original question asked how much the matting is reduced by. Using the sketch above, we can see the shaded area shows the matting after it has been cut to your specifications. You have already used your skills on adding fractions, from Section 6.2, to calculate the dimensions of the matting you need.

You can now use the skills on subtracting fractions, also from Section 6.2, to calculate how much of the matting is left.

If there are enough scraps left over, then you might find another arts and crafts project to do with them. Let's try to figure out how much of the matting is left over. Don’t worry, there won't be any more crafts today!

How much will be cut off from the bottom of the matting?

We know the original and final dimensions of the matting. The width of the original matting was inches and its length was inches; the width of matting we require is inches and the length we require is inches. What operation could we use now to determine how much was trimmed away from the bottom edge of the matting? Think subtraction!

The original width of the matting was inches and we need the width to be inches. So we need to cut off a piece from the bottom that is inches wide. Here is the calculation:

So, you trim inches from the bottom of the matting.

You could also find the solution using the calculator.

The calculator can be accessed on the left-hand side bar under Toolkit.

The original length of the matting was inches and we need the length to be inches. So we need to cut off a piece from the bottom that is inches wide.

Here is the calculation:

Checking with the calculator:

So, you trim inches from the right side of the matting.

There is quite a bit left over, so be sure to place the leftovers in your scrapbooking pile!

You have done a lot of math in this section, so now is probably time for a break before we cut out the middle of the matting for the photo.

This is the diagram from Section 7.1.2, which also shows the dimensions of the matting cut down to the required size and the size and position of the photograph.

The easiest way would be to use a pencil and ruler to mark the border of inches from the top, the bottom and the sides of the matting; then measure the width and length of the rectangular hole to ensure that it is very close to the inches width and the inches length you want for the photograph.

Now you are ready to cut out the hole. If you are careful, and use a craft knife and a ruler, you will have a rectangular piece of matting that you can use later for a smaller photograph!

Your last step is to buy the frame to put the photograph and matting in. You must now find the final size of the framed photo.

Did you start with the final dimensions of the matting? Did you remember to include the width of the frame?

Looking back at the information you collected, you chose the border of the frame to be of an inch all around. Thus, we must add to each side of the length of the matting.

The height we need for the frame is:

Check using your calculator:

The calculator can be accessed on the left-hand side bar under Toolkit.

Then, add to the top and bottom of the matting.

The length we need for the frame is: .

Check using your calculator:

So, we found that the overall dimensions of the framed photograph are approximately 17 inches by 29 inches. Now you can decide where you have the space to hang up your beautiful picture.

In order to decide where to hang up the picture, you need a space that is at least 17 inches in height and 29 inches in length.

When you complete a problem, you should always look back over your work. In this example, it is especially important, because you don’t want to incorrectly cut the matting. That could have cost you more money!

You should also consider which techniques you may be able to use in future problems. Here, breaking the problem down into small steps, using sketches, and using the real-world context of the problem helped. Perhaps you even could attempt to carry out this project in your home.

Another important strategy is discussing the problem with someone else. By explaining how you have approached the problem in your own words, or having another person provide insight and ask a question or two, you might clarify your thinking enough to find a way to move forward in your work.

Now you just have to decide in which room to hang up your old photograph. Lucky for you, you've already done the math!

In this section, we will try to deepen our understanding of the mathematical content that was discussed throughout the unit. If you find a problem difficult, feel free to discuss it with a friend. Don’t panic; just keep going.

Don’t forget that the idea of this section is to expose you to more math and to get you thinking about math in your daily life. If you believe that you don’t have the time to spend further exploring these topics, this is a section that you could treat as optional.

Now, get ready to practice the art of paper folding!

Let’s go back to the paper folding activity in Section 1 of Unit 6. If you still have your sheet of paper, then use it. Misplaced it? Don’t worry—just pull out a new sheet of paper.

How many times do you think you can fold a piece of paper in half?

Try it using your piece of paper. Really, it’s pretty fascinating! You can then try it using a tissue to compare your results.

In theory, you should be able to keep folding your paper in half forever. However, as you’ve just experienced, you eventually had to stop, because you weren’t able to fold it in half any more. This is due to the thickness of the paper. Most types of paper get stuck at seven folds. What about yours? Do you think a larger piece of paper could be folded more?

This video investigates it.

Interactive feature not available in single page view (see it in standard view).

In this unit, you might have found some portions easier than others. It’s not uncommon to get stuck on difficult problems.

The more you practice, the more your skills improve. It will help to work through the exercises for each section provided below.

160 people joined a new Fitness Center on the first day it opened. They were asked about the main reasons they had decided to join the center.

A quarter of the group wanted to take advantage of the discounted payment for those joining on the first day, three-eighths were advised to join by their physician and one-fifth were motivated by watching the Olympics.

(a) How many people had some other reason for joining?

(a) A quarter of 160 is .

So, 40 people joined on the first day to take advantage of the discount.

One-eighth of 160 is 20, so three-eighths of 160 is .

So, 60 people were advised to join the center by their physician.

One-fifth of 160 is .

So, 32 people were motivated by watching the Olympics.

Therefore, the total number of people who chose one of the three reasons is .

The number of people who gave some other reason was .

(b) What fraction of the people who joined gave some other reason for joining the center?

Take your solution from part (a), and express it as a fraction. You should now have .

This fraction can then be reduced by dividing each term by 4:

So, of the people joining gave some other reason.

Alternatively, you could have worked out the fraction of students that gave some other reason as follows:

Check by calculating

(You could also have used your calculator to work out these fractions.)

A recipe for an iced cake requires pounds of icing.

Two-thirds of the icing is to go on the sides of the cake, with one-third on the top. How much icing should be reserved for the sides of the cake? Give your answer in ounces.

There are several ways you can do this calculation. For example, you can calculate two thirds of as follows:

There are 16 ounces in a pound, so pounds is

So, 24 ounces should be reserved for the sides of the cake.

Alternatively, you might have converted the pounds to ounces. As there are 16 ounces in one pound, there are = = 36 ounces in pounds.

A Walmart employee is a shelf stacker. His time for stacking shelves is 50 minutes for an average set of shelves. If he is at work for 9 hours, and he has a 40 minute break in that time, how many sets of shelves will he be able to fill?

You need to calculate first how many hours he is working, and then divide that by how long it takes him to complete one set of shelves.

Alternatively, you could convert the 9 hours to minutes: minutes. So he is in the store for 540 minutes.

He has a break of 40 minutes, leaving his working time minutes.

It takes him 50 minutes to stack a set of shelves, so the number of shelves he can stack is sets of shelves.

Unit fractions are fractions which have a numerator of 1, for example and .

The Ancient Egyptians used only the fraction and unit fractions.

For example, could be written as + + .

(a) Can you express and as the sum of unit fractions? How did you work these out?

One way to work these out is to use a diagram split into quarters or eighths, with the required fraction shaded.

Alternatively, you can think about the problem practically: means '3 divided by 4'. So imagine three bars of chocolate being shared equally among four people. If you broke all the bars in half and gave one half to each person, there would still be two half bars left. These could be broken in half again, and a quarter given to each person. So each person would get a half and a quarter.

You can check by using the calculator.

The calculator can be accessed on the left-hand side bar under Toolkit.

(b) What is the answer when you add the unit fractions , and ?

(b) .

Did this answer surprise you?

(c) (More challenging) The number 60 was very important to the Ancient Egyptians (in a similar way to how the number 10 is important in the modern world). Can you use fractions with the denominator of 60 to find a sum of unit fractions which represent the fraction ?

(c) This is one answer:

You can check this answer and the one you got by using the calculator.

The calculator can be accessed on the left-hand side bar under Toolkit.

You might find it interesting to look at Egyptian Fractions on the internet.

Now that you have taken the time to work through these sections, try this short quiz! You may find that it will help you to monitor your progress, particularly if you took the quiz at the start of the unit as well.

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 more 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 to come back and review this unit!

Read through the list on the next page and think over all the work you have done in this unit. If there is a checkpoint that doesn’t seem familiar, skim your notes to jog your memory. Remember, your mathematical skills will develop and grow stronger over time. Just keep working at it!

You should feel confident to:

- Write numbers as fractions and decimals.
- Perform calculations with fractions and decimals.
- Understand how fractions are used in everyday life.
- Convert between fractions and decimals.
- Apply fractions and decimals to given scenarios.
- Revisit some of the strategies that help you to solve problems.
- Continue to appreciate how to write good mathematics.

Well done! Give yourself a math medal!

Let’s move onto Unit 8 and explore a few more ways to write numbers. You will work with percentages and ratios to improve your understanding of how numbers are used in communication and the business world. Click here for Unit 8.