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Time to look at textbook problems!
§5.1 is about writing ratios. There is not much to say about writing one number on top of another. But there a few things the textbook does to make this trivial task somewhat interesting. We'll look at three such complications.
Consider problem §5.1 # 21 (on page 291). Here the question asks for two ratios.
A question that asks for two ratios is a bit strange, but is nothing difficult. When making a ratio either order is allowed, and this problem asks for both. This time both pieces of information are provided.
Now look at problem §5.1 # 23.
Remember that we seldom simplify ratios. But we can play that game if the textbook asks us to do so, pretending the ratio is a normal fraction.
Now look at problem §5.1 # 30.
How do we simplify a fraction made of decimals?
(Seeing a fraction made of decimals poses a question. Do we need to review how to do long division with decimals? This is very similar to what we just did!)
The most common thing to do with a proportion is play a game involving "multiply in an x shape". Most of my math students have seen this already and are good at doing it. But we should still discuss the technique.
Consider these two pictures. In each, two ratios claim to be equal. But only the top problem's equality is true. The pictures claim you can check if ratios are equal by multiplying in an x shape.
Why does this trick work? In a group, think about pieces of pie until you develop an explanation that involves putting together 2 and 10, and 5 and 4.
What is happening when we check if two-fourths is equal to five-tenths by multiplying in an x shape? Why are we putting 2 and 10 together? Or 5 and 4 together?
Here is one explanation.
If I had 2 pieces that were one-fourths, and cut each into 10 parts, I would have 20 pieces that were one-fortieths.
Similarly, if I had 5 pieces that were one-tenths, and cut each into 4 parts, I would also have 20 pieces that were one-fortieths.
Both cutting processes give me 20 pieces that are one-fortieths.
In other words, the "multiply in an x shape" trick is simply telling you to find common denominators using the brute force method. To make this obvious, let's do side-by-side with 2/4 and 5/10 the "multiply in an x shape" game and the brute force method of finding common demoninators.
On pages 300-301 we can see the textbook's example problems for checking if ratios are equal.
Two ratios are proportional if they are equal.
The word "proportional" is just a fancy new term for the old concepts of "equal" or "equivalent fractions".
Let's do a problem about checking if a proportion is true, in which all of the numbers are either whole numbers or decimals.
Strangely, we need to do the "multiply in an x shape" trick twice if both diagonals of the "x shape" include fraction arithmetic.
Chapter 5 Test, Problem 10: Check if 7/8 is proportional to 63/72
Chapter 5 Test, Problem 11: Check if 1.3/3.4 is proportional to 5.6/15.2
Notice that most word problem situations allow more than one way to write the proportion! We'll examine this later in more detail.
Checking if the ratios in a potential proportion are really equal is only slightly interesting. Much more interesting is when we are told three of the four values in a proportion and must solve for the missing value. We still use the "multiply in an x shape" game. The process does not change if the ratios include decimals or mixed numbers.
On pages 301-303 we see that the technique involving cross multiplying also works if we only know three of the four numbers in the proportion and are solving for x. (Note that #47-58 involve fractions of fractions, which looks weird but works the same way as everything else.)
For now, just one more example problem.
There is an important warning about the "multiply in an x shape" game. The following warning is only for students who have been taught a certain "shortcut", who have been taught to include division with the multiplying. If you have not been taught this "shortcut" then the warning will not make sense. Please ignore it! It is not for you.
Some students know a supposed shortcut that allows solving for x in one step: multiply diagonally and then divide by the other number. It may seem faster to do this than to always write out the "multiply in an x shape" step.
Let us solve 8/12 = y/9 both ways to compare the differences.
You are advised to not use this shortcut! If the problem was even slightly harder the shortcut would hide options about how multiple ways to solve the problem. Don't build bad habits that will cause trouble in later classes.
Consider 8/3 = (y + ⅓)/2.
When we multiply in an x shape we get 8 × 2 = 3 × (y + ⅓)
We could change that into either 16 = 3 × (y + ⅓) or 16 = 3y + 1.
The options lead to different natural next steps. The "shortcut" always picks the first option. So the habit of always solving proportions using the shortcut will later on force your to follow one path (which might be the hard one) instead of noticing both options.
This is why in this class we clearly define:
Cross multiplying is the "multiply in an x shape" step for dealing with a proportion.
Cross multiplying is often (but not always) followed by a step involving division. Some textbooks and colleges combine everything into "cross multiplying", but this might lead to bad habits. (Our textbook uses the term "cross products".)
For now, work on good habits. Approach proportions by writing three steps.
Let's do more examples. Pick a homework problem on pages 304-305 and notice how we use the three steps
Chapter 5 Test, Problem 12: Solve: 9/4 = 27/x
Chapter 5 Test, Problem 13: Solve: 150/2.5 = x/6
Chapter 5 Test, Problem 14: Solve: x/100 = 27/64
Chapter 5 Test, Problem 15: Solve: 68/y = 17/25
Just as the previous sections involving checking the correctness of proportions before we tried to solve for the missing value with proportions, now we will check if word problems are correctly put into proportions before we try to solve any word problems.
We begin by examining the patterns that differentiate correct and incorrect proportions. Below are four situations, each involving a pair of events. For each situation four possible proportions are listed. In groups, use cross-multiplying to check which proportions are correct. When most groups are done we will discuss what patterns people found.
The pattern your group should have found was that the two events needs to be "kept together" symmetrically, either vertically or horizontally.
If the two events are spread out upside down compared to each other then the proportion will not be correct.
If the two events are spread out diagonally then the proportion will not be correct.
The pattern your group should have found while checking if a setup was correct is that the two events needs to be "kept together" symmetrically.
Most students remember this with the rule The labels on the right must match the labels on the left—do not flip them!
Here is a "working backwards" question: what are some word problems that could be built from a proportion? First add labels to make the ratios into rates. Then compose the word problem question.
Create a word problem from each of these proportions:
Here are two more examples of proportion word problems. As we solve them look for the symmetry we just discussed.
Suzie can read 12 pages in sixteen minutes. How many pages can she read in five hours?
Scott can do three test problems in eleven minutes. How long would it take him to finish a test with twenty problems?
Recall our six-step problem solving process.
Let's augment this list of six steps to include the details of solving a proportion word problem.
All we have to do is fit in the three steps for solving proportions!
Recall that we usually write two lines for the cross multiplying step. The second line is where we multiply within the side of the equation that has two numbers and no letter.
Let's do some examples from §5.4.
Martha drinks 5 cups of coffee every 4 days. How many cups of coffee is this per year?
A few problems in §5.4 are really tricky. These are the "catch and release" problems. Everyone's natural intuition about labels for rates is of absolutely no help in creating "symmetrical" labels for the two rates in these proportions. So don't feel bad that these are hard. They are tricky for everyone. Let's look at one example.
Example 14 (§5.4 # 29)
Chapter 5 Test, Problem 16: An ocean liner traveled 432 kilometers in 12 hours. At this rate, how far would it travel in 42 hours?
Chapter 5 Test, Problem 17: A watch loses 2 minutes in 10 hours. At this rate, how much will it lose in 24 hours?
Chapter 5 Test, Problem 18: On a map, 3 inches represents 225 miles. If two cities are 7 inches apart on the map, how far are they apart in reality?
Chapter 5 Test, Problem 21: A grocery store special sells ingredients for a traditional turkey dinner for eight people for $33.81. How much should it cost if that deal applied to a dinner for fourteen people?
Knowing how to make any value grow or shink by a certain amount is called scaling. That certain amount is called the scale factor.
Problems about scaling can be solved using arithmetic. Always multiply or divide by the scale factor. Problems about scaling are never simple addition or subtraction problems.
Problems about scaling can also be solved using a proportion. This is the safe way, because the proportion tells you where to multiply and divide.
Let's consider three examples. They are described only using arithmetic. How could we do them with a proportion?
In this example the scale factor is provided.
An elementary school student is doing a measurement project at the playground when she notices something surprising. At that time, every shadow is three times the height of its object. If she is 4' 2" tall, how long is her shadow?
The scale factor is 3.
4' 2" × 3 = 12' 6" long
Notice we did not need to "carry" from inches to feet. What if we did? How long is the shadow of her 5' 8" teacher?
In the next example the scale factor is not provided. It is a fact that everyone is assumed to know.
Tom spends $2 each day on coffee. Theta spends $3 each day on coffee. How much does each spend per year on coffee?
The scale factor is 365 because there are 365 days in a year.
Tom: $2 × 365 = $730 per year
Theta: $3 × 365 = $1,095 per year
In the next example we must find the scale factor before dividing.
A web page designer needs to use an image that is 1,600 pixels wide and 1,024 pixels high. He needs to shrink the image so it is 500 pixels wide. What is the proportionally shrunken height?
First find scale factor.
old width ÷ new width = 1,600 pixels ÷ 500 pixels = 3.2
Then apply the scale factor to the height.
1,024 pixels ÷ 3.2 = 320 pixels high
Each time we used division, do you see why that was the right thing to do?
(Later we might consider an alternative solution that uses a proportion.)
Notice that the first situation involved scaling because there were two objects (the girl and her shadow). The second situation involved scaling because something was measured differently (moving from a daily to annual amount). The third situation involved scaling because an object changed (its size shrunk).
Some applications involve two "scale factors". For example, you will learn that simple interest problems involve scaling the principal up by both the interest rate and the time.