I'm taking a class after school to help me on the standardized tests, especially the SAT which I'm going to take next year, and there's one math topic that seems to be in more questions than any other topic even though I don't remember learning much about it in school. I'm talking about ratios. If you wouldn't mind, I have two questions for you. First: Why are they considered important? Second: Can you explain what I need to know about them because they look easy but I get confused with them and get a lot of them wrong, even when I'm almost certain I finally got the idea?
Sincerely, Robin B,
High School student breaking out in ratios!
I'm impressed! Your question shows you've noticed something I've long found surprising: I believe that when one pages through most of the popular textbooks and sees how much space is devoted to each topic, almost always ratios are the topic that is more tested than taught. I've discussed this with a range of people involved in the field of curriculum and testing, and there seems to be no clear consensus on why this is the case. I suspect that most adults (including teachers) think the topic is simple, even easier than it truly is, or at least simpler than it can be to a young person with little real world practice using different kinds of math at once. With ratios, you might have to do some multiplying, dividing, problem-solving, estimating with percents and decimals, and demonstrate an understanding of fractions with a flexibility that takes practice. All in a single question!
Anyway, your second question is even more interesting: How should one think when a ratio comes up?
Here's my answer: Think of shrink-wrapped packages.
"What?" You might say, "Shrink what?"
Shrink wrapping is the thin, transparent plastic that manufacturers and packagers use to seal groups of products into a single bunch so they cannot be separated and have to be purchased together. Think: you'd like to buy a certain battery, say a triple A, and you only need one for a clock you just bought. But when you get to the store, (most stores, anyway,) you find that it is difficult to come across a single packaged battery that you can buy on its own. Instead, your choices are a pack of 2 or a pack of 4 or a pack of 6 or a pack of 8 or a pack of 12, etc. And they are in a cardboard package with that clear shrink wrapping across the front: So you can see the batteries but not separate them.
An example of how shrink-wrapping can require a buyer to purchase more than he wants occurred to me when I wanted a certain color tube of paint, and the art supply store was closed. I was able to get the paint at a department store, but I had to buy it in a shrink-wrapped set with a full set of 8- 12 other colors. That is shrink-wrapping. Okay, back to ratios. The nice thing about ratios as test questions is that they tend to be fairly easy to spot. They are almost always presented in one of these four forms:
3 to 4
'For every three ____, there are four ___."
So regardless of the words or form, all of these mean the same thing. There are two different kinds of things, and they exist in a relationship: for every group of 7 of the two items, there will be three of the first and four of the second.
So if a man has only red neckties and blue neckties, and he has them in the red to blue ratio of 3 to 4, then if you organize them to help see the pattern: out of every seven you will find three red ties and 4 blue ties.
Note: A ratio of 3/4 red to blue is NOT the same thing as 3/4 red and 1/4 BLUE. NO,
IT IS NOT.
You have to do something to it to turn it into something you can use to solve problems and that would be a real fraction.
And that's how students get confused: They misunderstand the fractional look of the 3/4 math symbolism. Here is what you do: You think of the numbers 3 and 4 as shrink-wrapped together, so even if you only want one of them you must buy seven things like this: in a 3:4 ratio with three of the first item mentioned and four of the second item mentioned. When it's given to you in that fractional-looking form, you add the top of the "fraction" and the bottom of the "fraction" (though never forget that a ratio depicted in that form is not really a true fraction). And when you get that sum of the top and bottom numbers, it is to be used as a new bottom, the total quantity of items in one shrink-wrapped package, and if it is presented in one of the other forms, you still just add the two numbers together in a ratio of 3/4 as follows:
3items+4items = 7items wrapped together in a total pack of 7, three of which are red and four of which are blue. .
7? What?? Why??
At this point it is helpful to draw a box-chart with the number of each color's representation in the package written over the seven:
Seven what?? Well, the order of the words in a ratio is extremely important. EXTREMELY. Because the ratio will be read in the order of the words, top to bottom of the fractional style,
"3 to 4, red to blue" or "3 red to 4 blue" can be written as 3/4 , red to blue, meaning there are 3 red ties and 4 blue ties sealed into each pack of 7 ties (if organized that way); and, therefore, 4 ties will be blue out of every 7 ties. Therefore, there would be 3/7 red and 4/7 blue.
Remember to try to think of them as though they were shrink-wrapped in groups of seven...
3/7 of the ties will be red, 4/7 will be blue.
So if the man with only red ties and blue ties has seven hundred ties altogether, you do this: Draw the chart:
You figure out what 3/7 of 700 equals, to get the number of red ties in the collection.
3/7 x 700
(of means times, and times means of)
(So: 3/7 x 700/1 = (After you notice that seven goes into seven once,
canceling that out, and seven goes into 700, 100 times, reducing that 700 to 100.)
3 x 100
So, the man has 300 red ties in his drawer.
And you do 4/7 of 700 to figure out the number of blue ties.
(remember, 'of' means 'times')
So, 4/7 of 700 =
Okay, so the two most common types of ratio questions involve being given the ratio, such as 6:7 green fish to orange fish, and being given the total number of fish, say, for example 39, and being asked to figure out how many of the fish are green or orange.
Each sealed pack looks like this:
(Six green fish and seven orange fish.)
Equals thirteen all together.
So, if there are 39 all together, as the question states there is, then there must be three packets of 13.
(Because 13 + 13 +13 = 39)
And now it is easy to SEE that there are 6G + 6G + 6G = 18 Green fish
And 7 Orange fish + 7 Orange fish + seven Orange fish = 21 Orange fish.
Checking our work: 18 + 21 = 39. YES!
If a collection of feathers has exactly seven white feathers to every two blue feathers, and there are no other feathers in the collection besides blue feathers and white feathers, how many feathers altogether COULD the collection have?
Think: They are arranged in shrink-wrapped packages of 7 +2, or 9.
So the answer has to be divisible by nine with no remainders.
You can either divide the choices by nine to see which one works OR, YOU CAN USE A VERY USEFUL TRICK to help you spot it in a second or two. Numbers, of course, form all kinds of patterns, and knowing just a little bit about some of those patterns can lead to all kinds of easier ways to figure problems out than most people are ever taught. So, in this case, for example, it is very helpful to know that any number that is divisible by nine has a nine or a multiple of nine as the sum of its digits.
In other words, if you look at 330, and add up those three digits, the first three, the second three, and the zero, you get a sum of 6. 6 is NOT divisible by nine, so neither is 330.
Next you might try 331. 3+3+1= 7, which is not divisible by nine.
Next try 333. 3+3+3 = 9. 9 is divisible by 9... BINGO!
And, of course you can check the other two choices to make sure that they do not work.
Again, those seem to be the two most common types that appear on tests, and they appear a lot, and once you get a little practice with ratios, they become quite easy to get correct. (And even easy to check by plugging your answer back into the question and seeing if it works out,)
BUT, before saying goodbye, I should also give you a little warning: There ARE LOTS of different kinds of questions that can be tricky, and are made even trickier because they LOOK just like the same old basic ratio questions we covered. BE CAREFUL because some people who write questions for the big important tests you mentioned find ratio questions to be such fun to write that they then double up their own fun by getting creative and trying anything they can to come up with new twists on the basic questions, AND THOSE TRICKS ARE ALMOST INVISIBLE, or at least hard to spot unless you are paying AWESOME attention.
As you might guess, I collect test questions from around the world, and, like anyone who collects things they enjoy, I, of course, have my favorites. Many of the questions I consider among my 'favorites' have one thing in common: they look so simple and easy that almost every student taking the test is tempted to just try to figure it out in their head, not even bothering to use any of the methods they do know, which would have made the question almost as simple as it appears to be. Instead, though, it seems easy enough to bang out or 'crunch' in your head. And you do arrive at an answer. The trouble, of course, is that that excellent answer is wrong.
Yes my virtual friend, it's wrong.
Here's comes a question like that from a Chinese math test that is given to all (or almost all) of that country's sixth graders. Now, don't get thrown by the fact that it's a ratio of three numbers. It's solved very similarly to any of the 2-number ratios we did so far, only now you add all three numbers together to get the total in the "package". Here it is:
Yahn, David and Nathan shared $900 in the ratio of 1: 2 : 6. How much more money would Yahn receive if the ratio was changed to 2 : 3 : 5?
And so, even doing the first step in your head, you see that the numbers are chosen to make your work and your answer neat and easy: $900 is divided up among three people, yes, but it is also divided into 9 parts, or bundles. Nine? Where'd that number come from?
Here's where: Recall, you add up all the numbers set out in the ratio to find out how many are shrink-wrapped together to form one 'sealed ratio'. So here we have 1 + 2 + 6.
So, you add them up to form the bottom of the ratio's fractions that you would use to find out how many red ties, or blue ties (or miniature boxes of cereal -- or whatever the items at issue happen to be). So, 9 parts for $900, means each part is worth $100. But, notice after the change, the new ratio is 2:3:5, which adds up to 10 parts.
So, the same $900 is now to be divided into 10 parts.
900 divided by 10 = 90, which means that each part equals $90. So, when it was
Y D N
1: 2 : 6
Yahn received $100 x 1 = $100.
But now, well, I'm going to leave it up to you to complete!
Hope that helps,