4.11 SAT Math - Ratios / Rates / Proportions

Ratios

A ratio sets up a relationship between two groups.  Ratios can be written as fractions, as numbers separated by a colon, or as words: or 2 boys to 3 girls. 

-SAT Math Hint: If a ratio is given to you in any form other than a fraction, set it up as a fraction immediately, the first term going on top, the second on the bottom.
Some ratio problems will be simply asking you to set up that fraction and/or solve for what's asked:

3. If the ratio of a to b is 2 to 3, what is the ratio of 4a to 4b?

Simply set up the fractions:  and then solve for the next part:

More often, however, ratios will be a bit more complicated by giving you a different ratio than you need in order to solve.  What does that mean?  It means they'll give you the ratio of boys to girls in the class, and then ask for the total number of kids.  Or they'll give you the ratio of oranges to lemons to apples, and then ask for the number of citrus fruits.

These aren't that difficult to solve on the SAT exam as long as you remember that a bunch of parts added up equal a total, or that a total can be subtracted into a bunch of pieces.  Simple, right?

Try this one:

8.  The ratio of slices of pizza to sub sandwich pieces to buffalo wings at the football party was five to four to seven.  If there were 52 sub sandwich pieces at the party, how many other choices were available?

To solve, you'll need to add up the ratios to determine the total number of ratio pieces before you can convert those ratios into real numbers.

-SAT Math Hint: Ratios are not real numbers!!  Do not mix ratios with real numbers in one fraction.  Think of ratios as little bigots.  They don't like hanging out with real numbers, so don't mix them.  Keep them separated by an equals sign until it's time to solve.

So the total number of pieces the food is divided into for the ratio is 16.  Since they gave us sub sandwich information, that's what we'll use to set up our first equation.  What's the ratio of sub sandwiches to total food? .  So what's the real number of sub sandwiches to total food?  if f is our total food pieces.  Now we set it up with an equals sign:  and cross-multiply to solve. 4f = 52•16     f = 208   .  Are we done?  Nope.  Reread the bottom of the question.  They're not asking for total food, they're asking for other food choices, which means we need to subtract those sub sandwiches from the total food number we got because we don't want to include them.  So 208-52=156.  That's our answer.

Sometimes with ratios you'll be given two separate ratios and asked to get a third one.  This is a little easier than it looks.

14. If the ratio of y to z is 6 to 5 and the ratio of x to y is 7 to 3, what is the ratio of z to x?
The key to these is, again, to set them up properly.  First, let's set up our fractions:

What you're going to want to do is use the ability to cross cancel to get your third ratio.  Let's just work with the variable ratios first to demonstrate.  If we multiply them together, what happens?

The ys can cross cancel, right?  So we're left with



Now we'll do the same with the numbers given:


Are we done?  Reread the bottom of the question.  What do we have?

What are they asking for? z/x.  So just flip your answer over.  So the answer is 5/14.  But you can bet 14/5 will be an answer choice option, so watch yourself.  They're sneaky, kids.  Very sneaky.

Proportions

Proportions are another kind of ratio problem.  In a proportion, they're giving you two ratios that work together to ask you to solve.  In a word problem, look for a pair of things that gets repeated, things like "If it takes the watchmaker 4 hours to make one watch, how long will it take him to make 5 watches?" Hours and watches, hours and watches.  A pair of things repeated.

Proportions, like ratios, just need to be set up properly.  The numerators should both refer to one thing, the denominators to the other.  So in the above example, you should have it set up as .  Then solve for your variable.
-SAT Math Tip: Be careful of mixed units.  If one part of the question is given to you in feet and the other in inches, you will need to convert one to the other before you can continue.
7. Marta usually runs 12 miles in 2 hours.  If she has only has 30 minutes to run today, how many miles will she run?
First, set up your proportion:



This, unfortunately, is not a correct proportion as the denominators refer to different things.  So you have to convert one of the time units to the other before you can cross-multiply to solve.  It doesn't really matter which one you do, just do whichever you feel is easier.  We're going to convert hrs to minutes for this example.  So how many minutes are in an hour?  60.  So 60•2 = 120.  Our new, correct, proportion is:

 

-SAT Math Tip: Do NOT multiply the numerator by 60 also!  We're just writing the denominator in an equivalent term, not trying to get an equivalent fraction.  2 hrs and 120 minutes are the exact same thing, so we don't want to change anything else.

Now, cross-multiply to solve: 12•30 = 120•x     x = 3.

These kind of proportions are called direct proportions or direct variation.  They are the most common kind you will see.

There is another kind of proportion called indirect proportion.  It can also be called inverse proportion, indirect variation or inverse variation.  It's all the same thing.  These types of problems are based on the concept that sometimes things vary indirectly, that is, sometimes if you increase one thing the other thing decreases.  For example, say you hired someone to come clean your house.  By herself, it would take her 4 hours.  Say she brought 3 people to help.  A direct proportion would suggest that with 4 people it would take 16 hours, but that doesn't make any sense in real life, right?  Something like that should go faster with more people.  That's an indirect proportion.

Indirect proportions will be tested in word problems just like that.  If you get a word problem that you know to be a proportion, give yourself a second to figure out whether it's direct or indirect.  Here are some examples.  Which ones are direct and which are indirect?

A fast typist can type 100 words per minute.  How many minutes will it take to type 24,500 words?

A hose fills a bucket at a rate of 4 gallons per hour.  How long will it take a 180 gallon bucket to be filled if 3 identical hoses are used at the same time?

Farmer O'Brien has 2 sheep that can eat his pasture in 10 days.  If he buys 3 more sheep, how many days will it take the flock to eat the pasture?

A scale model is built of the new building at a scale of 1 inch represents 8 feet.  If the model is 7.25 inches tall, how tall will the real building be?

If you are given an indirect proportion in a word problem, you're going to set it up opposite to the way a regular proportion works.  So where a regular proportion could be written as , an indirect proportion will be set up as y1x1 = y2x2.  You're going to multiply the two things together rather than divide.  So looking at one of the examples above,
Farmer O'Brien has 2 sheep that can eat his pasture in 10 days.  If he buys 3 more sheep, how many days will it take the flock to eat the pasture?
you would set it up as 2•10 = 5•d (Why 5?  Because he bought 3 more sheep.  That means he now has five total.)  So the answer will be 4 days. 

-Hint: If you're not sure you set up the equation properly, that is, you thought it was direct but you're not sure, check your answer to see if it makes sense.  Does it make sense that more sheep would clear a pasture faster?  Yes it does.

[By the way, the answers to the above examples were direct, indirect, indirect, direct.]

Starting Score of 600 or Above:

Another way to think of direct/indirect variation is to define it as direct if y/x equals a constant k, or indirect if yx = k.  Sometimes, variation will be tested in this equation form, such as

18. If y varies directly as x/2, find x when k=-3 andy = 4.

Here, you will need to know the above definitions.  Simply put in the variance they gave  you into the proper equation, in this case direct:   Then solve with the numbers they gave you:

Rate

A rate is simply a ratio having to do with time.  "The car drove at a rate of 35 mph," "The runner ran at a rate of 1 mile per 8 minutes," "The sewing machine sewed 40 stitches per second."  The word "per" usually means "divided by," so simply put the thing done on top and the time on the bottom.  Rates are usually used as part of proportion questions, or in word problems needing the standard distance formula.

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