1. LA EDUCACIÓN ESPECIAL EN LA HISTORIA RECIENTE DE ESPAÑA
1.1. LA EDUCACIÓN ESPECIAL A TRAVÉS DE LA LEY GENERAL DE 1970 11
Multiply by Actual Number
= +
=
× × = ×
= = =
We’ll use the Ratio Box to keep track of the information. Start by plugging in the ratio numbers from the problem, and add them up to get your ratio whole.
You have one actual number, namely 35 members, which is the actual whole. Fill in that box, and figure out the multiplier. The ratio whole is 5 (2 + 3), and you must multiply 5 times 7 to get the actual whole, 35. To keep the ratio intact, the same multiplier must be applied to all parts. Thus, the actual number of men is 2 × 7 = 14.
MEN WOMEN TOTAL
Ratio 2 3 5
Multiply by 7 7 7
Actual
Number 14 21 35
= +
=
× × = ×
= = =
Let’s try ratios with data sufficiency.
2. Rachel throws a cocktail party for her friends. At the party, she serves martinis, screwdrivers, and boilermakers. How many martinis did Rachel serve at the party?
(1) Rachel served martinis, screwdrivers, and boilermak-ers in a ratio of 5:7:9 respectively.
(2) Rachel served a total of 35 screwdrivers.
Statement (1) gives you a ratio, and nothing else. Remember that a ratio alone tells you nothing about the actual amounts. If you need proof, draw a ratio box and try to find the multiplier. You won’t be able to! Narrow it down to BCE.
Now look at Statement (2), forgetting all about Statement (1). That Rachel served 35 screwdrivers does not tell us anything about the number of martinis or boilermakers she might have served. Eliminate (B). Now consider the statements together. Can you find a multiplier and fill in your ratio box? Since you know the ratio number for screwdrivers is 7, and the actual number of screwdrivers is 35, the multiplier must be 5 (35 = 5 × 7). The multiplier applies to all parts of the ratio, so you have enough information to determine the number of martinis served. However, don’t waste time solving a data sufficiency question. Once you realize that you CAN solve it, you know that the statements together are sufficient. The answer is (C), and you don’t need to fill in the whole box.
Assignment 3
Here’s another type of ratio problem:
3. If 3u=5v, then the ratio of 5u to v is
Since the problem involves variables, let’s Plug In. What are some easy numbers to plug in for u and v? How about u = 5 and v = 3? In that case, the ratio of 5u to v would be 5 5× to 3, or 25: 3, so the best answer is (E).
Proportions
Some problems set two ratios equal to one another. These fixed relationships are called proportions. For example, the relationship between hours and minutes is fixed, so we can set up a proportion:
The key to doing a proportion problem is to set one ratio equal to another, making sure to keep your units in the same places.
Let’s try an example:
4. On a certain map, Washington, D.C., and Montreal are 4 inches apart. If Washington, D.C., and Montreal are actually 500 miles apart, and if the map is drawn to scale, then 1 inch represents how many miles on the map?
125 150 250 375 500
The information tells you that the ratio of inches to miles is 4: 500. Those are equivalent units, since 4 inches on the map is the same as 500 miles. Put those equal amounts in a fraction, like this:
4 500
inches
miles. Next, set up the other side, keeping units in the same places in the fraction:
A
VERAGESOn the GMAT, average is also called arithmetic mean, or simply mean. All averages are based on this equation:
Average Total Number of Things
=
Drawing an Average Pie will help you organize your information.
Average Total
# of things
Here’s how the Average Pie works. The total is the sum of the numbers you are averaging. The number of things is the number of quantities you are averaging, and the average is, of course, the average.
Let’s apply the Average Pie to a simple example. Say you wanted to find the average of 3, 7, and 8. You would add up the numbers and then divide by 3
3 7 8 3
18
3 6
+ + = =
. Here’s how to organize the same information in the average pie:
6 18
3
.. ..
The horizontal line across the middle means divide. If you have the total and the number of things, divide to get the average. If you have the total and the average, divide to solve for the number of things. If you have the average and the number of things, multiply to get the total. As you will see, the key to most average questions is finding the total. You may also need more than one pie if a problem involves multiple averages. Draw a separate pie for every average in a problem.
The Average Pie is a great tool because it:
• Organizes the information clearly.
• Allows you to solve for one piece of the pie when you have the other two.
• Helps you to focus on what else you need to solve the problem, which is great for data sufficiency.
Assignment 3
Let’s try a problem.
1. The average (arithmetic mean) weight of three people is 160 pounds. If one of these people weighs 200 pounds, what is the average weight, in pounds, of the two remaining people?
731
3 140
160 240 480
Draw an Average Pie, and fill in the first set of information. You know the average of three things is 160, so can you multiply to get the total (480). Notice that you can cross off (E) as a trap. The question asks for the average weight of the two remaining people, so you’re not done yet. Draw another Average Pie for the next average in the problem. There are two remaining people, but you don’t know their total weight. However, you can get their total weight by subtracting 200 from the total for all three (480). Since 480 – 200 = 280, you know the total weight of the other two people is 280. To get the average, just divide your new total by the number of people,
280
2 pounds 140
people = . Choice (B) is the answer.
R
ATESRate problems are very similar to average problems. They often ask about average speed or distance traveled. Other rate problems ask about how fast someone works or how long it takes to complete a task. All of these problems involve this important relationship:
Rate =
Distance
Time or Rate =
Amount of Work Time
Because this relationship is identical to that in the average formula, you can use the Rate Pie to organize your information:
Rate Distance
Time Rate
Work
Time
The Rate Pie has the same advantages as the Average Pie. Whenever you have two pieces of the pie, you can solve for the third. Let’s put the Rate Pie to work.
1. It takes Mike 1 hour and 30 minutes to commute from home to work at an average speed of 40 miles per hour. If Mike returns home along the same route at an average speed of 45 miles per hour, how long does the return trip take?
1 hour, 15 minutes 1 hour, 20 minutes 1 hour, 25 minutes 1 hour, 30 minutes 1 hour, 35 minutes
Draw a Rate Pie, and fill in the information you know about the first trip:
Mike’s rate was 40 mph, and the time it took was 1 hour and 30 minutes, or 1.5 hours. Now you can solve for total distance by multiplying: 40 × 1.5 = 60 miles.
That’s the same distance he’ll need to travel on the way home, so make another Rate Pie for the return trip, and fill in that piece. Now you also know the return trip average speed is 45 miles per hour, so go ahead and put that in too. So the time for the return trip can be found by
60 45
4 3 11
= = 3 hours, or one hour and twenty minutes. The answer is (B).
Let’s try another one, this time with work.
2. A machine at the golf ball factory can produce 16 golf balls in 5 minutes. If several of these machines work independently and each machine performs at the same rate, how many machines are needed to produce 32 golf balls per minute?
3 6 8 10 13
Assignment 3
Put the information you have into a Rate Pie. You know that one machine produces 16 balls in 5 minutes, so fill in those pieces. That gives you a rate of
16
5 = . balls per minute.3 2 Then set up a proportion, since each machine works at the same rate,
1
3 2 32
machine balls
machines balls
. = x
. Cross-multiply and you get x =
32 1 3 2× =10
. , choice (D).