Similar to Cause and Effect Reasoning, Conditional Reasoning, and Formal
Logic, the concept of Numbers and Percentages is featured in many LSAT
stimuli. Although most people are comfortable working with numbers or
percentages because they come up so frequently in daily life (for example in
balancing a checking account, dividing a bar tab, or adding up a grocery bill),
the makers of the LSAT often prey upon several widely-held misconceptions:
Misconception #1: Increasing percentages automatically lead to
increasing numbers.
Most people assume that if a percentage becomes larger, the number that
corresponds to that percentage must also get larger. This is not
necessarily true because the overall size of the group under discussion
could get smaller. For example, consider the following argument: “Auto
manufacturer X increased their United States market share from 10%
last year to 25% this year. Therefore, Company X sold more cars in the
United States this year than last.” This is true if the size of the U.S. car
market stayed the same or became larger. But if the size of the U.S. car
market decreased by enough, the argument would not be true, as in the
following example:
Last Year This Year
Total number of cars 1000 200
sold in the United States
X’s market share 10% 25%
X’s total car sales 100 50
in the United States
Thus, even though auto manufacturer X’s market share increased to
25%, because the size of the entire market decreased significantly, X
actually sold fewer cars in the United States.
Misconception #2: Decreasing percentages automatically lead to
decreasing numbers.
This misconception is the opposite of Misconception #1. Just because a
percentage decreases does not necessarily mean that the corresponding
number must become smaller. Reversing the years in the previous
example proves this point.
Misconception #3: Increasing numbers automatically lead to increasing
percentages.
Just as increasing percentages do not automatically translate into
increasing numbers, the reverse is also true. Consider the following
example: “The number of bicycle-related accidents rose dramatically
from last month to this month. Therefore, bicycle-related accidents must
make up a greater percentage of all road accidents this month.” This
conclusion can be true, but it does not have to be true, as shown by the
following example:
Last Month This Month
Number of bicycle-related 10 30
accidents
Total number of road accidents 100 600
Percentage of total accidents 10% 5%
that are bicycle-related
Thus, even though the number of bicycle-related accidents tripled, the
percentage of total road accidents that were bicycle-related dropped
because the total number of road accidents rose so dramatically.
Misconception #4: Decreasing numbers automatically lead to decreasing
percentages.
This misconception is the opposite of Misconception #3. Just because a
number decreases does not necessarily mean that the corresponding
percentage must become smaller. Reversing the months in the previous
example proves this point.
Misconception #5: Large numbers automatically mean large
percentages, and small numbers automatically mean
small percentages.
In 2003, Porsche sold just over 18,000 cars in the United States. While
18,000 is certainly a large number, it represented only about 1/5 of 1%
of total U.S. car sales in 2003. Remember, the size of a number does not
reveal anything about the percentage that number represents unless you
know something about the size of the overall total that number is drawn
from.
Misconception #6: Large percentages automatically mean large
numbers, and small percentages automatically mean
small numbers.
This misconception is the reverse of Misconception #5. A figure such as
90% sounds impressively large, but if you have 90% of $5, that really
isn’t too impressive, is it?
Numerical situations normally hinge on three elements: an overall total, a
number within that total, and a percentage within the total. LSAT problems will
often give you one of the elements, but without at least two elements present,
you cannot make a definitive judgment about what is occurring with another
element. When you are given just percentage information, you cannot make a
judgment about numbers. Likewise, when you are given just numerical
information you cannot make a judgement about percentages.
In a moment, we will explore this idea by examining several LSAT questions.
But first, you must be able to recognize number and percentage ideas when they
appear on the LSAT:
Words used to introduce numerical ideas:
Amount
Quantity
Sum
Total
Count
Tally
Words used to introduce percentage ideas:
Percent
Proportion
Fraction
Ratio
Incidence
Likelihood
Probability
Segment
Share
Three words on the percentage list—“incidence, “likelihood,” and
“probability”—bear further discussion. Each of these words relates to the
chances that an event will occur, and when the LSAT makers uses phrases such
as “more likely” or “less likely” they are telling you that the percentage chances
are greater than 50% or less than 50%, respectively. In fact, a wide variety of
phrases can be used to introduce percentage ideas, including such disparate
phrases as “more prone to” or “occurs with a high frequency.”
With these indicators in mind, please take a moment to complete the following
question:
1. From 1973 to 1989 total energy use in this country increased less than 10 percent. However, the use of energy in this country during this same period grew by more than 50 percent, as did the gross national product—the total value of all goods and services produced in the nation.
If the statements above are true, then which one of
the following must also be true?
(A) Most of the energy used in this country in
1989 was electrical energy.
(B) From 1973 to 1989 there was a decline in the
use of energy other than electrical energy in
this country.
(C) From 1973 to 1989 there was an increase in
the proportion of energy use in this country
that consisted of electrical energy use.
(D) In 1989 electrical energy constituted a larger
proportion of the energy used to produce the
gross national product than did any other
form of energy.
(E) In 1973 the electrical energy that was
produced constituted a smaller proportion of
the gross national product than did all other
forms of energy combined.
Like the vast majority of Must Be True problems, the stimulus does not contain
a conclusion. We are given the following facts, however:
From 1973 to 1989 total energy use increased less than 10%.
During this same period, the use of electrical energy grew by more than
50%.
During this same period, the gross national product (GNP) grew by
more than 50%.
A careful examination of the second sentence reveals that there is no stated
connection between the growth of the GNP and the increase in the use of
electrical energy. If you assume that the use of electrical energy somehow
caused the growth of the GNP, you are guilty of making an unwarranted causal
assumption. Because there is no stated connection between the two other than
they both grew by more than 50%, any answer that attempts to connect the two
is incorrect. Answer choices (D) and (E) can both be eliminated by this
reasoning.
Now that we recognize that the GNP issue is only a red herring, let us examine
the percentages that are given in the stimulus. The 50% increase in electrical
energy gives the impression that the jump must have been substantial. But we
know from Misconception #6 that a large percentage does not automatically
mean a large number. For example, in this problem it is possible that the 50%
increase in electrical energy use was a jump from 2 units to 3 units. The
possibility that electrical energy use in 1973 was a relatively small percentage of
overall energy use directly undermines answer choices (A), as shown by the
following example:
1973 1989
Total energy use 100 109
(in units)
Electrical energy use 10 15
(in units)
Percentage of total energy 10% 13+%
use that was electrical
A close analysis of the chart also reveals that answer choice (B) can be
eliminated. In the example, the use of energy other than electrical energy rose
from 90 units to 94 units.
Although the example disproves both answer choice (A) and (B), obviously
you do not have time to make a chart during the test to examine each possibility,
so is there a faster way to eliminate the first two answers? Yes—consider the
previous discussion point that information about percentages does not tell us
about the numbers. With that idea in mind, because the stimulus contains only
percentage information (even though there are two percentages), you should be
very suspicious of answer choice (A) (which states that the number of electrical
units used was greater) and answer choice (B) (which states that the use of nonelectrical
energy declined) since they both contain numerical information. At the
same time, you should be attracted to an answer such as (C) because it contains
only percentage information, and as it turns out, answer choice (C) is correct.
Because the misconceptions discussed earlier have a predictable effect when
you try to make inferences, you can use the following general rules for Must Be
True questions:
1. If the stimulus contains percentage or proportion information only, avoid
answers that contain hard numbers.
Example Stimulus Sentence:
The car market share of Company X declined this year.
Avoid answers which say:
Company X sold a smaller number of cars this year.
Company X sold a greater amount of cars this year.
2. If the stimulus contains only numerical information, avoid answers that
contain percentage or proportion information.
Example Stimulus Sentence:
Company Y sold fewer computers this year.
Avoid answers which say:
Company Y now has a lower share of the computer market.
Company Y now possesses a greater proportion of the
computer market.
3. If the stimulus contains both percentage and numerical information, any
answer choice that contains numbers, percentages, or both may be true.
Please keep in mind that these rules are very general. You must read the
stimulus closely and carefully to determine exactly what information is present
because the makers of the LSAT are experts at camouflaging or obscuring
important information in order to test your ability to understand complex
argumentation.
Please take a moment to complete the following question:
2. The number of North American children who are
obese—that is, who have more body fat than do 85
percent of North American children their age—is
steadily increasing, according to four major studies
conducted over the past 15 years.
If the finding reported above is correct, it can be
properly concluded that
(A) when four major studies all produce similar
results, those studies must be accurate
(B) North American children have been
progressively less physically active over the
past 15 years
(C) the number of North American children who
are not obese increased over the past 15
years
(D) over the past 15 years, the number of North
American children who are underweight has
declined
(E) the incidence of obesity in North American
children tends to increase as the children
grow older
Like the previous question, this is a Must Be True question with a stimulus that
does not contain a conclusion. But, this stimulus does provide information about
both the numbers and percentages of obese children, and so you can end up
with an answer that has either a number or a percentage (though a numerical
answer is more likely since the percentage is fixed at a constant 15% in the
stimulus).
