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# Logic and Venn Diagrams for CBEST Exam Study Guide

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Updated on Mar 29, 2011

You deserve a break after all your hard work on math problems. This lesson is shorter than the others; unless logic problems give you a lot of trouble, you can probably spend less than half an hour on this lesson

### If Problems

If problems are among the easiest problems on the test if you know how to work them. A genuine If problem begins with the word If and then gives some kind of rule. Generally, these problems mention no numbers. In order for the problem to be valid, the rule has to be true for any numbers you put in.

#### Sample If Question

The following is a typical if problem. Experiment with this problem to see how the answer is always the same no matter what measurements you choose to use.

1. If the length and width of a rectangle are doubled, the area is
1. doubled.
2. halved.
3. multiplied by 3.
4. multiplied by 4.
5. divided by 4.

First of all, choose a length and width for your rectangle, like 2' by 3'. The area is 2 × 3, or 6 square feet. Now double the length and the width and find the area: 4 × 6 = 24 square feet. Twenty-four is 4 times 6, so choice d must be the answer. Try a few different numbers for the original length and width to see how easy these types of questions can be.

#### Practice

Try another one:

1. If a coat was reduced 20% and then further reduced 20%, what is the total percent discount off the original price?
1. 28%
2. 36%
3. 40%
4. 44%
5. 50%
##### Hot Tip

When choosing numbers for if problems, choose small numbers. When working with percents, start with 100.

Since this question concerns percents, make the coat's beginning price \$100. A 20% discount will reduce the cost to \$80. The second time 20% is taken off, it is taken off \$80, not \$100. Twenty percent of 80 is 16. That brings the cost down to \$64 (80 – 16 = 64). The original price of the coat, 100, minus 64 is 36. One hundred down to 64 is a 36% reduction. So two successive discounts of 20% equal not a 40%, but a 36% total reduction.

### Venn Diagrams

Venn diagrams provide a way to visualize groups in relationship to each other. Words such as some, all, and none commonly appear in these types of questions.

In Venn diagram problems, you are given two or more categories of objects. First, draw a circle representing one of the categories. Second, draw another circle representing the other category. Draw the second circle according to these rules:

1. If the question says that ALL of a category is the second category, place the second circle around the first.
2. Example: All pigs (p) are animals (a).

3. If the question says that SOME of a category is the second category, place the second circle so that it cuts through (overlaps) the first circle.
4. Example: Some parrots (p) are talking birds (t).

5. If the question says NO, meaning that none of the first category is in the second category, make the second circle completely separate from the first.
6. Example: No cats (c) are fish (f).

#### Sample Venn Diagram Question

1. All bipeds (B) are two headed (TH).Which diagram shows the relationship between bipeds and two-headed?

The question says ALL, so the two-headed circle surrounds the circle denoting bipeds. The answer is choice d.

### More Than Two Categories

Should there be more than two categories, proceed in the same way.

Example: Some candy bars (c) are sweet (s), but no bananas (b) are candy bars.

The sweet circle will cut through the candy bar circle. Since the problem did not specify where bananas and sweet intersect, bananas can have several positions. The banana circle can be outside both circles completely:

The banana circle can intersect the sweet circle:

Or the banana circle can be completely inside the sweet circle but not touching the candy bar shape:

##### Hot Tip

Even when there are no pictures of Venn diagrams in the answers, you can often solve this type of problem by drawing the diagram one way and visualizing all the possible positions of the circles given the facts in the problem.