MATH SHORTCUTS to the SAT*
For the U.S. College Entrance Exam

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Sample Chapters

Shortcut #21 When a triangle can be formed from three given sides

A triangle can be formed from three sides only if the longest side is less than the sum of the other two sides.

Exercises: Could the following lengths form a triangle?

1) 3, 5, 7
2) 3, 5, 9
3) 7, 9, 12
4) 3, 3, 6
5) 3, 3, 3,
6) 3, 4, 5
7) 5, 8, 10
8) 1, 1, 5

Answers: 1) Yes; 2) No; 3) Yes; 4) No; 5) Yes; 6) Yes; 7) Yes; 8) No

Shortcut #33: Intersections of two categories

An intersection of two categories is the quantity in both categories.
To solve this type of problem perform the following two steps:

1st) Determine the number in both categories.

2nd) Determine the number in only a single category (do this for each category).

Example 1

There are 9 students in the Math class. There are 5 students in the French class. If 3 students take both math and French, what is the total number of students taking one or both classes?

Step 1
Draw two overlapping circles. One circle represents the number of students taking math and the other circle represents the number of students taking French. The overlapping section represents the number of students taking both math and French. The non-overlapping sections represent the number of students taking only one of these classes.

Step 2
In the overlapping section, fill in the number of students taking both math and French ( 3 students ).

Step 3
In the non-overlapping sections, fill in the number of students taking only math or only French. Remember that the total number of students in the math class must be 9 and the total number of students in the French class must be 5. The number of students taking only math is 6 (9 ­ 3 = 6). The number of students taking only French is 2 (5 ­ 3 = 2).

The picture now shows that 9 students are in the math class ( 6 + 3 = 9 ) and 5 students are in the French class (2 + 3 = 5). The picture also shows that 6 students take only math, 2 students take only French and 3 students take both math and French.

Step 4
The total number of students in one or both classes is the sum of the three numbers in this picture. The total number of students in one or both classes is: 6 + 3 + 2 = 11.

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Example 2

Of 13 students, each student takes French or Spanish or both classes. 10 students are in the French class and 7 students are in the Spanish class. How many students are in both classes? Choices: 1, 2, 3, 4, 5

Step 1
Draw two overlapping circles. One circle represents the number of students in the French class and the other circle represents the number of students in the Spanish class. The overlapping section represents the number of students in both classes. The non-overlapping sections represent the number of students taking only one class or the other.

Step 2
Choose a likely answer for the number of students in both classes and write this number in the overlapping section. First, 2 will be chosen.

Step 3
Fill in the remaining number of students taking only one class or the other. Remember that the French class must have a total of 10 students and the Spanish class must have a total of 7 students. If two students are in both classes, then the number of students taking only French is 8 (10 ­ 2 = 8) and the number of students taking only Spanish is 5 (7 ­ 2 = 5).

The picture now shows that the French class has 10 students ( 8 + 2 = 10 ) and the Spanish class has 7 students ( 5 + 2 = 7 ). The picture also shows that 8 students take only French, 5 students take only Spanish and 2 students take both French and Spanish.

Step 4
The choice of 2 students taking both classes will be correct only if the total number of students taking one or both classes is 13. The picture shows a total of 15 students ( 8 + 2 + 5 = 15 ), therefore the choice of 2 students taking both classes is incorrect.

Step 5
Choose another possible answer. Now the choice of 4 students taking both classes will be made. Write a 4 in the overlapping section.

Step 6
Fill in the remaining number of students taking only one class or the other. Remember that the total number of students in the French class must be 10 and the total number of students in the Spanish class must be 7. If four students take both classes, then the number of students taking only French is 6 (10 ­ 4 = 6) and the number of students taking only Spanish is 3 (7 ­ 4 = 3).

Step 7

The picture now shows that the French class has 10 students (6 + 4 = 10) and that the Spanish class has 7 students (3 + 4 = 7). The picture also shows that 6 students take only French, 3 students take only Spanish and 4 students take both French and Spanish. If the sum of these three numbers is 13, then the correct answer has been found. 6 + 4 + 3 = 13, therefore the choice of 4 students taking both French and Spanish is correct.

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Exercises:

1. In the art class, there are 16 students. In the math class, there are 7 students. If 5 students are in both classes, what is the total number of students in one or both classes?

2. In the French class, there are 52 students. In the Italian class, there are 37 students. If 10 students are in both classes, what is the total number of students in one or both classes?

3. In the art class, there are 30 students. In the history class, there are 27 students. If 6 students are in both classes, what is the total number of students in one or both classes?

4. Of 20 students, each takes French or Spanish or both. 15 students are in the French class and 9 students are in the Spanish class. How many students are in both classes?
Choices: 0, 2, 4, 5, 6

5. Of 12 students, each takes math or history or both. 10 students are in the math class and 3 students are in the history class. How many students are in both classes?
Choices: 0, 1, 2, 3, 4

6. Of 22 students, each takes history or math or both. 12 students are in the history class and 12 students are in the math class. How many students are in both classes?
Choices: 0, 1, 2, 5, 12

Answers:

1. 18
2. 79
3. 51
4. 4
5. 1
6. 2

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Last Revised: 10 May 2001
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