SAT Mc Graw Hill 2011

244 MCGRAW-HILL’S SAT

Lesson 3: Finding Patterns

Repeating Patterns

Finding patterns means looking for simple

rules that relate the parts of a problem. One

key to simplifying many SAT math problems is

exploiting repetition. If something repeats, you

usually can cancelor substituteto simplify.

If 5x^2 + 7x+ 12 = 4x^2 + 7x+ 12, then what is the value

of x?

This question is much simpler than it looks at

first because of the repetition in the equation. If

you subtract the repetitive terms from both sides of

the equation, it reduces to 5x^2 = 4x^2. Subtracting 4x^2

from both sides then gives x^2 = 0, so x= 0.

Patterns in Geometric Figures

Sometimes you need to play around with the

parts of a problem until you find the patterns or

relationships. For instance, it often helps to

treat geometric figures like jigsaw puzzle pieces.

The figure above shows a circle with radius 3 in

which an equilateral triangle has been inscribed.

Three diameters have been drawn, each of which in-

tersects a vertex of the triangle. What is the sum of

the areas of the shaded regions?

This figure looks very complicated at first. But

look closer and notice the symmetryin the figure. No-

tice that the three diameters divide the circle into six

congruent parts. Since a circle has 360°, each of the

central angles in the circle is 360° ÷ 6 = 60°. Then no-

tice that the two shaded triangles fit perfectly with the

other two shaded regions to form a sector such as this:

Moving the regions is okay because it doesn’t

change their areas. Notice that this sector is 1/3 of the

entire circle. Now finding the shaded area is easy. The

total area of the circle is a= r^2 = (3)^2 = 9. So the

area of 1/3 of the circle is 9/3 = 3.

Patterns in Sequences

Some SAT questions will ask you to analyze a

sequence. When given a sequence question,

write out the terms of the sequence until you

notice the pattern. Then use whole-number di-

vision with remainders to find what the ques-

tion asks for.

1, 0, −1, 1, 0, −1, 1, 0, −1,...

If the sequence above continues according to the

pattern shown, what will be the 200th term of the

sequence?

Well, at least you know it’s either 1, 0, or −1, right?

Of course, you want a better than a one-in-three

guess, so you need to analyze the sequence more

deeply. The sequence repeats every 3 terms. In 200

terms, then the pattern repeats itself 200 ÷ 3 = 66

times with a remainder of 2. This means that the

200th term is the same as the second term, which is 0.

What is the units digit of 27^40?

The units digit is the “ones” digit or the last digit.

You can’t find it with your calculator because when

2740 is expressed as a decimal, it has 58 digits, and your

calculator can only show the first 12 or so. To find the

units digit, you need to think of 27^40 as a term in the

sequence 27^1 , 27^2 , 27^3 , 27^4 ,.... If you look at these

terms in decimal form, you will notice that the units

digits follow a pattern: 7, 9, 3, 1, 7, 9, 3, 1,.... The

sequencehas a repeating pattern of four terms. Every

fourth term is 1, so the 40th term is also 1. Therefore,

the units digit of 27^40 is 1.

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