Cracking the SAT subject test in math 2, 2nd edition

Cracking the SAT Subject Test in Math 2, 2nd Edition CHAPTER 6 FUNDAMENTALS DRILL EXPLANATIONS Drill 1 PEMDAS and Your Calculator 1 5 2 35 3 12 4 35 5 0 6 2 5 or 7 0 333 or Drill 2 Word Problem Transl[.]

CHAPTER 6: FUNDAMENTALS DRILL EXPLANATIONS

Drill 1: PEMDAS and Your Calculator

1 5

2 35

3 12

4 35

5 0

6 2.5 or

7 0.333…or

Drill 2: Word Problem Translation

1

2

3

4

6

7

Drill 3: Percent Change

2 B You already know that the difference is going to be 25 Set up

your formula: So, adding 25 gallons to 150 gallons is a 16.67% increase

3 B The decrease from 5 to 4 is a 20% decrease, and the increase

from 4 to 5 is a 25% increase So the difference is 5%

4 C Using the formula 12 = (150 ÷ x)100, you get x = 1,250, so 1,250

must have been the original amount 1,250 + 150 = 1,400 Be careful to read “after the deposit” in the question

18 B Plug In! Make n = 3 f(3) = (3) + 2 = 3 and f(6) = (6) + 2 = 4.

Drill 4: Repeated Percent Change

20 D Using the formula, you get 1,000(1.0512) = 1,795.856

25 E For this question, you need to compute 100 annual increases of

8%, so you must multiply the starting amount, 120,000, by

1.08100, which gives you 263,971,350.8—which can also be written as 2.6 × 108 If you weren’t sure, multiply the answers in your calculator to check your work

28 C This question is a little trickier For each annual decrease of 4%,

you must multiply by 0.96 The easiest way to solve the question is to start with 2,000 and keep multiplying until the result is less than 1,000—just count the number of decreases it takes The seventeenth annual decrease makes it less than 1,000, so the sixteenth is the last one that is not less than 1,000

—and 1995 + 16 = 2011 You can also use logs, which we’ll get to

in a little while

40 D Plug In and Plug In The Answers! Make the starting value of the

account 100 To find the resulting price you would use 100(1+0.025)n , where n is the number of years Plug in each answer for n to determine which one equals 200 The answer

which is closest to 200 is (D)

Drill 5: Averages

1 There were 9 people at dinner

2 All told, 4,500 apples were picked

3 The average height of a chess club member is 5.5 feet

Drill 6: Multiple Average Questions

18 D Nineteen donations averaging $485 total $9,215 Twenty

donations averaging $500 total $10,000 The difference is

$785

20 A The Tribune received 80 letters in the first 20 days and 70 for

the last 10 days That’s a total of 150 in 30 days, or 150 ÷ 30 = 5

21 A One day in five out of a year is 73 days At 12 a day, that’s 876

umbrellas sold on rainy days The rest of the year (292 days) is clear At 3 umbrellas a day, that’s another 876 umbrellas A total of 1,752 umbrellas in 365 days makes a daily average of 4.8 umbrellas

42 E Plug In! Make x = 100 and n = 5 If her average score on the

first 5 tests is 100, then her total score to this point is 100 ∙ 5 =

500 On the last 3 tests, she scores an average of 120, making

120 ∙ 3 = 360 points for the last three tests She therefore has a total of 860 points over 8 tests for an average of

for all tests Use the chosen variables to check all five answers and only (E) works

Drill 7: Irrational Numbers

33 A Type each answer into your calculator, then use MATH->FRAC

to see if the result is equal to any simple fraction To test (C) and (D), use the change of base formula: (See the section on logarithms in Chapter 12.) The only answer that does NOT result in an integer or simple fraction is (A), so (A) is irrational

35 C Plug In! Make x = 3 Plug x = 3 into each answer choice and use

MATH->FRAC to see if it is equivalent to a fraction The only answer which becomes a fraction is (C)

Drill 8: Exponents

1 b = 3 (1 root)

2 x = 11, −11 (2 roots)

3 n = 2 (1 root)

4

5 x = 3, −3 (2 roots)

6 x = −2 (1 root)

7 d = 3, −3 (2 roots)

8 n = any real number except for 0; everything to the 0 power is 1

(infinitely many roots)

9 C Remember that the top is the exponent and the bottom is the

root So in your calculator, put in 4^(3/2) This is the same as cubing 4 and then taking the square root of your result (or taking the square root of 4 and then cubing that result) Any way you slice it, the answer is 8

10 D Remember your rules A negative exponent means flip it And a

fractional exponent means the top is the exponent and the bottom is the root You may have ended up with , which isn’t wrong It just happens to not be in the answers The only

correct answer is

11 A Flip and square it: Don’t forget your parentheses in

your calculator

12 E Flip and take the third root

13 A In your calculator it goes, either as is (with parentheses) or if

you need to take more steps, just remember that the numerator

is the exponent and the denominator is the root Or notice that must be less than and use POE

14 D What’s anything raised to the zero power? That’s right, 1

Comprehensive Fundamentals Drill

1 A Be careful to follow PEMDAS! Start with the innermost

parentheses in the numerator: Multiply by 4 and

thankfully easier, but start with the parentheses: 2 + 9 = 11; multiplying by 3 gives you 33; adding 49 gives you

Finally, 2.55 ÷ 9.06 = 0.281

2 E Plug In! Make x = 3 and y = 7 The only choice that is even is

(E)

10 B Because “is” means “equals,” you can break this problem into

two pieces: what comes before the “is” and what comes after

“The cube root of the sum of one-third m and n” means you multiply m by , add that product to n, then take the cube root

of the entire operation: You can eliminate (C), (D), and (E), because they do not include the equivalent of this

expression This expression must be equal to “the cube of one-third of the sum of m and n.” You must add m and n, multiply

that sum by , then take that result to the power of

The only answer that has accurate equivalents to both of these expressions is (B)

15 B Use fractional exponents in order to combine these two

radicals Square roots are equivalent to the power, and cube roots are the power, so Next, distribute the fraction within the parentheses:

When you multiply with the same base, add the exponents, but to add fractions you need a common denominator Make the denominators equal to 6 and

Because the exponents are improper fractions, you can “pull

out” from the radical by rewriting the exponent as an integer plus a fraction: Finally, you can rewrite the fractional exponents as under the radical

, (B)

18 A If the average of a, b, c, d, and e is 86, then the sum of of a, b, c,

d, and e is 86.5 = 430 The sum of a, b, and c is 84.3 = 252, and

the sum of c, d, and e is 82.3 = 246 If you add the sum of a, b, and c to the sum of c, d, and e, you get (a + b + c) + (c + d + e) =

498 Because c occurs twice in this equation, if you subtract a +

b + c + d + e = 430 from (a + b + c) + (c + d + e) = 498, you are

left with c = 68.

20 D Plug In! Make the edge length 2, which makes the area of each

face 4 and the total surface area x = 24 Therefore, the volume

of cube A will be 23 = 8 For cube B, the total surface area will

be 24 ∙ 4 = 96; if you divide by 6 you find the area of each face is

16 = 4 gives you the edge of cube B The volume of cube B

will be 43 = 64 To find percent change, you take the difference divided by the original and multiply by , which is (D)

23 C When you are adding (or subtracting) with exponents, you can

factor out the largest common factor of the two terms If you do

this to the numerators and denominators of both fractions, you get You can cancel the binomials in both equations, leaving Simplify both fractions and add

to get  , (C)

35 D Plug In The Answers! The formula for repeated percent change

is original(1 − rate)number of changes The answers represent the number of changes, so if you start with (C), you find that 4395(1

− 0.10)7 = 2102.11 Close, but not small enough; eliminate (A), (B), and (C) Try (D) 4395(1 − 0.10)8 = 1891.90 There are no lesser answers which work, so choose (D)

37 C Plug In! Make x = −3, y = −2, and z = −1 All of the answers are

true except (C)

43 C Plug In The Answers! If you need a 400 mL solution which is at

45% concentration, you need of solution in your final product Try (C) If you use 120 mL of the 80% concentration, you will have mL of solution This

leaves 280 mL of the 30% solution, which contributes

mL of solution 96 + 84 = 180 mL of solution in your final product, so your answer is (C)

45 E Plug In! Make x = 5, y = 4, and z = 2 If 5 elements have an

average of 4, then the sum of those elements is 20 If 3 of those elements have an average of 2, then their total is 6, which means the other 2 elements add up to 14 The average of the remaining 2 elements therefore must be 7 The only answer choice which equals 7 is (E)