Art of Problem Solving Textbooks Do You Know Prealgebra?

Art of Problem Solving Textbooks
Do You Know
Prealgebra?
This diagnostic test consists of two parts, Fundamentals and Problem Solving. If your
student can solve nearly all of the Fundamentals problems and at least half of the Problem
Solving problems, then the Art of Problem Solving Prealgebra text would only serve as a review
for your student. We recommend using the following process to assess whether or not your student
has already mastered the contents of the book:
Step 1: The student should attempt all of the questions below without a calculator and without
any help. There is no time limit.
Step 2: Check the student’s responses using the answer key at the end of this document.
Step 3: The student should be given a second chance on the problems that he or she answered
incorrectly.
1. Exponent basics. Compute:
Fundamentals
(a) 2 5 (b) (−5) 2 − 4 2 (c) 3 0 + 3 −2
2. Exponent laws. Express each of the following as a power of 2:
(a) 2 4 · 2 3
3. Number Theory basics.
(b) 28
2 4
(a) Find the least common multiple of 54 and 24.
(b) Find the greatest common divisor of 288 and 684.
(c) Find the least common multiple of 72, 120, and 28.
4. Fraction arithmetic. Compute:
(a) 2 7 1
+ −
3 6 4
(b) 3 8 7
· ·
4 5 6
1
(c) 10 ÷ 2
3
(d) 6 1
− 31
4 3
(c) (2 4 ) 3

Art of Problem Solving Textbooks
Do You Know
Prealgebra?
5. Fraction and Decimal conversion. Convert the following decimals to fractions in simplest
form:
(a) 0.625 (b) 3.64 (c) 0.8
Convert the following fractions to decimals:
(d) 13
4
(e) 7
20
6. Linear Equations. Solve each of the following equations:
(a) 3r − 4 = 16 − 7r (b)
7. Ratio and Rates.
2x − 3
5
4 − 3x
=
7
(f) 5
6
(c) 2 − t
�
= 3 5 −
4 t
�
6
(a) The ratio of boys to girls at a summer camp is 4 to 5. If the total number of students
at the camp is 108, then how many boys are at the camp?
(b) The ratio of teachers to students in a particular school is 1 to 11. The ratio of female
students to the total number of students is 4 to 9. If there are 396 female students, then
how many teachers are there?
(c) A train is traveling 1 mile every 75 seconds. If the train continues at this rate, then how
far will it travel in two hours?
8. Percent.
(a) 18% of 50 is what percent of 24?
(b) If there are 240 boys in a school with a total of 960 students, then what percent of the
students are girls?
(c) Katie wants a fancy new music player that costs $300. Katie can buy it in her home
state and pay 8% sales tax, or she can drive to a neighboring state and pay only 5% sales
tax. How much does Katie save on the player if she drives to the neighboring state?
9. Square Roots. Simplify each of the following:
(a) √ 81
(b) √ �
(c) 1
144
7
9
√
120
(d) √
3
2
(e) √ 75 + √ 27 − √ 192
(f) 3 √ 18 + 7 √ 8

Art of Problem Solving Textbooks
Do You Know
Prealgebra?
Problem Solving
10. What is the value of the sum 5 + 10 + 15 + · · · + 95 + 100?
11. Express 2 5 · 8 3 · 16 2 as a power of 4.
12. The four-digit number A55B is divisible by 36. What is the sum of A and B?
13. Two-fifths of the students in Central Middle School are boys. One-third of the girls have
blond hair and one-quarter of the boys have blond hair. What fraction of the students in
Central Middle School have blond hair?
14. Douglas writes down his favorite number, which is a two-digit positive integer. He then turns
the number into a three-digit number by writing a 7 at the end of his favorite number. This
new number is 385 more than Douglas’s favorite number. What is Douglas’s favorite number?
15. Kayla adds the same number to both the numerator and denominator of the fraction 1
10 . Her
resulting fraction equals 2
3 . What number did she add to both the numerator and denominator
of her original fraction?
16. Five workers together can build a road in 20 days. Suppose every worker works at the same
rate. Three workers work on the road for 10 days before eleven more workers join them. How
much longer will it take the fourteen workers to finish the road?
17. A northbound train from Miami to Jacksonville made the 324-mile journey at an average
speed of 50 miles per hour. On its southbound return trip, it made the journey at an average
speed of 40 miles per hour. To the nearest tenth of a mile per hour, what was the train’s
average speed for the 648-mile roundtrip journey?
18. At Annville Junior High School, 30% of the students in the Math Club are in the Science
Club, and 80% of the students in the Science Club are in the Math Club. There are 15
students in the Science Club. How many students are in the Math Club?
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The answers to Do You Know Prealgebra are below.
1. (a) 32; (b) 9; (c) 1 1
9
2. (a) 2 7 ; (b) 2 4 ; (c) 2 12
or 10
9
3. (a) 216; (b) 36; (c) 2520
4. (a) 19
12
5. (a) 5 91
; (b)
8 25
7 7
or 1 ; (b) ; (c) 15; (d) 211
12 5 12
Art of Problem Solving Textbooks
Do You Know
Prealgebra?
or 35
12
8
or 316;
(c) ; (d) 3.25; (e) 0.35; (f) 0.83
25 9
6. (a) r = 2; (b) x = 41
; (c) t = 52
29
7. (a) 48 boys; (b) 81 teachers; (c) 96 miles
8. (a) 37.5%; (b) 75%; (c) $9
9. (a) 9; (b) 12; (c) 4
3
10. 1050
11. 4 11
12. 8
13. 3
10
14. 42
15. 17
16. 5 days
17. 44.4 miles per hour
18. 40
or 1 1
3 ; (d) 2√ 10; (e) 0; (f) 23 √ 2
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