Algebra/Chapter 1/Exercises

A set of exercises related to concepts from Chapter 1.

This set contains 104 exercises (including the Conceptual Questions)

Conceptual Questions

Q1.1 (Alien Society) Imagine you came across an alien from a distant planet, where all of its civilians have a solid grasp of English and had three fingers on each hand. Though this civilization is intelligent, they had never learned about what "counting" is or how to do it. In addition to that, they had never learned about what a "number" is, what they're called, or what they looked like. Think about how you might teach this civilian about counting and numbers so that they can go back to their planet to teach their people of this knowledge. What tools might you use to explain the idea? What are some of the concepts you'd want to get across? What are some of the difficulties that might arise from this task?

Q1.2 (What is a Number?) Define what a "number" is in your own words. Define what a "numeral" is in your own words.

Q1.3 (Sign of Zero) Is the number zero positive, negative, or neither? Explain your reasoning.

Q1.4 (Difference of Decimals) What is the difference between "ten" and "one-tenth"?

Q1.5 (Picture Perfect) Suppose the number line actually existed physically. Would you be able to take a photo of the entire number line if you backed away far enough?

Q1.6 (Explaining the Writing of Numbers) Explain in your own words how you write numbers, both in word form and with numerical symbols.

Q1.7 (Largest Number Possible) What is the largest and smallest three-digit number you can write using the digits 0, 8, and 4? Use each digit only once, and explain how you obtained your results. If you wrote these numbers to the right of a decimal point, what is the largest number you can make.

Q1.8 (A Million) A million is one thousand thousands. Explain how this is so.

Q1.9 (Reading it Wrong) Explain what is wrong with reading "50,002" as "fifty-thousand and two". Explain what is wrong with reading "2.203" as "two and two hundred and three thousanths".

Q1.11 (Number Associations) What whole numbers are associated with each word?

1. zilch
2. duo
3. decade
4. a pair
5. naught
6. trio
7. four score
8. century

Q1.12 (Problem with Fractions) Why can't we say that 3/5 of the figure below have been shaded in?

Image: 300 pixels

Q1.13 (Large Numbers) Determine if the following is true: "The more digits a number has, the larger it is".

Q1.14 (Signs) A fast-food menu has the cost of a hamburger listed as .99¢. Explain what is wrong with this.

Q1.15 (Operations on the Number Line) Determine the performed operation that is being represented in each diagram.

Q1.16 (Inverse Operations) What is the inverse operation of “I put my shoes on today, and I walk out of my house”?

Q1.17 (Decimal Operations) Explain how addition with decimals is comparable to addition with whole numbers, how are they different? Do the same thing with multiplication with decimals.

Q1.18 (Powers of 1) Find $1^{2}$, $1^{3}$, and $1^{4}$. What can you assume about any power of 1?

Q1.19 (Zeroes) How many zeroes would you need to write the number $10^{1000}$?

Q1.20 (Steps of the Order of Operations) In your own words, explain the four steps of the order of operations.

Q1.21 (Steps of the Order of Operations II) Does the Order of Operations indicate that you perform Addition before Subtraction? Does it indicate that you perform Multiplication before Division? Explain your reasoning for both questions.

Q1.22 (First Step) Determine the first step you would take to evaluate the following expressions. Explain your reasoning.

1. $15-3*4$
2. $13+(20/5)*2$
3. $8+2(10-4)$

Q1.23 (Viral Math Expression) The seemingly simple expression below has stumped many people across the Internet. Some will argue the answer is 9, while others will argue it is 1. However, there is a fundamental issue with the way that the expression is written, leading to these two different answers, can you figure out what it is?

$6/2(1+2)$

Q1.24 (Listing Prime and Composite Numbers)

1. List the first 10 prime numbers.
2. List the first 10 composite numbers.

Q1.25 (Prime or Composite?) Determine if the following numbers are prime, composite, or neither.

Q1.26 (Even Prime Number) Explain why 2 is the only even prime number.

Q1.27 (Consecutive Numbers) What is the LCM of two consecutive numbers? What is the GCF of two consecutive numbers?

Q1.28 (Infinite Decimal Expansions) Suppose the numerator of a fraction is 142. What numbers should be in the denominator for the fraction's decimal expansion to be finite? What numbers should be in the denominator for the fraction's decimal expansion to be infinite?

Q1.29 (SI Units) What are the SI units for length, mass, and time?

Q1.30 (Appropriate Prefixes) For each scenario, determine which metric prefix on the meter is most appropriate.

Exercises

Section 1.1

1.1 (Locating Numbers) Order the set of numbers below from least to greatest. Afterwards, draw a number line, and then figure out where they might be located on it.

$2.1,-4,\ {\frac {1}{2}},\ 3.99,\ -{\frac {3}{4}},\ -0.25,\ 0.0001,\ -2.3$

1.2 (Comparing Numbers) For each given pair of numbers, determine which of the two is larger.

1. 4, 100
2. 9, 9.0001
3. -7, -2
4. -5, 0
5. 100, 100

1.3 (Weighing Bull Sharks) A biologist is studying bull shark populations. She records the weights of four sharks, in pounds, that she has caught. Order the bull sharks from lightest to heaviest.

Bull Shark Weights

Shark	Weight
Shark 1	130.5 kg
Shark 2	213.2 kg
Shark 3	97.7 kg
Shark 4	97.1 kg

1.4 (Place Values) Find the place value of the number 5 in each of the following numbers.

1. 5,000,000
2. 0.5
3. 105
4. 3572896
5. 123,456,789
6. 0.000005
7. 8051
8. 85,931
9. 800,026

1.5 (Writing Numbers) Translate the following to mathematical symbols

1. eleven
2. two-hundred seventy
3.
4.
5.
6.

1.6 (Writing Numbers in Words) Write the following numbers in words

1. 9
2. 10
3. 274
4. 8,322
5. 1,000,000,009
6. 1,343,234,985
7. 0.01

1.7 (Numbers in Expanded Form) In the number 7,893, there are "7 thousands", "8 hundreds", "9 tens", and "3 ones". We therefore say that a number is in expanded form when it is written as follows:

7 thousands + 8 hundreds + 9 tens + 3 ones
or
7000 + 800 + 9 + 3

Write the following numbers in expanded form:

1. 473
2. 6852
3. 73,016
4. 570,003
5. 3,519,803
6. 48,000,061
7. 37.89
8. 124.575
9. 7496.5467
10. 6.40941

1.8 (Fraction Diagrams) Write a fraction to describe what part of the diagrams below are shaded. Write a fraction to describe what part of the diagrams aren't shaded in.

1.9 (Fruit Basket) A basket of fruit holds 5 mangoes, 7 apples, 12 oranges, and 20 pomegranates.
1. What fraction of the fruits in the basket are apples?
2. What fraction of the fruits in the basket are not oranges?
3. What fraction of the fruits in the basket are oranges or pomegranates?

Section 1.2

1.10 (Expressions) Simplify the following expressions involving basic operations.

1.11 (Make 1000 out of 8) Eight digits “8” are written together, like below, and plus signs “+” are inserted in between to get the sum of 1000. Where were the plus signs added?

$8\ \ \ 8\ \ \ 8\ \ \ 8\ \ \ 8\ \ \ 8\ \ \ 8\ \ \ 8$

1.12 (Unknown Sum) In the addition problem below, A, B, and C each represent three different digits. What are the digits?

${\begin{array}{r}AAA\\+\ BBB\\\hline AAAC\end{array}}$

1.13 (Unknown Product) A six-digit number with 1 as its left-most digit is three times bigger when we put the one at the end of the number instead. What number is this?

1.14 (Fractions and Decimals) Use long division to find the decimal expansion of each fraction.

1. ${\frac {27}{100}}$	2. ${\frac {1}{11}}$	3. ${\frac {142}{4}}$	4. ${\frac {2}{11}}$
5. ${\frac {14}{13}}$	6. ${\frac {7000}{9}}$	7. ${\frac {2}{11}}$	8. ${\frac {6555555}{3}}$

1.15 (Terminating and Repeating Decimals) You may notice from Problem 1.13 that when you convert a fraction to a decimal, you will sometimes get what is called a repeating decimal. Take for example the fraction ${\frac {3}{11}}$.

${\frac {3}{11}}=0.272727272727...$

The decimal form of ${\frac {3}{11}}$ consists of the two digits 2 and 7 in an infinitely repeating sequence. To simplify things, instead of writing the above, we denote it as 0.27.

1. Use this bar notation to write each of the repeating decimals from Problem 1.14.
2. We see that in the fraction above, the fractional part repeats after two digits. We say that this number has a period of 2. Likewise, we say that the number ${\frac {1}{7}}$ has a period of 6, because the number repeats after 6 digits. From the numbers below, which of them has the largest period?

1.16 (Turtles) Tony and Aaron go to the park. They can see 17 turtles sunning themselves on an island in the middle of a pond around the lake. As Tony and Aaron circle the lake to count the turtles they hear 24 plops as they scare more turtles from the shore. How many turtles do Tony and Aaron know live in the lake?

1.17 (Bricks) A bricklayer stacks bricks in 3 rows, with 9 bricks in each row. On top of each row, there is a stack of 6 bricks. How many bricks are there in total?

A set of stamps (For Problem 1.18)

1.18 (Stamp Collection) The picture to the right shows stamps, arranged in four groups of four. How many stamps are in that image? While you can count them individually, there is a much faster way of getting the total.

1.19 (Marbles) Lana has 3 bags with the same amount of marbles in them, totaling 12 marbles. Markus has 3 bags with the same amount of marbles in them, totaling 18 marbles. How many more marbles does Markus have in each bag?

1.20 (On Allison's Street) Allison's house is on the same street as the library, post office, and supermarket, as shown in the diagram below. The distance from Allison's house to each of the three buildings is different. Based on this information, at which point is Allison's house located?

Section 1.3

1.21 (Exponential Form) Write the following in exponential form.

1. 8 * 8 * 8 * 8
2. 16 * 16 * 16
3. 7 * 7 * 7 * 7 * 7
4. 24 * 24 * 24
5. 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2

1.22 (Exponent Expressions) Evaluate the following exponents.

1.23 (Hydra) Lerna’s Hydra is a mythological character that appears in some stories such as the 12 Tasks of Hercules. The Hydra was a one-headed monster but when it is cut off, 2 more heads grow in its place. If a hero were to try and kill it by cutting off all of its heads every day, how many heads would the Hydra have on the third day? And at the end of 10 days of trying to kill it?

1.24 (Bank Account) Nick deposits $2 into a bank account on the first day, $4 on the second day, and $8 on the third day. He will continue to double the deposit each day. How much will he deposit on the tenth day?

1.25 (Image Enlargement) Suppose you are enlarging an image that is initially 300 pixels wide on your computer. Each time you press a button on a program it, its width doubles. If you enlarge the image four times, how wide will it be?

1.26 (Radical Expressions) Evaluate the following radicals.

1.27 (Coffee Table) A square coffee table has an area of 196 square inches. What is the length of one side of the table?

1.28 (Root of 37) To the nearest tenth, what is the square root of 37?

1.29 (Root of 2000) Between what two whole numbers is ${\sqrt {(}}2000)$?

Section 1.4

1.30 (Integer Expressions) Evaluate each expression.

1.31 (Comparing Absolute Values) For each given pair of numbers, determine which of the two is larger.

1.32 (Multiple-Choice Test) To discourage random guessing on a multiple choice exam, a professor assigns 4 points for a correct answer, -2 points for an incorrect answer, and -1 point for leaving the question blank. What is the score for a student who had 18 correct answers, 9 incorrect answers, and had left 2 questions blank?

Section 1.5

1.33 (Factors) Find all of the factors of the following numbers

1.34 (Prime Factorization) Find the prime factorization of the following numbers.
1. 693

1.35 (Least Common Multiple) Find the least common multiple of the following sets of numbers.

1.36 (Greatest Common Factor) Find the greatest common factor of the following sets of numbers.

1.37 (Using Divisibility Rules) Use divisibility tests to find the remainder of the following quotients:

1.38 (Italian Restaurant) An Italian restaurant receives a shipment of 95 veal cutlets. If it takes 4 cutlets to make a dish, how many cutlets will the restaurant have left over after making as many dishes as possible? What is the maximum number of dishes the restaurant can make with the shipment?

Section 1.6

1.39 (Mixed Fractions) Write the following improper fractions as mixed fractions.

1.40 (Fractions Operations) Simplify the following expressions involving fractions.

1.41 (Sharing Pizza) Billy's family ordered a large pizza. His father had ${\frac {1}{6}}$ of it, and his mother had ${\frac {1}{5}}$ of what remained. Later on, Billy's sister ate some pizza, and then Billy had the remaining pizza when there was exactly a half of what they started with (Billy is a large kid). What fraction of what their parents had left for her did the sister have?

Section 1.7

1.42 (Decimal Operations) Simplify the following expressions involving decimals.

1.43 (Percentage Questions) Answer the following questions involving percents.

1.44 (Conversions) Convert the following fractions into percentages.

1.45 (Distance from the Moon and Earth) The distance between the moon and the Earth is approximately 238,900 miles. Write this value in scientific notation.

1.46 (Radius of an Atom) The radius of an atom is approximately 1.6 x 10-10 meters. Write this value in standard form.

Section 1.8

1.47 (Using the Order of Operations) Simplify the following expressions using the Order of Operations.
1. $2+6-7$
2. $6+10*2$
3. $12$ ÷ $4*6$
4. $\ |3-16|$
5. $12*4$ ÷ $6$
6. $10+3^{2}$
7. $11-2^{2}$
8. $(10-3)^{2}$
9. ${\sqrt {(}}15-6)$
10. $(-2)^{2}(3)^{3}$
11. $4*2^{3}-11$
12. $36$ ÷ $2$ ÷ $6$
13. $18+36$ ÷ $4*3$
14. $\ |1-{\sqrt {3}}|+|2-{\sqrt {2}}|-|{\sqrt {3}}-{\sqrt {2}}|$
15. ${\frac {(12-5)^{2}+2^{3}}{3(13-6)-(8+7)}}$
16. $(12-9)^{3}+420$ ÷ $[3*5+(19-13)]$
17. $\ {\frac {|-(-5)|-|3|}{-3}}$
18. $\ {\frac {2|3*2^{2}-1|-10|-2|}{6}}$
19. $\ {\frac {(5-6)^{2}-2|3-7|}{89-3*5^{2}}}$
20. $4${$18-[(10-8)+2^{3}]$}$-${$[485-168+392]+45-24*16$} ÷ $8$

1.48 (Find the Mistakes) Find the mistake in each of the following, then explain how the expression should be solved correctly.

Section 1.9

1.49 (Unit Conversions) Perform the indicated conversions.

1.50 (Soda Can) How many milliliters of soda are in a 12.0 fl oz can? (1 fl oz = 29.6 mL)

1.51 (Soda Bottle) A bottle of soda has a volume of 16.0 fl oz. How many gallons does the bottle contain?

1.52 (Pancake Mix) A certain recipe for pancakes calls for 4 tablespoons of baking powder to make the batter. How many cups of baking powder is 4 tablespoons?

1.53 (Paperclip) A paperclip weighs 0.03 oz. How much does a paperclip weigh in grams?

1.54 (Nanoseconds) A light-nanosecond is the distance light travels in 1 ns. Convert 1 ft to 1 light-nanosecond.

Section 1.10

1.55 (Significant Figures) Count the number of significant figures in the following numbers.

1.56 (Operations with Significant Figures) Perform the following operations with the correct numbers of significant figures.

Section 1.11

1.57 (Measures of Center and Spread) Find the mean, median, mode, and range of the following data sets:

Section 1.12

1.58 (Patterns) Determine the next item in the patterns below:

Reason and Apply

1.59 (Count the 24ths) Without performing division, how many ${\frac {1}{24}}$'s are in ${\frac {2}{3}}?$

1.60 (Half the Difference) Find a rational number which is one-half the difference between ${\frac {5}{11}}$ and ${\frac {4}{11}}$

1.61 (Negative Negative Negative Negative...)

1. What is $-(-(-2))$?
2. What is $-(-(-(-2)))$?
3. What if there were 20 minus signs in front of the 2?
4. What if there were 75 minus signs in front of the 2?

1.62 (The Turtle and the Wall) A turtle is 2 feet away from a wall. It then moves halfway to the wall and stops. Afterwards, it then moves one-half the remaining distance before it stops again. If it continues to move one-half the remaining distance to the wall, how far will it be from the wall after moving a fifth time?

1.63 (Rulers) Look at the diagram of a ruler below.

1. How many tick marks are between 0 and 1?
2. What number is the arrow pointing to?

1.64 (Page Numbers) How many numerals are required to number all of the pages of a book containing 450 pages?

1.65 (Operations with Repeating Decimals) Calculate:

1. 0.55555... + 0.66666...
2. 0.99999... + 0.11111...
3. 1.11111... - 0.22222...
4. 0.33333... * 0.66666...
5. 1.22222... * 0.81818...

1.66 (Twin Primes) Twin primes are two numbers that are prime that differ from two. Two such numbers are 17 and 19. Find three more pairs of numbers besides 17 and 19 that are twin primes.

1.67 (Huge Powers) Put the following in order from smallest to largest.

${2}^{800}$, ${3}^{500}$, ${5}^{400}$, ${6}^{300}$

1.68 (Percentage of Squares) What percent of all of the numbers between 1 and 1,000,000 are square numbers?

Challenge Problems

1.69 (Digits out of 12) What is the largest multiple of 12 that can be written with the digits from 0 to 9 exactly once?

1.70 (Which is Bigger?) Which is bigger? ${10}^{250}$ or ${6}^{300}$?

1.71 (One to Ten) To get yourself thinking about this, try this simple mathematical game:

Take the numbers 1 through 10 on the left side of an equation, and pick a number for the right side.

Example: 1 2 3 4 5 6 7 8 9 10 = 1

Now put operators between those numbers. Only use parentheses when necessary.

Example: 1 + 2 - 3 + 4 - 5 + 6 + 7 + 8 - 9 - 10 = 1

1. Change the number on the right-hand side. Can you generate an expression for this number? If not, can you prove why not?
2. Does this change if you change the order of the numbers?

1.72 (Diffy Squares) Draw a square. One each of the corners of that square, write the numbers 7, 5, 9, and 2. Now, draw a second square around the first one so that it it goes through each of the four corners. At each corner of the second square, write the difference of the numbers at the closest corners of the smaller square: 7-5 = 2, 9-5 = 4, 9-2 = 7, and 7-2 = 5.

Repeat this process until you come to a pattern of four numbers that do not change.

1. What is the pattern?
2. Try this same procedure with another set of four starting numbers. Do you end up with the same pattern?
3. Explain what happened.

1.73 (The Binary Number System)

1. Convert the following numbers to binary.
2. Convert the following numbers to decimals.

1.74 (The Hexadecimal Number System)

1. Convert the following numbers to hexadecimals.
2. Convert the following numbers to decimals.