# Algebra/Chapter 1/Exercises

A set of exercises related to concepts from Chapter 1.
This set contains 153 problems (24 Conceptual Questions + 123 Exercises + 6 Projects)

## 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 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".
Q1.10 (Problem with Fractions) Why can't we say that 3/5 of the figure below have been shaded in?
Q1.11 (Large Numbers) Determine if the following is true: "The more digits a number has, the larger it is".
Q1.12 (Signs) A fast-food menu has the cost of a hamburger listed as .99¢. Explain what is wrong with this.
Q.13 (Division Symbols) Write three symbols that can be used for division.
Q1.14 (Reciprocal of Zero) Does the number 0 have a reciprocal? Explain.
Q1.15 (Which is Larger?) Explain how to determine which fraction is larger, ${\frac {7}{16}}$, or ${\frac {1}{2}}$
Q1.16 (LCM vs LCD) Explain the difference between the LCM and LCD.
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 (Steps of the Order of Operations) In your own words, explain the four steps of the order of operations.
Q1.19 (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.
Q1.20 (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?
$6/2(1+2)$
Q1.21 (Even Prime Number) Explain why 2 is the only even prime number.
Q1.22 (Consecutive Numbers) What is the LCM of two consecutive numbers? What is the GCF of two consecutive numbers?
Q1.23 (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?
Q1.24 (SI Units) What are the SI units for length, mass, and time?

## 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. 800,025
(★) 1.5 (Writing Numbers) Translate the following to mathematical symbols
1. eleven 2. two-hundred seventy 3. three-million two-hundred-thirty-four-thousand five-hundred sixty-seven
(★) 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:
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 (Reading Meters) The amount of electricity used in a household is measured in kilowatt-hours (kwh). Determine the reading on the meter shown below. (When the pointer is between two numbers, read the lower number.)
(★) 1.9 (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.
(★) 1.10 (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

(★) 1.11 (Expressions) Simplify the following expressions involving basic operations.
1. 2 + 5 2. 7 - 4 3. 16 - 9
(★) 1.12 (Operations on the Number Line) Determine the performed operation that is being represented in each diagram.
(★★) 1.13 (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?
$8\ \ \ 8\ \ \ 8\ \ \ 8\ \ \ 8\ \ \ 8\ \ \ 8\ \ \ 8$
(★★) 1.14 (Unknown Sum) In the addition problem below, A, B, and C each represent three different digits. What are the digits?
${\begin{array}{r}AAA\\+\ BBB\\\hline AAAC\end{array}}$
(★★) 1.15 (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?
(★) 1.16 (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?
(★) 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 (Pine Tree Garden) A landscape designer intends to plant pine trees 14 feet apart to form a windscreen along one side of a flower garden. How many trees are needed if the length of the flower garden is 896 feet?
(★★) 1.21 (Page Numbers) How many numerals are required to number all of the pages of a book containing 450 pages?
(★★) 1.22 (Cereal Factory) A cereal factory produces 153,600 ounces of cereal each day. Each box contains 16 ounces. The boxes are packaged in bundles of 6 boxes per bundle, then stacked on pallets, 25 bundles to a pallet. Finally, the pallets are loaded onto trucks, 24 pallets to each truck
1. How many boxes of cereal are produced each day? 2. How many bundles are produced? 3. How many pallets are needed? 4. How many trucks are needed? Explain your reasoning.

### Section 1.3

(★) 1.23 (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
(★) 1.24 (Exponent Expressions) Evaluate the following expressions.
1. $3^{2}$ 2. $0^{150}$ 3. $7228529837^{0}$ 4. $5^{3}$
(★) 1.25 (Radical Expressions) Evaluate the following expressions.
(★) 1.26 (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?
(★) 1.27 (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?
(★) 1.28 (Tearing Paper) You tear a piece of paper in half. Then, you tear each remaining sheet of paper in half again. You tear the collection of papers 5 times over all. When you are done, how many scraps of paper do you have?
(★) 1.29 (Powers of 1) Find $1^{2}$, $1^{3}$, and $1^{4}$. What can you assume about any power of 1?
(★) 1.30 (Powers of 10) Find $10^{2}$, $10^{3}$, and $10^{4}$. What can you assume about any power of 10?
(★) 1.31 (Zeroes) How many zeroes would you need to write the number $10^{1000}$?
(★★) 1.32 (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?
(★) 1.33 (Coffee Table) A square coffee table has an area of 196 square inches. What is the length of one side of the table?
(★★) 1.34 (Relatives) Everybody is born to $2^{1}$ biological parents. Our parents each had $2^{1}+2^{1}$ biological parents. We can say that our grandparents are $2^{2}$ mathematically as the number of our ancestors doubles with each generation we go back.
1. How many times would 2 be multiplied to determine the number of great grandparents? 2. How many times would 2 be multiplied to determine the number of great-great grandparents? 3. How many people would be our 2$^8$ ancestors?
(★★) 1.35 (Root of 37) To the nearest tenth, what is the square root of 37?
(★★) 1.36 (Root of 2000) Between what two whole numbers is ${\sqrt {}2000)$?

### Section 1.4

(★) 1.37 (Listing Prime and Composite Numbers)
1. List the first 10 prime numbers. 2. List the first 10 composite numbers.
(★) 1.38 (Prime or Composite?) Determine if the following numbers are prime, composite, or neither.
1. 2 2. 3 3. 13 4. 15 5. 1,873 6. 29 7. 42 8. 101 9. 93
10. -10 11. 1 12. ${\frac {5}{3}}$ 13. ${\frac {6}{3}}$ 14. 2.5 15. 0
(★) 1.39 (Factors) Find all of the factors of the following numbers
(★) 1.40 (Prime Factorization) Find the prime factorization of the following numbers. 1. 693
(★) 1.41 (Least Common Multiple) Find the least common multiple of the following sets of numbers.
(★) 1.42 (Greatest Common Factor) Find the greatest common factor of the following sets of numbers.
(★) 1.43 (Divisibility Table) Refer to the table below. If the given number is divisible by 2, 3, 4, 5, 6, 7, 8, 9, or 10, enter a checkmark in the appropriate box.
| | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 2940 | | | | | | | | | |
| 5850 | | | | | | | | | |
(★) 1.44 (Using Divisibility Rules) Use divisibility tests to find the remainder of the following quotients:
(★) 1.45 (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?
(★) 1.46 (Girl Scouts) Diana needs to buy beads for her Girl Scout troop. The glass beads come 25 to a package and the wooden beads come 35 to a package. What is the least number of packages she will need to buy to ensure that each of the girl scouts gets the same number of glass beads and the same number of wooden beads?
(★★) 1.47 (456395) Without performing division, what will be the remainder of 456395 when it is divided by 9? Explain your reasoning.
(★★) 1.48 (Soup Bowls) Four bowls; each holding 12 16, 28, and 36 ounces respectively; can hold an exact number of full ladles of soup.
1. If there is no spillage, what is the greatest size ladle (in ounces) that one can use for all four bowls? 2. How many ladles will it take to fill all four bowls?
(★★) 1.49 (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.
(★★) 1.50 (Perfect Numbers) A perfect number is any number whose factors (excluding the number itself) adds up to it. For example, the number 6 is perfect because 1 + 2 + 3 = 6. Is 496 also a perfect number? Why or why not?
(★★★) 1.51 (Digits out of 12) What is the largest multiple of 12 that can be written with the digits from 0 to 9 exactly once?
(★★) 1.52 (Sieve of Eratosthenes) The Sieve of Eratosthenes is an ancient algorithm used to find the prime numbers up to any given limit. It is one of the most efficient ways to find small prime numbers.
1. Write the numbers 1 to 100 as shown in the grid below.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
| 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |
| 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 |
| 41 | 42 | 43 | 20 | 25 | 30 | 35 | 40 | 45 | 50 |
| 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 |
| 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 |
| 71 | 72 | 73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 |
| 81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 |
| 91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |
2. Cross out the number 1. 3. Circle the first prime, 2. Afterwards, cross out all of the multiples of 2. 4. Circle the next prime, 3. Cross out all of the multiples of 3. 5. The number 4 has already been crossed out. Cross out the number 5, then cross out all of the multiples of 5 that have not already been crossed out. 6. The number 6 has already been crossed out. Cross out the number 7, then cross out all of the multiples of 7 that have not already been crossed out. 7. Do you need to check the multiples greater than 7? Why or why not? 8. How many prime numbers are there between 1 and 100? 9. Use this technique to determine how many prime numbers there are between 1 and 250.

### Section 1.5

(★) 1.53 (Integer Expressions) Evaluate each expression.
1. (-4) + 2 2. (-6) - 1
(★) 1.54 (Negative Division) Which of the numbers below equals -7?
${\frac {-7}{1}},{\frac {7}{-1}},{\frac {7}{1}},-{\frac {7}{1}},{\frac {-7}{-1}},-{\frac {-7}{-1}},-{\frac {-7}{1}}$
(★★) 1.55 (Comparing Absolute Values) For each given pair of numbers, determine which of the two is larger.
(★★) 1.56 (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?
(★) 1.57 (Temperature) The temperature at 6 p.m. was 31°F. By midnight, it had dropped 40 °F. What was the temperature at midnight?
(★) 1.58 (Melting Points) Mercury is the only metal which is a liquid at room temperature. Its melting point is –39°C.
1. Other elements such as gallium, cesium, and francium also have relatively low melting points. Their melting points are 29°C, 28°C, and 27°C respectively. How much warmer are the melting points (i) gallium, (ii) cesium, and (iii) francium compared to that of mercury?
(★★) 1.59 (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?
(★) 1.60 (Atoms) Atoms are composed of protons, neutrons, and electrons. Protons have a positive charge (+1), neutrons have a neutral charge (0), and electrons have a negative charge (-1). Refer to the diagrams below to answer the following questions.
1. In Atom A, how many protons, electrons, and neutrons are there? 2. What is the net charge of Atom A? 3. In Atom B, how many protons, electrons, and neutrons are there? 4. What is the net charge of Atom B?

### Section 1.6

(★) 1.61 (Mixed Fractions) Write the following improper fractions as mixed fractions.
(★) 1.62 (Fractions Operations) Simplify the following expressions involving fractions.
(★) 1.63 (Figure Drawing) As an aid in drawing the human body, artists divide the body into three parts. Each part is then expressed as a fraction of the total body height. For example, the torso is ${\frac {4}{15}}$ of the body height, and ${\frac {3}{5}}$ their body is below the waist. What fraction of body height is the head?
(★★) 1.64 (Count the 24ths) Without performing division, how many ${\frac {1}{24}}$'s are in ${\frac {2}{3}}?$
(★★) 1.65 (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?
(★★) 1.66 (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?
(★★) 1.67 (Chain of Fractions) Simplify the expression below. Write the final answer in exponential form:
${\frac {2}{1003}}\times {\frac {4}{1002}}\times {\frac {6}{1001}}\times ...\times {\frac {2002}{3}}\times {\frac {2004}{2}}\times {\frac {2006}{1}}$
(★★) 1.68 (Half the Difference) Find a fraction which is one-half the difference between ${\frac {5}{11}}$ and ${\frac {4}{11}}$
(★★★) 1.69 (Time Signatures) In musical notation, notes are written on a staff, a set of five horizontal lines that are equally spaced. The notes of a musical composition are grouped into measures, or bars.
(★★★) 1.70 (Problem from the Middle Ages) From the Middle Ages through the nineteenth century, it was common for the aristocracy to support court mathematicians and challenge other courts to contests between the mathematicians. In one of these contests, Fibonacci and Leonardo of Pisa were given the following problem.
Find a fraction that is a square and has the following property: If 5 is added onto the fraction, the new fraction is still a square, and if 5 is subtracted from the original fraction, the new fraction is still a square.

### Section 1.7

(★) 1.71 (Decimal Expressions) Simplify the following expressions involving decimals.
(★) 1.72 (Percentage Questions) Answer the following questions involving percentages.
1. What number is 25% of 640? 2. What number is 36% of 120?
(★) 1.73 (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. ${\frac {6555555}{3}}$ |
(★) 1.74 (Conversions) Convert the following fractions into percentages.
(★) 1.75 (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.
(★) 1.76 (Radius of an Atom) The radius of an atom is approximately 1.6 x 10-10 meters. Write this value in standard form.
(★) 1.77 (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?
(★★) 1.78 (Itchy's Fleas) Itchy the Dog has 1,000,000 fleas on her. Her anti-flea shampoo claims that it will leave at most 1% of the original number of fleas. What is the minimum number of fleas the shampoo will kill?
(★★) 1.79 (Percentage of Squares) What percent of all of the numbers between 1 and 1,000,000 are square numbers?
(★★) 1.80 (Terminating and Repeating Decimals) You may notice from Problem 1.73 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. Use this bar notation to write each of the repeating decimals from Problem 1.73.
(★★) 1.81 (Periods of Fractions) We see that in the fraction from below, the fractional part repeats after two digits.
${\frac {3}{11}}=0.272727272727...$
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. ${\frac {1}{3}}$ 2. ${\frac {1}{6}}$ 3. ${\frac {1}{7}}$ 4. ${\frac {1}{9}}$ 5. ${\frac {1}{14}}$
(★★) 1.82 (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...

### Section 1.8

(★) 1.83 (Classifying Numbers) Identify the type(s) of numbers each number belongs to.
| 1. $\ 12$ | 2. $\ -7$ | 3. $\ 0$ | 4. $\ {\frac {1}{7}}$ | 5. $\ {\frac {16}{13}}$ |
| 6. $\ -8.08$ | 7. $\ -{\frac {24}{365}}$ | 8. $\ 0.1231231234...$ | 9. $\ (10)^{2}$ | 10. $\ 0.5{\overline {678}}$ |
| 11. $\ 2.0{\overline {505}}$ | 12. $\ {\frac {1}{2}}$ | 13. $\ {\sqrt {17}}$ | 14. $\ {\frac {37}{0}}$ | 15. $\ \pi$ |
| 16. $\ {\sqrt {9}}$ | 17. $\ {\sqrt {\frac {1}{36}}}$ | 18. $\ -5000$ | 19. $\ -{\sqrt {36}}$ | 20. $\ 5+{\sqrt {2}}$ |
| 21. $\ {\sqrt {\frac {294}{6}}}$ | 22. $\ 1.23456789101112...$ | 23. $\ {\frac {\sqrt {100-81}}{\sqrt {76}}}$ | 24. $\ (10.25)^{2}$ | 25. $\ {\sqrt {0.33}}$ |

### Section 1.9

(★) 1.84 (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. $8+2(10-4)$
(★) 1.85 (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. $(12-9)^{3}+420$ ÷ $[3*5+(19-13)]$
(★★) 1.86 (Using the Order of Operations II) Simplify the following expressions using the Order of Operations.
1. $\ {\frac {|-(-5)|-|3|}{-3}}$ 2. $\ {\frac {2|3*2^{2}-1|-10|-2|}{6}}$ 3. $\ {\frac {(5-6)^{2}-2|3-7|}{89-3*5^{2}}}$ 4. $4${$18-[(10-8)+2^{3}]$}$-${$[485-168+392]+45-24*16$} ÷ $8$
(★★) 1.87 (Find the Mistakes) Find the mistake in each of the following, then explain how the expression should be solved correctly.

### Section 1.10

(★) 1.88 (Appropriate Prefixes) For each scenario, determine which metric prefix on the meter is most appropriate.
(★) 1.89 (Soda Can) How many milliliters of soda are in a 12.0 fl oz can? (1 fl oz = 29.6 mL)
(★) 1.90 (Soda Bottle) A bottle of soda has a volume of 16.0 fl oz. How many gallons does the bottle contain?
(★) 1.91 (Pounds) A pound is equal to 16 ounces. How many pounds are in 435 ounces?
(★) 1.92 (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?
(★) 1.93 (Paperclip) A paperclip weighs 0.03 oz. How much does a paperclip weigh in grams?
(★) 1.94 (Nanoseconds) A light-nanosecond is the distance light travels in 1 ns. Convert 1 ft to 1 light-nanosecond.
(★) 1.95 (Hummingbird) A hummingbird's wings beats 65 times per second. How many wingbeats does a hummingbird do per minute?
(★★) 1.96 (Tiny Organisms) The table below shows the dimensions of small organisms.
| Name of Organism | Length | Width |
| Dust Mite | 0.42 millimeters | 0.25 millimeters |
| Bacteria | 2 micrometers | 0.5 micrometers |
| Virus | 0.3 micrometers | 15 nanometers |
1. Write all of the dimensions listed in meters. 2. Which organism is the longest? 3. Which organism is the widest? 4. Is it possible to answer parts 2 and 3 without converting any of the values? Explain.
(★★) 1.97 (Rough Diamond) The largest single rough diamond ever found, the Cullinan Diamond, weighed 3106 carats. One carat is equivalent to the mass of 0.20 grams.
1. What is the mass of this diamond in milligrams? 2. What is the diamond's weight in pounds?
(★★) 1.98 (Heartbeats) On average, the heart of a healthy adult, while resting, can range from 60 to 100 heart beats.
1. How many times does a resting healthy adult's heart beat in a day, on the lower range? 2. How many times does a resting healthy adult's heart beat in a day, on the higher range?
(★★) 1.99 (50, 50, 50, 50 Birthday) Bentley says that he is 50 years, 50 months, 50 weeks and 50 days old. How old will he be on his next birthday?
(★★) 1.100 (Square Garden) Susan’s garden has an area of 125 square yards ($yd^{2}$). To figure out how much soil she needs to purchase, Susan needs to know the area of her garden in square feet ($ft^{2}$). What is the area of Susan’s garden in square feet?
(★★★) 1.101 (Volume Conversion) The volume of a box is 78.8 cubic centimeters ($cm^{3}$). What is this volume in cubic feet ($ft^{3}$)?
(★★★) 1.102 (Solar System Model) A science museum wants to make a scale model of the Solar System. The diagram below shows the real distances between the Earth, Moon, and Sun.
1. In the model, it has been decided that the Moon will be 15 cm apart from the Earth. How far away must the Sun be from the Earth in the model? 2. Is this a good scale for the model? If not, suggest a better scaling.

### Section 1.11

(★) 1.103 (Significant Figures) Count the number of significant figures in the following numbers.
(★) 1.104 (Operations with Significant Figures) Perform the following operations with the correct numbers of significant figures.
(★) 1.105 (Rounding to Significant Figures) Round the following numbers up to 3 significant figures.
(★) 1.106 (Reporting Significant Figure Operations) Suppose two measurements have the following information:
* The first measurement has 3 decimal places and 5 significant figures.
* The second measurement has 1 decimal place and 4 significant figures.
1. How should the sum be reported? 2. How should the difference be reported? 3. How should the product be reported? 4. How should the quotient be reported?
(★) 1.107 (Speed of Light) The speed of light is 983,571,072 m/s.
1. Round this value to the nearest million. 2. Round this value to the nearest ten million. 3. Round this value to the nearest hundred million. 4. Of the three estimates, figure out which one is the most accurate, and which is the most precise.
(★) 1.108 (Fermi Estimation) Use Fermi Estimation to answer the following questions
1. How many grains of rice are in a 10kg bag?

### Section 1.12

(★) 1.109 (Measures of Center) Find the mean, median, and mode of each set of data.
1. 5, 8, 12, 4, 8, 9, 11, 2 2. 24, 26, 37, 24, 16, 44, 26, 34, 24 3. 15, 48, 89, 74, 25, 36, 57, 51, 17, 22 4. 2, 6, 8, 7, 8, 2, 2, 9, 10
(★) 1.110 (Measures of Spread) Find the range, variance, and standard deviation of each set of data. Assume that each data set represents a population.
(★★) 1.111 (Creating a Box Plot) Refer to the following data set to answer the following questions.
1. What is the mean of the data set? 2. What is the median of the data set? 3. What is the mode of the data set? 4. What is the range of the data set? 5. What is the variance of the data set? 6. What is the standard deviation of the data set? 7. What are the quartiles of the data set? 8. Create a box-and-whisker plot of the data set.
(★★) 1.112 (Reading a Box Plot I) Consider the box-and-whisker plot to answer the following questions.
(★★) 1.113 (Reading a Box Plot II) The box plots show the daily morning temperatures for the months of January, February and March in Upstate New York. Use this data to answer the following questions.
(★★) 1.114 (Reading a Data Set and a Box Plot) Consider the data below to answer the following questions.

### Section 1.13

(★) 1.115 (Patterns) Determine the next two items in the sequences below. Explain your reasoning.
1. 9, 7, 5, 3... 2. 0, 1, 4, 9, 16, 25...
(★) 1.116 (Hugs) Eight people are at a group therapy session. Everyone hugs everyone once. How many hugs take place?
(★) 1.117 (Pocket Change) Carrie has ten pockets and 46 dollar bills. She wants to have a different amount of money in each pocket. Can she do it?
(★) 1.118 (Weighing Coins) You have five coins, no two of which weigh the same. In seven weighings on a balance scale, can you put the coins in order from lightest to heaviest? That is, can you determine which coin is the lightest, next lightest, . . . , heaviest.
(★★) 1.119 (Pipes) Certain pipes are sold in lengths of 6 inches, 8 inches, and 10 inches. How many different lengths can you form by attaching three sections of pipe together?
(★★) 1.120 (Cutting Pizza) How can you cut a pizza into 11 slices by only cutting it 4 times?
(★★) 1.121 (Chess Board) How many squares are possible on an 8x8 chess board. (It is not 64, as there are more)
(★★★) 1.122 (Chess Board II) How many rectangles of any size and shape can you find on a standard 8 × 8 chess board?
(★★★) 1.123 (Broken Clock) This clock has been broken into three pieces. If you add the numbers in each piece, the sums are consecutive numbers. (Consecutive numbers are whole numbers that appear one after the other, such as 1, 2, 3, 4 or 13, 14, 15.)
1. Can you break another clock into a different number of pieces so that the sums are consecutive numbers? Assume that each piece has at least two numbers and that no number is damaged (e.g. 12 isn’t split into two digits 1 and 2).
2. Find every possibly way to break the clock into some number of pieces so that the sums of the numbers on each piece are consecutive numbers. Justify that you have found every possibility.

## Projects

P1.1 (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?
P1.2 (The 24 Game)
P1.3 (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.
P1.4 (The Binary Number System) Typically, numbers are represented in base-10, meaning they are represented with the digits 0 to 9. We call these decimal numbers. Contrary to decimal numbers, binary numbers, or numbers represented in base-2, use bits, 0 and 1. For example:
We can convert between the two bases by realizing that base-10 relies on powers of 10. From this, this means that base-2 relies on powers of 2. The first place being $2^{1}$, the second place being $2^{2}$, the third place being $2^{3}$, and so forth.
1. Convert the following numbers to binary. 2. Convert the following numbers to decimals.
P1.5 (The Hexadecimal Number System) There are 16 different digits that make up hexadecimal numbers, numbers that are of base-16. These are the digits 0 to 9, as well as the letters A, B, C, D, E, and F.
1. Convert the following numbers to hexadecimals. 2. Convert the following numbers to decimals.
P1.6 (Magic Squares)