Skills Refresher: Prealgebra & Unit Conversion (1.3 – 1.6) — Intermediate Algebra

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1 Skills Refresher: Prealgebra & Unit Conversion (1.3 – 1.6) — Intermediate Algebra

1.3 Order of Operations

When simplifying expressions, it is important to do so in the correct order. Consider the problem 2 + 5 ⋅ 3 done two different ways:

Method 1: Add first	Method 2: Multiply first
Add: 2 + 5 ⋅ 3	Multiply: 2 + 5 ⋅ 3
Multiply: 7 ⋅ 3	Add: 2 + 15
Solution: 21	Solution: 17

The previous example illustrates that if the same problem is done two different ways, it will result in two different solutions. However, only one method can be correct. It turns out the second method is the correct one. The order of operations ends with the most basic of operations, addition (or subtraction). Before addition is completed, do all repeated addition, also known as multiplication (or division). Before multiplication is completed, do all repeated multiplication, also known as exponents. When something is supposed to be done out of order, to make it come first, put it in parentheses (or grouping symbols). This list, then, is the order of operations used to simplify expressions.

Key Takeaways: Order of Operations

1st Brackets (Grouping)

2nd Exponents

3rd Multiplication and Division (Left to Right)

4th Addition and Subtraction (Left to Right)

Multiplication and division are on the same level because they are the same operation (division is just multiplying by the reciprocal). This means multiplication and division must be performed from left to right. Therefore, division will come first in some problems, and multiplication will come first in others. The same is true for adding and subtracting (subtracting is just adding the opposite).

Often, students use the word BEMDAS to remember the order of operations, as the first letter of each operation creates the word (written as B E MD AS). Remember BEMDAS to ensure that multiplication and division are done from left to right (same with addition and subtraction).

Example 1.3.1

Evaluate $2+3(9-4)^2$ using the order of operations.

$\begin{array}{rl} 2+3(9-4)^2 & \text{Parentheses first} \\ \\ 2+3(5)^2 & \text{Exponents} \\ \\ 2+3(25)&\text{Multiply} \\ \\ 2+75 & \text{Add} \\ \\ 77&\text{Solution} \end{array}$

It is very important to remember to multiply and divide from left to right!

Example 1.3.2

Evaluate $30\div 3 \cdot 2$ using the order of operations.

$\begin{array}{rl} 30 \div 3 \cdot 2 & \text{Divide first (left to right)} \\ \\ 10\cdot 2 & \text{Multiply} \\ \\ 20 & \text{Solution} \end{array}$

If there are several sets of parentheses in a problem, start with the innermost set and work outward. Inside each set of parentheses, simplify using the order of operations. To make it easier to know which left parenthesis goes with which right parenthesis, different types of grouping symbols will be used, such as braces { }, brackets [ ], and parentheses ( ). These all do the same thing: they are grouping symbols and must be evaluated first.

Example 1.3.3

Evaluate $2\{8^2-7\left[32 - 4(3^2 + 1)\right](-1)\}$ using the order of operations.

$\begin{array}{rl} 2 \{8^2-7\left[32 - 4(3^2 + 1)\right](-1) \} & \text{Innermost parentheses, exponents first} \\ \\ 2 \{8^2- 7\left[32 - 4(9 + 1)\right](-1) \} & \text{Add inside those parentheses} \\ \\ 2 \{8^2 - 7\left[32 - 4(10)\right](-1) \}& \text{Multiply inside innermost parentheses} \\ \\ 2 \{8^2-7\left[32 - 40\right](-1) \}& \text{Subtract inside those parentheses} \\ \\ 2 \{8^2-7\left[-8\right](-1) \} & \text{Exponents next} \\ \\ 2 \{64 - 7\left[-8\right](-1) \} &\text{Multiply left to right} \\ \\ 2 \{64 + 56(-1) \}& \text{Finish multiplying inside the parentheses} \\ \\ 2 \{64 - 56 \} & \text{Subtract inside parentheses} \\ \\ 2 \{8 \} & \text{Multiply} \\ \\ 16 & \text{Solution} \end{array}$

As Example 1.3.3 illustrates, it can take several steps to complete a problem. The key to successfully solving order of operations problems is to take the time to show your work and do one step at a time. This will reduce the chance of making a mistake along the way.

There are several types of grouping symbols that can be used besides parentheses, brackets, and braces. One such symbol is a fraction bar. The entire numerator and the entire denominator of a fraction must be evaluated before reducing. Once the fraction is reduced, the numerator and denominator can be simplified at the same time.

Example 1.3.4

Evaluate $\dfrac{2^4-(-8)\cdot 3}{15\div 5-1}$ using the order of operations.

$\begin{array}{rl} \dfrac{2^4-(-8)\cdot 3}{15\div 5-1} & \text{Evaluate the exponent in the numerator and divide in the denominator} \\ \\ \dfrac{16-(-8)\cdot 3}{3-1} & \text{Multiply in the numerator, subtract in the denominator} \\ \\ \dfrac{16-(-24)}{2} & \text{Add in the numerator} \\ \\ \dfrac{40}{2} & \text{Divide} \\ \\ 20 & \text{Solution} \end{array}$

Another type of grouping symbol is the absolute value. Everything inside a set of absolute value brackets must be evaluated, just as if it were a normal set of parentheses. Then, once the inside is completed, take the absolute value—or distance from zero—to make the number positive.

Example 1.3.5

Evaluate $1 + 3 | -4^2- (-8) | + 2 | 3 + (-5)^2 |$ using the order of operations.

$\begin{array}{rl} 1 + 3 | -4^2- (-8) | + 2 | 3 + (-5)^2| & \text{Evaluate the exponents} \\ \\ 1+3|-16-(-8)| + 2|3+25| & \text{Add inside the absolute values} \\ \\ 1+3|-8| + 2|28| & \text{Evaluate absolute values} \\ \\ 1+3(8)+2(28) & \text{Multiply left to right} \\ \\ 1+24+56 & \text{Add left to right} \\ \\ 81 & \text{Solution} \end{array}$

Key Takeaways: Exponents

The above example also illustrates an important point about exponents:

Exponents are only considered to be on the number they are attached to.
This means that, in the expression −4², only the 4 is squared, giving us −(4²) or −16.
But when the negative is in parentheses, such as in (−5)², the negative is part of the number and is also squared, giving a positive solution of 25.

1.4 Properties of Algebra

When doing algebra, it is common not to know the value of the variables. In this case, simplify where possible and leave any unknown variables in the final solution. One way to simplify expressions is to combine like terms.

Like terms are terms whose variables match exactly, exponents included. Examples of like terms would be $3xy$ and $-7xy,$ $3a^2b$ and $8a^2b,$ or −3 and 5. To combine like terms, add (or subtract) the numbers in front of the variables and keep the variables the same.

Example 1.4.1

Simplify $5x - 2y - 8x + 7y.$

$\begin{array}{rl} 5x - 8x \text{ and } -2y + 7y & \text{Combine like terms} \\ \\ -3x + 5y & \text{Solution} \end{array}$

Example 1.4.2

Simplify $8x^2 - 3x + 7 - 2x^2 + 4x - 3.$

$\begin{array}{rl} 8x^2 - 2x^2, -3x + 4x, \text{ and } 7 - 3 & \text{Combine like terms} \\ \\ 6x^2 + x + 4 & \text{Solution} \end{array}$

When combining like terms, subtraction signs must be interpreted as part of the terms they precede. This means that the term following a subtraction sign should be treated like a negative term. The sign always stays with the term.

Another method to simplify is known as distributing. Sometimes, when working with problems, there will be a set of parentheses that makes solving a problem difficult, if not impossible. To get rid of these unwanted parentheses, use the distributive property and multiply the number in front of the parentheses by each term inside.

$\text{Distributive Property: } a(b + c) = ab + ac$

Several examples of using the distributive property are given below.

Example 1.4.3

Simplify $4(2x-7).$

$\begin{array}{rl} 4(2x-7)& \text{Multiply each term by } 4. \\ \\ 8x-28 & \text{Solution} \end{array}$

Example 1.4.4

Simplify $-7(5x-6).$

$\begin{array}{rl} -7(5x-6) & \text{Multiply each term by }-7. \\ \\ -35x+42 & \text{Solution} \end{array}$

In the previous example, it is necessary to again use the fact that the sign goes with the number. This means −6 is treated as a negative number, which gives (−7)(−6) = 42, a positive number. The most common error in distributing is a sign error. Be very careful with signs! It is possible to distribute just a negative throughout parentheses. If there is a negative sign in front of parentheses, think of it like a −1 in front and distribute it throughout.

Example 1.4.5

Simplify $-(4x-5y+6).$

$\begin{array}{rl} -(4x-5y+6) & \text{Negative can be thought of as }-1. \\ \\ -1(4x-5y+6) & \text{Multiply each term by }-1. \\ \\ -4x+5y-6 & \text{Solution} \end{array}$

Distributing throughout parentheses and combining like terms can be combined into one problem. Order of operations says to multiply (distribute) first, then add or subtract (combine like terms). Thus, do each problem in two steps: distribute, then combine.

Example 1.4.6

Simplify $3x-2(4x-5).$

$\begin{array}{rl} 3x-2(4x-5) & \text{Distribute }-2, \text{ multiplying each term.} \\ \\ 3x-8x+10 & \text{Combine like terms }3x-8x. \\ \\ -5x+10 & \text{Solution} \end{array}$

Example 1.4.7

Simplify $5+3(2x-4).$

$\begin{array}{rl} 5+3(2x-4) & \text{Distribute 3, multiplying each term.} \\ \\ 5+6x-12 & \text{Combine like terms }5-12. \\ \\ -7+6x & \text{Solution} \end{array}$

In Example 1.4.6, −2 is distributed, not just 2. This is because a number being subtracted must always be treated like it has a negative sign attached to it. This makes a big difference, for in that example, when the −5 inside the parentheses is multiplied by −2, the result is a positive number. More involved examples of distributing and combining like terms follow.

Example 1.4.8

Simplify $2(5x-8)-6(4x+3).$

$\begin{array}{rl} 2(5x-8)-6(4x+3) & \text{Distribute 2 into the first set of parentheses and }-6\text{ into the second.} \\ \\ 10x-16-24x-18 & \text{Combine like terms }10x-24x\text{ and }-16-18. \\ \\ -14x-34 & \text{Solution} \end{array}$

Example 1.4.9

Simplify $4(3x-8)-(2x-7).$

$\begin{array}{rl} 4(3x-8)-(2x-7) & \text{The negative sign in the middle can be thought of as }-1. \\ \\ 4(3x-8)-(2x-7) & \text{Distribute 4 into the first set of parentheses and }-1\text{ into the second.} \\ \\ 12x-32-2x+7 & \text{Combine like terms }12x-2x\text{ and }-32+7. \\ \\ 10x-25& \text{Solution} \end{array}$

1.5 Terms & Definitions for Algebra

Digits can be defined as the alphabet of the Hindu–Arabic numeral system that is in common usage today. This alphabet is: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Written in set-builder notation, digits are expressed as:

$\text{Set of digits is }\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$

Natural numbers are often called counting numbers and are usually denoted by $\mathbb{N}.$ These numbers start at 1 and carry on to infinity, which is denoted by the symbol ∞. Writing the set of natural numbers in set-builder notation gives:

$\text{Set of natural numbers }(\mathbb{N})\text{ is }\{1, 2, 3, 4, 5, \dots \infty\}$

Whole numbers include the set of natural numbers and zero. Whole numbers are generally designated by $\mathbb{W}.$ In set-builder notation, the set of whole numbers is denoted by:

$\text{Set of whole numbers }(\mathbb{W})\text{ is }\{0, 1, 2, 3, 4, 5, \dots \infty\}$

Integers include the set of all whole numbers and their negatives. This means the set of integers is composed of positive whole numbers, negative whole numbers, and zero (fractions and decimals are not integers). Common symbols used to represent integers are $\mathbb{Z}$ and $\mathbb{J}.$ For this textbook, the symbol $\mathbb{Z}$ will be used to represent integers.

$\text{Set of integers }(\mathbb{Z})\text{ is }\{-\infty, \dots , -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, \dots \infty\}$

Rational numbers include all integers and all fractions, terminating decimals, and repeating decimals. Every rational number can be written as a fraction $\dfrac{a}{b},$ where $a$ and $b$ are integers. Rational numbers are denoted by the symbol $\mathbb{Q}.$ In set-builder notation, the set of rational numbers $\mathbb{Q}$ can be informally written as:

$\text{Set of rational numbers }(\mathbb{Q})\text{ is }\{{\text{all numbers defined by }\dfrac{a}{b},\text{ where }a \text{ and } b\text{ are integers}\}$

Irrational numbers include any number that cannot be defined by the fraction $\dfrac{a}{b},$ where $a$ and $b$ are integers. These are numbers that are non-repeating or non-terminating. Classic examples of irrational numbers are pi $(\pi)$ and the square roots of 2 and 3. The symbol for irrational numbers is commonly given as $\mathbb{I}$ or $\mathbb{H}.$ For this textbook, the symbol $\mathbb{I}$ will be used. In set-builder notation, the set of irrational numbers $\mathbb{I}$ can be informally written as:

$\text{Set of irrational numbers }(\mathbb{I})\text{ is }\{\text{all non-repeating or non-terminal numbers}\}$

Real numbers include the set of all rational numbers and irrational numbers. The symbol for real numbers is commonly given as $\mathbb{R}.$ In set-builder notation, the set of real numbers $\mathbb{R}$ can be informally written as:

$\text{Set of real numbers }(\mathbb{R})\text{ is }\{\text{all rational and irrational numbers}\}$

Numbers that may not yet have been encountered are imaginary numbers (commonly $i,$ sometimes $j$ ) and complex numbers $(\mathbb{C}).$ These numbers will be properly defined later in the textbook.

Imaginary numbers $(i)$ include any real number multiplied by the square root of −1.

Complex numbers $(\mathbb{C})$ are combinations of any real number, imaginary number, or a sum and difference of them.

Consecutive integers are integers that follow each other sequentially. Examples are:

$\begin{array}{l} 1, 2, 3, 4, \dots \\ 89, 90, 91, 92, \dots \\ -45, -44, -43, -42, \dots \end{array}$

Consecutive even or odd integers are numbers that skip the odd/even sequence to just show odd, odd, odd, or even, even, even. Examples are:

$\begin{array}{rlll} \text{Consecutive odds:} & 1, 3, 5, 7, \dots &\text{ or }& -5, -3, -1, 1, \dots \\ \text{Consecutive evens:} & 4, 6, 8, 10, \dots & \text{ or } & -4, -2, 0, 2, \dots \end{array}$

Prime numbers are numbers that cannot be divided by any integer other than 1 and itself. The following is a list of all the prime numbers that are found between 0 and 1000. (Note: 1 is not considered prime.)

$\begin{array}{l} \phantom{10}2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, \\ 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, \\ 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, \\ 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, \\ 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, \\ 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, \\ 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, \\ 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, \\ 941, 947, 953, 967, 971, 977, 983, 991, 997 \end{array}$

Squares are numbers multiplied by themselves. A number that is being squared is shown as having a superscript 2 attached to it. For example, 5 squared is written as 5², which equals 5 × 5 or 25.

Perfect squares are squares of whole numbers, such as 1, 4, 9, 16, 25, 36, and 49. They are found by squaring natural numbers. The following is the list of perfect squares using numbers up to 20:

$\begin{array}{llll} 1^2=1\hspace{0.5in}&\phantom{1}6^2=36\hspace{0.5in}&11^2=121\hspace{0.5in}&16^2=256 \\ \\ 2^2=4&\phantom{1}7^2=49&12^2=144&17^2=289 \\ \\ 3^2=9&\phantom{1}8^2=64&13^2=169&18^2=324 \\ \\ 4^2=16&\phantom{1}9^2=81&14^2=196&19^2=361 \\ \\ 5^2=25&10^2=100&15^2=225&20^2=400 \end{array}$

Cubes are numbers multiplied by themselves three times. A number that is being cubed is shown as having a superscript 3 attached to it. For example, 5 cubed is written as 5³, which equals 5 × 5 × 5 or 125.

Perfect cubes are cubes of whole numbers, such as 1, 8, 27, 64, 125, 216, and 343. They are found by cubing natural numbers. The following is the list of perfect cubes using numbers up to 20:

$\begin{array}{llll} 1^3=1\hspace{0.5in}&\phantom{1}6^3=216\hspace{0.5in}&11^3=1331\hspace{0.5in}&16^3=4096 \\ \\ 2^3=8&\phantom{1}7^3=343&12^3=1728&17^3=4913 \\ \\ 3^3=27&\phantom{1}8^3=512&13^3=2197&18^3=5832 \\ \\ 4^3=64&\phantom{1}9^3=729&14^3=2744&19^3=6859 \\ \\ 5^3=125&10^3=1000&15^3=3375&20^3=8000 \end{array}$

Percentage means parts per hundred. A percentage can be thought of as a fraction $\dfrac{a}{b},$ where $a,$ the numerator, is the number to the left of the % sign, and $b,$ the denominator, is 100. For example: $42\% = \dfrac{42}{100} = 0.42.$

Absolute values. The absolute value of an expression $a,$ denoted $| a |,$ is the distance from zero of the number or operation that occurs between the absolute value signs. For example:

$| -4 | = 4 \text{ or } | -9 | = 9$

Examples of absolute values of simple operations are:

$\begin{array}{ccccc} |-8+6|=2&\text{since}&-8+6=-2&\text{and}&|-2|=2 \\ \\ &&\text{or}&& \\ \\ |-8\times 5|=40&\text{since}&-8\times 5=-40&\text{and}&|-40|=40 \end{array}$

Set-builder notation follows standard patterns and is as follows:

Begin the set with a left brace {
A vertical bar | means “such that”
End the set with a right brace }

So to say $X$ is an integer, write this as:

$\{X | X\text{ is an integer}\}$

This means “the set of $X,$ such that $X$ is an integer.”

Another way of writing this is to use the symbols that mean “element of” and “not an element of.”

“Element of” is shown by the symbol ∈, and “not an element of” is shown by the element symbol with a line drawn through it, ∉.

In simplest terms, if something is an element of something else, it means that it belongs to or is part of it. For example, a set of numbers called $A$ can only be made up of any natural number $(\mathbb{N}),$ like 4, 6, 9, and 15. This can be stated as $\{A | A \in \mathbb{N}\},$ which reads as “the set of $A,$ such that $A$ is an element of the natural number system.”

“Not an element of” can be used to state that the set cannot contain excluded values. For example, say there is a set $C$ of all numbers $\mathbb{R}$ except counting numbers $\mathbb{N}.$ This can be written as:

$\{C | C \in \mathbb{R,}\text{ but }C \notin \mathbb{N}\}$

This can be read as “the set of $C,$ such that $C$ is an element of the set of all real numbers, excluding those numbers that are natural numbers.”

Sets of numbers giving excluded values can be seen throughout this textbook. The standard example is to exclude values that would result in a denominator of zero. This exclusion avoids division by zero and getting an undefinable answer.

The empty set. Sometimes, a set contains no elements. This set is termed the “empty set” or the “null set.” To represent this, write either { } or Ø.

1.6 Unit Conversion Word Problems

One application of rational expressions deals with converting units. Units of measure can be converted by multiplying several fractions together in a process known as dimensional analysis.

The trick is to decide what fractions to multiply. If an expression is multiplied by 1, its value does not change. The number 1 can be written as a fraction in many different ways, so long as the numerator and denominator are identical in value. Note that the numerator and denominator need not be identical in appearance, but rather only identical in value. Below are several fractions, each equal to 1, where the numerator and the denominator are identical in value. This is why, when doing dimensional analysis, it is very important to use units in the setup of the problem, so as to ensure that the conversion factor is set up correctly.

Example 1.6.1

If 1 pound = 16 ounces, how many pounds are in 435 ounces?

$\begin{array}{rrll} 435\text{ oz}&=&435\text{ \cancel{oz}}\times \dfrac{1\text{ lb}}{16\text{ \cancel{oz}}} \hspace{0.2in}& \text{This operation cancels the oz and leaves the lbs} \\ \\ &=&\dfrac{435\text{ lb}}{16} \hspace{0.2in}& \text{Which reduces to } \\ \\ &=&27\dfrac{3}{16}\text{ lb} \hspace{0.2in}& \text{Solution} \end{array}$

The same process can be used to convert problems with several units in them. Consider the following example.

Example 1.6.2

A student averaged 45 miles per hour on a trip. What was the student’s speed in feet per second?

$\begin{array}{rrll} 45 \text{ mi/h}&=&\dfrac{\text{45 \cancel{mi}}}{\text{\cancel{hr}}}\times \dfrac{5280 \text{ ft}}{1\text{ \cancel{mi}}}\times \dfrac{1\text{ \cancel{hr}}}{3600\text{ s}}\hspace{0.2in}&\text{This will cancel the miles and hours} \\ \\ &=&45\times \dfrac{5280}{1}\times \dfrac{1}{3600} \text{ ft/s}\hspace{0.2in}&\text{This reduces to} \\ \\ &=&66\text{ ft/s}\hspace{0.2in}&\text{Solution} \end{array}$

Example 1.6.3

Convert 8 ft³ to yd³.

$\begin{array}{rrll} 8\text{ ft}^3&=&8\text{ ft}^3 \times \dfrac{(1\text{ yd})^3}{(3\text{ ft})^3}&\text{Cube the parentheses} \\ \\ &=&8\text{ }\cancel{\text{ft}^3}\times \dfrac{1\text{ yd}^3}{27\text{ }\cancel{\text{ft}^3}}&\text{This will cancel the ft}^3\text{ and replace them with yd}^3 \\ \\ &=&8\times \dfrac{1\text{ yd}^3}{27}&\text{Which reduces to} \\ \\ &=&\dfrac{8}{27}\text{ yd}^3\text{ or }0.296\text{ yd}^3&\text{Solution} \end{array}$

Example 1.6.4

A room is 10 ft by 12 ft. How many square yards are in the room? The area of the room is 120 ft² (area = length × width).

Converting the area yields:

$\begin{array}{rrll} 120\text{ ft}^2&=&120\text{ }\cancel{\text{ft}^2}\times \dfrac{(1\text{ yd})^2}{(3\text{ }\cancel{\text{ft}})^2}&\text{Cancel ft}^2\text{ and replace with yd}^2 \\ \\ &=&\dfrac{120\text{ yd}^2}{9}&\text{This reduces to} \\ \\ &=&13\dfrac{1}{3}\text{ yd}^2&\text{Solution} \\ \\ \end{array}$

The process of dimensional analysis can be used to convert other types of units as well. Once relationships that represent the same value have been identified, a conversion factor can be determined.

Example 1.6.5

A child is prescribed a dosage of 12 mg of a certain drug per day and is allowed to refill his prescription twice. If there are 60 tablets in a prescription, and each tablet has 4 mg, how many doses are in the 3 prescriptions (original + 2 refills)?

$\begin{array}{rrll} 3\text{ prescriptions}&=&3\text{ \cancel{pres.}}\times \dfrac{60\text{ \cancel{tablets}}}{1\text{ \cancel{pres.}}}\times \dfrac{4\text{ \cancel{mg}}}{1\text{ \cancel{tablet}}}\times \dfrac{1\text{ dosage}}{12\text{ \cancel{mg}}}&\text{This cancels all unwanted units} \\ \\ &=&\dfrac{3\times 60\times 4\times 1}{1\times 1\times 12}\text{ or }\dfrac{720}{12}\text{ dosages}&\text{Which reduces to} \\ \\ &=&60\text{ daily dosages}&\text{Solution} \\ \\ \end{array}$

Metric and Imperial (U.S.) Conversions

Distance

$\begin{array}{rrlrrl} 12\text{ in}&=&1\text{ ft}\hspace{1in}&10\text{ mm}&=&1\text{ cm} \\ 3\text{ ft}&=&1\text{ yd}&100\text{ cm}&=&1\text{ m} \\ 1760\text{ yds}&=&1\text{ mi}&1000\text{ m}&=&1\text{ km} \\ 5280\text{ ft}&=&1\text{ mi}&&& \end{array}$

Imperial to metric conversions:

$\begin{array}{rrl} 1\text{ inch}&=&2.54\text{ cm} \\ 1\text{ ft}&=&0.3048\text{ m} \\ 1\text{ mile}&=&1.61\text{ km} \end{array}$

Area

$\begin{array}{rrlrrl} 144\text{ in}^2&=&1\text{ ft}^2\hspace{1in}&10,000\text{ cm}^2&=&1\text{ m}^2 \\ 43,560\text{ ft}^2&=&1\text{ acre}&10,000\text{ m}^2&=&1\text{ hectare} \\ 640\text{ acres}&=&1\text{ mi}^2&100\text{ hectares}&=&1\text{ km}^2 \end{array}$

Imperial to metric conversions:

$\begin{array}{rrl} 1\text{ in}^2&=&6.45\text{ cm}^2 \\ 1\text{ ft}^2&=&0.092903\text{ m}^2 \\ 1\text{ mi}^2&=&2.59\text{ km}^2 \end{array}$

Volume

$\begin{array}{rrlrrl} 57.75\text{ in}^3&=&1\text{ qt}\hspace{1in}&1\text{ cm}^3&=&1\text{ ml} \\ 4\text{ qt}&=&1\text{ gal}&1000\text{ ml}&=&1\text{ litre} \\ 42\text{ gal (petroleum)}&=&1\text{ barrel}&1000\text{ litres}&=&1\text{ m}^3 \end{array}$

Imperial to metric conversions:

$\begin{array}{rrl} 16.39\text{ cm}^3&=&1\text{ in}^3 \\ 1\text{ ft}^3&=&0.0283168\text{ m}^3 \\ 3.79\text{ litres}&=&1\text{ gal} \end{array}$

Mass

$\begin{array}{rrlrrl} 437.5\text{ grains}&=&1\text{ oz}\hspace{1in}&1000\text{ mg}&=&1\text{ g} \\ 16\text{ oz}&=&1\text{ lb}&1000\text{ g}&=&1\text{ kg} \\ 2000\text{ lb}&=&1\text{ short ton}&1000\text{ kg}&=&1\text{ metric ton} \end{array}$

Imperial to metric conversions:

$\begin{array}{rrl} 453\text{ g}&=&1\text{ lb} \\ 2.2\text{ lb}&=&1\text{ kg} \end{array}$

Temperature

Fahrenheit to Celsius conversions:

$\begin{array}{rrl} ^{\circ}\text{C} &= &\dfrac{5}{9} (^{\circ}\text{F} - 32) \\ \\ ^{\circ}\text{F}& =& \dfrac{9}{5}(^{\circ}\text{C} + 32) \end{array}$

Fahrenheit to Celsius conversion scale. Long description available.

Celsius to Fahrenheit conversion scale. [Long Description]

Long Descriptions

Celsius to Fahrenheit conversion scale long description: Scale showing conversions between Celsius and Fahrenheit. The following table summarizes the data:

Celsius	Fahrenheit
−40°C	−40°F
−30°C	−22°F
−20°C	−4°F
−10°C	14°F
0°C	32°F
10°C	50°F
20°C	68°F
30°C	86°F
40°C	104°F
50°C	122°F
60°C	140°F
70°C	158°F
80°C	176°F
90°C	194°F
100°C	212°F

[Return to Celsius to Fahrenheit conversion scale]

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1.3 Order of Operations

1.5 Terms & Definitions for Algebra

1.6 Unit Conversion Word Problems

Metric and Imperial (U.S.) Conversions

Distance

Area

Volume

Mass

Temperature

Long Descriptions

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