Math Articles

Cheat Sheet / Updated 09-14-2023

Some of the most important things to remember in AS-level and A-level maths are the rules for differentiating and integrating expressions. This cheat sheet is a handy reference for what happens when you differentiate or integrate powers of x, trigonometric functions, exponentials or logarithms – as well as the rules you need for what to do when they’re combined!

Article / Updated 09-13-2023

A pie chart, which looks like a divided circle, shows you how a whole object is cut up into parts. Pie charts are most often used to represent percentages. For example, the following figure is a pie chart representing Eileen’s monthly expenses. You can tell at a glance that Eileen’s largest expense is rent and that her second largest is her car. Unlike a bar graph, the pie chart shows numbers that are dependent upon each other. For example, if Eileen’s rent increases to 30% of her monthly income, she’ll have to decrease her spending in at least one other area. Here are a few typical questions you may be asked about a pie chart: Individual percentages: What percentage of her monthly expenses does Eileen spend on food? Find the slice that represents what Eileen spends on food, and notice that she spends 10% of her income there. Differences in percentages: What percentage more does she spend on her car than on entertainment? Eileen spends 20% on her car but only 5% on entertainment, so the difference between these percentages is 15%. How much a percent represents in terms of dollars: If Eileen brings home $2,000 per month, how much does she put away in savings each month? First notice that Eileen puts 15% every month into savings. So you need to figure out 15% of $2,000. Solve this problem by turning 15% into a decimal and multiplying: 0.15 2,000 = 300 So Eileen saves $300 every month.

Article / Updated 08-28-2023

When you want to count up how many things are in a set, you have quite a few options. When the set contains too many elements to count accurately, you look for some sort of pattern or rule to help out. Here, you practice the multiplication property. If you can do task one in m1 ways, task two in m2 ways, task three in m3 ways, and so on, then you can perform all the tasks in a total of m1 · m2 · m3 . . . ways. Sample questions How many ways can you fly from San Francisco to New York City, stopping in Denver, Chicago, and Buffalo, if the website offers four ways to fly from San Francisco to Denver, six ways to fly from Denver to Chicago, two ways to fly from Chicago to Buffalo, and three ways to fly from Buffalo to New York City? 144. Multiply 4 x 6 x 2 x 3 = 144. This method doesn’t tell you what all the routes are; it just tells you how many are possible so you know when you’ve listed all of them. (Better get to work on that.) How many ways can you write a password if the first symbol has to be a digit from 1 to 9; the second, third, and fourth symbols have to be letters of the English alphabet; and the last symbol has to be from the set {!, @, #, $, %, ^, &, *, +}? 1,423,656. You multiply 9 x 26 x 26 x 26 x 9 = 1,423,656. This system allows a lot of passwords, but most institutions make you use eight or more characters, which makes the number of possibilities even greater. Practice questions If you have to take one class in each subject, how many different course loads can you create if you have a choice of four math classes, three history classes, eight English classes, and five science classes? How many different ice-cream sundaes can you create if you have a choice of five ice-cream flavors, three sauces, and five sprinkled toppings if you choose one of each type? How many different automobiles can you order if you have a choice of six colors, four interiors, two trim options, three warranties, and two types of seats? How many different dinners can you order if you have a choice of 12 appetizers, 8 entrees, 5 potatoes, 6 desserts, and a choice of soup or salad? Following are answers to the practice questions: The answer is 480. Multiply: 4 x 3 x 8 x 5 = 480. The answer is 75. Multiply: 5 x 3 x 5 = 75. The answer is 288. Multiply: 6 x 4 x 2 x 3 x 2 = 288. The answer is 5,760. Multiply: 12 x 8 x 5 x 6 x 2 = 5,760. Don’t forget that soup or salad is two choices for that selection.

Article / Updated 08-14-2023

As you work through pre-calculus, adopting certain tasks as habits can help prepare your brain to tackle your next challenge: calculus. In this article, you find ten habits that should be a part of your daily math arsenal. Perhaps you’ve been told to perform some of these tasks since elementary school — such as showing all your work — but other tricks may be new to you. Either way, if you remember these ten pieces of advice, you’ll be even more ready for whatever calculus throws your way. Figure Out What the Problem Is Asking Often, you’ll find that reading comprehension and the ability to work with multiple parts that comprise a whole is an underlying property of a math problem. That’s okay — that’s also what life is all about!! When faced with a math problem, start by reading the whole problem or all the directions to the problem. Look for the question inside the question. Keep your eyes peeled for words like solve, simplify, find, and prove, all of which are common buzz words in any math book. Don’t begin working on a problem until you’re certain of what it wants you to do. For example, take a look at this problem: The length of a rectangular garden is 24 feet longer than the garden’s width. If you add 2 feet to both the width and the length, the area of the garden is 432 square feet. How long is the new, bigger garden? If you miss any of the important information, you may start to solve the problem to figure out how wide the garden is. Or you may find the length but miss the fact that you’re supposed to find out how long it is with 2 feet added to it. Look before you leap! Underlining key words and information in the question is often helpful. This can’t be stressed enough. Highlighting important words and pieces of information solidifies them in your brain so that as you work, you can redirect your focus if it veers off-track. When presented with a word problem, for example, first turn the words into an algebraic equation. If you’re lucky and are given the algebraic equation from the get-go, you can move on to the next step, which is to create a visual image of the situation at hand. And, if you’re wondering what the answer to the example problem is, you’ll find out as you read further. Draw Pictures (the More the Better) Your brain is like a movie screen in your skull, and you’ll have an easier time working problems if you project what you see onto a piece of paper. When you visualize math problems, you’re more apt to comprehend them. Draw pictures that correspond to a problem and label all the parts so you have a visual image to follow that allows you to attach mathematical symbols to physical structures. This process works the conceptual part of your brain and helps you remember important concepts. As such, you’ll be less likely to miss steps or get disorganized. If the question is talking about a triangle, for instance, draw a triangle; if it mentions a rectangular garden filled with daffodils for 30 percent of its space, draw that. In fact, every time a problem changes and new information is presented, your picture should change, too. If you were asked to solve the rectangular garden problem from the previous section, you’d start by drawing two rectangles: one for the old, smaller garden and another for the bigger one. Putting pen or pencil to the paper starts you on the way to a solution. Plan Your Attack — Identify Your Targets When you know and can picture what you must find, you can plan your attack from there, interpreting the problem mathematically and coming up with the equations that you’ll be working with to find the answer: Start by writing a statement. In the garden problem from the last two sections, you’re looking for the length and width of a garden after it has been made bigger. With those in mind, define some variables: Let x = the garden’s width now. Let y = the garden’s length now. Now add those variables to the rectangle you drew of the old garden. But you know that the length is 24 feet greater than the width, so you can rewrite the variable y in terms of the variable x so that y = x + 24. You know that the new garden has had 2 feet added to both its width and length, so you can modify your equations: Let x + 2 = the garden’s new width. Let y + 2 = x + 24 + 2 = x + 26 the garden’s new length. Now add these labels to the picture of the new garden. By planning your attack, you’ve identified the pieces of the equation that you need to solve. Write Down Any Formulas If you start your attack by writing the formula needed to solve the problem, all you have to do from there is plug in what you know and then solve for the unknown. A problem always makes more sense if the formula is the first thing you write when solving. Before you can do that, though, you need to figure out which formula to use. You can usually find out by taking a close look at the words of the problem. In the case of the garden problem from the previous sections, the formula you need is that for the area of a rectangle. The area of a rectangle is A = lw. You’re told that the area of the new rectangle is 432 square feet, and you have expressions representing the length and width, so you can replace A = lw with 432 = (x + 26)(x + 2). As another example, if you need to solve a right triangle, you may start by writing down the Pythagorean Theorem if you know two sides and are looking for the third. For another right triangle, perhaps you’re given an angle and the hypotenuse and need to find the opposite side; in this situation, you’d start off by writing down the sine ratio. Show Each Step of Your Work Yes, you’ve been hearing it forever, but your third-grade teacher was right: Showing each step of your work is vital in math. Writing each step on paper minimizes silly mistakes that you can make when you calculate in your head. It’s also a great way to keep a problem organized and clear. And it helps to have your work written down when you get interrupted by a phone call or text message — you can pick up where you left off and not have to start all over again. It may take some time to write every single step down, but it’s well worth your investment. Know When to Quit Sometimes a problem has no solution. Yes, that can be an answer, too! If you’ve tried all the tricks in your bag and you haven’t found a way, consider that the problem may have no solution at all. Some common problems that may not have a solution include the following: Absolute-value equations This happens when the absolute value expression is set equal to a negative number. You may not realize the number is negative, at first, if it’s represented by a variable. Equations with the variable under a square-root sign If your answer has to be a real number, and complex numbers aren’t an option, then the expression under the radical may represent a negative number. Not allowed. Quadratic equations When a quadratic isn’t factorable and you have to resort to the quadratic formula, you may run into a negative under the radical; you can’t use that expression if you’re allowed only real answers. Rational equations Rational expressions have numerators and denominators. If there’s a variable in the denominator that ends up creating a zero, then that value isn’t allowed. Trig equations Trig functions have restrictions. Sines and cosines have to lie between –1 and 1. Secants and cosecants have to be greater than or equal to 1 or less than or equal to –1. A perfectly nice-looking equation may create an impossible answer. On the other hand, you may get a solution for some problem that just doesn’t make sense. Watch out for the following situations: If you’re solving an equation for a measurement (like length or area) and you get a negative answer, either you made a mistake or no solution exists. Measurement problems include distance, and distance can’t be negative. If you’re solving an equation to find the number of things (like how many books are on a bookshelf) and you get a fraction or decimal answer, then that just doesn’t make any sense. How could you have 13.4 books on a shelf? Check Your Answers Even the best mathematicians make mistakes. When you hurry through calculations or work in a stressful situation, you may make mistakes more frequently. So, check your work. Usually, this process is very easy: You take your answer and plug it back into the equation or problem description to see if it really works. Making the check takes a little time, but it guarantees you got the question right, so why not do it? For example, if you go back and solve the garden problem from earlier in this chapter by looking at the equation to be solved: 432 = (x + 26)(x + 2), you multiply the two binomials and move the 432 to the other side. Solving x2 + 28x – 380 = 0 you get x = 10 and x = –38. You disregard the x = –38, of course, and find that the original width was 10 feet. So, what was the question? It asks for the length of the new, bigger garden. The length of the new garden is found with y = x + 26. So, the length (and answer) is that the length is 36 feet. Does this check? If you use the new length or 36 and the new width of x + 2 = 12 and multiply 36 times 12 you get 432 square feet. It checks! Practice Plenty of Problems You’re not born with the knowledge of how to ride a bike, play baseball, or even speak. The way you get better at challenging tasks is to practice, practice, practice. And the best way to practice math is to work the problems. You can seek harder or more complicated examples of questions that will stretch your brain and make you better at a concept the next time you see it. Along with working along on the example problems in this book, you can take advantage of the For Dummies workbooks, which include loads of practice exercises. Check out Trigonometry Workbook For Dummies, by Mary Jane Sterling, Algebra Workbook and Algebra II Workbook For Dummies, both also by Mary Jane Sterling, and Geometry Workbook For Dummies, by Mark Ryan (all published by Wiley), to name a few. Even a math textbook is great for practice. Why not try some (gulp!) problems that weren’t assigned, or maybe go back to an old section to review and make sure you’ve still got it? Typically, textbooks show the answers to the odd problems, so if you stick with those you can always double-check your answers. And if you get a craving for some extra practice, just search the Internet for “practice math problems” to see what you can find! For example, if you search the Internet for “practice systems of equations problems” you’ll find more than a million hits. That’s a lot of practice! Keep Track of the Order of Operations Don’t fall for the trap that always is lying there by performing operations in the wrong order. For instance, 2 – 6 × 3 doesn’t become –4 × 3 = –12. You’ll reach those incorrect answers if you forget to do the multiplication first. Focus on following the order of PEMDAS every time, all the time: Parentheses (and other grouping devices) Exponents and roots Multiplication and Division from left to right Addition and Subtraction from left to right Don’t ever go out of order, and that’s an order! Use Caution When Dealing with Fractions Working with denominators can be tricky. It’s okay to write: But, on the other hand: Also, reducing or cancelling in fractions can be performed incorrectly. Every term in the numerator has to be divided by the same factor — the one that divides the denominator. So, it’s true that: because But , because the factor being divided out is just 4, not 4x. And, again, it has to be the same factor that’s dividing, throughout. Can you spot the error here? Some poor soul reduced each term and the term directly above it — separately. Nope, doesn’t work that way. The correct process is to factor the trinomials and then divide by the common factor:

Video / Updated 08-09-2023

A surface of revolution is a three-dimensional surface with circular cross sections, like a vase or a bell, or a wine bottle. This article, and the video, show you how to find its area. For these problems, you divide the surface into narrow circular bands, figure the surface area of a representative band, and then just add up the areas of all the bands to get the total surface area. The following figure shows such a shape with a representative band. What’s the surface area of a representative band? Well, if you cut the band and unroll it, you get sort of a long, narrow rectangle whose area, of course, is length times width. Surface of Revolution: A surface generated by revolving a function, y = f (x), about an axis has a surface area — between a and b — given by the following integral: By the way, in the above explanation, you might be wondering why the width of the rectangular band is It’s because the little band width is slanted instead of horizontal (in which case it would be just dx). The fact that it’s slanted makes it work like the hypotenuse of a little right triangle. The fancy-looking expression for the width of the band comes from working out the length of this hypotenuse with the Pythagorean Theorem. That should make you feel a lot better! If the axis of revolution is the x-axis, r will equal f (x) — as shown in the above figure. If the axis of revolution is some other line, like y = 5, it’s a bit more complicated — something to look forward to. Now try a problem: What’s the surface area — between x = 1 and x = 2 — of the surface generated by revolving about the x-axis? A surface of revolution — this one’s shaped sort of like the end of a trumpet. Take the derivative of your function. Now you can finish the problem by just plugging everything into the formula, but you should do it step by step to reinforce the idea that whenever you integrate, you write down a representative little bit of something — that’s the integrand — then you add up all the little bits by integrating. Figure the surface area of a representative narrow band. Add up the areas of all the bands from 1 to 2 by integrating.

Article / Updated 08-07-2023

To find the volume of a prism or cylinder, you can use the following formula, where Ab is the area of the base and h is the height: V = Ab x h Practice questions Find the volume of a prism with a base that has an area of 6 square centimeters and a height of 3 centimeters. Figure out the approximate volume of a cylinder whose base has a radius of 7 millimeters and whose height is 16 millimeters. Answers and explanations 18 cubic centimeters V = Ab x h = 6cm2 x 3cm = 18cm3 Approximately 2,461.76 cubic millimeters First, use the area formula for a circle to find the area of the base: Ab = π x r2 ≅ 3.14 x (7mm)2 = 3.14 x 49mm2 = 153.86mm2 Plug this result into the formula for the volume of a cylinder: V = Ab x h = 153.86mm2 x 16mm = 2,461.76mm3

Video / Updated 07-14-2023

The nice thing about finding the area of a surface of revolution is that there’s a formula you can use. Memorize it and you’re halfway done. To find the area of a surface of revolution between a and b, watch this video tutorial or follow the steps below: This formula looks long and complicated, but it makes more sense when you spend a minute thinking about it. The integral is made from two pieces: The arc-length formula, which measures the length along the surface The formula for the circumference of a circle, which measures the length around the surface So multiplying these two pieces together is similar to multiplying length and width to find the area of a rectangle. In effect, the formula allows you to measure surface area as an infinite number of little rectangles. When you’re measuring the surface of revolution of a function f(x) around the x-axis, substitute r = f(x) into the formula: For example, suppose that you want to find the area of revolution that’s shown in this figure. Measuring the surface of revolution of y = x3 between x = 0 and x = 1. To solve this problem, first note that for So set up the problem as follows: To start off, simplify the problem a bit: You can solve this problem by using the following variable substitution: Now substitute u for 1+ 9x4 and for x3 dx into the equation: Notice that you change the limits of integration: When x = 0, u = 1. And when x = 1, u = 10. Now you can perform the integration: Finally, evaluate the definite integral:

Article / Updated 07-10-2023

Some expressions contain only multiplication and division. When this is the case, the rule for evaluating the expression is pretty straightforward. When an expression contains only multiplication and division, evaluate it step by step from left to right. The Three Types of Big Four Expressions Expression Example Rule Contains only addition and subtraction 12 + 7 – 6 – 3 + 8 Evaluate left to right. Contains only multiplication and division 18 ÷ 3 x 7 ÷ 14 Evaluate left to right. Mixed-operator expression: contains a combination of addition/subtraction and multiplication/division 9 + 6 ÷ 3 1. Evaluate multiplication and division left to right. 2. Evaluate addition and subtraction left to right. Suppose you want to evaluate this expression: 9 × 2 ÷ 6 ÷ 3 × 2 Again, the expression contains only multiplication and division, so you can move from left to right, starting with 9 x 2: = 18 ÷ 6 ÷ 3 × 2 = 3 ÷ 3 × 2 = 1 × 2 = 2 Notice that the expression shrinks one number at a time until all that’s left is 2. So 9 × 2 ÷ 6 ÷ 3 × 2 = 2 Here’s another quick example: −2 × 6 ÷ −4 Even though this expression has some negative numbers, the only operations it contains are multiplication and division. So you can evaluate it in two steps from left to right (remembering the rules for multiplying and dividing with negative numbers): = −2 × 6 ÷ −4 = −12 ÷ −4 = 3 Thus, −2 × 6 ÷ −4 = 3

Article / Updated 07-10-2023

Cross-multiplication is a handy tool for finding the common denominator for two fractions, which is important for many operations involving fractions. In the following practice questions, you are asked to cross-multiply to compare fractions to find out which is greater or less. Practice questions 1. Find the lesser fraction: 2. Among these three fractions, which is greatest: Answers and explanations 1. Of the two fractions, Cross-multiply to compare the two fractions: Because 35 is less than 36, 2. Of the three fractions, Use cross-multiplication to compare the first two fractions. Because 21 is greater than 20, this means that 1/10 is greater than 2/21, so you can rule out 2/21. Next, compare 1/10 and 3/29 by cross-multiplying. Because 30 is greater than 29, 3/29 is greater than 1/10. Therefore, 3/29 is the greatest of the three fractions.

Article / Updated 07-10-2023

The Big Four math operations — addition, subtraction, multiplication, and division — let you combine numbers and perform calculations. Certain operations possess properties that enable you to manipulate the numbers in the problem, which comes in handy, especially when you get into higher math like algebra. The important properties you need to know are the commutative property, the associative property, and the distributive property. Understanding what an inverse operation is is also helpful. Inverse operations Inverse operations are pairs of operations that you can work "backward" to cancel each other out. Two pairs of the Big Four operations — addition, subtraction, multiplication, and division —are inverses of each other: Addition and subtraction are inverse operations of each other. When you start with any value, then add a number to it and subtract the same number from the result, the value you started with remains unchanged. For example: 2 + 3 = 5 so 5 – 3 = 2 7 – 1 = 6 so 6 + 1 = 7 Multiplication and division are inverse operations of each other. When you start with any value, then multiply it by a number and divide the result by the same number (except zero), the value you started with remains unchanged. For example: 3 × 4 = 12 so 12 ÷ 4 = 3 10 ÷ 2 = 5 so 5 × 2 = 10 The commutative property An operation is commutative when you apply it to a pair of numbers either forwards or backwards and expect the same result. The two Big Four that are commutative are addition and subtraction. Addition is commutative because, for example, 3 + 5 is the same as 5 + 3. In other words 3 + 5 = 5 + 3 Multiplication is commutative because 2 × 7 is the same as 7 × 2. In other words 2 × 7 = 7 × 2 The associative property An operation is associative when you can apply it, using parentheses, in different groupings of numbers and still expect the same result. The two Big Four operations that are associative are addition and multiplication. Addition is associative because, for example, the problem (2 + 4) + 7 produces the same result as does the problem 2 + (4 + 7). In other words, (2 + 4) + 7 = 2 + (4 + 7) No matter which pair of numbers you add together first, the answer is the same: 13. Multiplication is associative because, for example, the problem 3 × (4 × 5) produces the same result as the problem (3 × 4) × 5. In other words, 3 × (4 × 5) = (3 × 4) × 5 Again, no matter which pair of numbers you multiply first, both problems yield the same answer: 60. The distributive property The distributive property connects the operations of multiplication and addition. When multiplication is described as "distributive over addition," you can split a multiplication problem into two smaller problems and then add the results. For example, suppose you want to multiply 27 × 6. You know that 27 equals 20 + 7, so you can do this multiplication in two steps: First multiply 20 × 6; then multiply 7 × 6. 20 × 6 = 1207 × 6 = 42 Then add the results. 120 + 42 = 162 Therefore, 27 × 6 = 162.