Pre-Algebra Articles

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 04-04-2023

A polyhedron is the three-dimensional equivalent of a polygon, which is a shape that has only straight sides. Similarly, a polyhedron is a solid that has only straight edges and flat faces (that is, faces that are polygons). The most common polyhedron is the cube. As you can see, a cube has 6 flat faces that are polygons — in this case, all of the faces are square — and 12 straight edges. Additionally, a cube has 8 vertexes (corners). The above figure shows a few common polyhedrons. The above figure shows a special set of polyhedrons called the five regular solids. Each regular solid has identical faces that are regular polygons. Notice that a cube is a type of regular solid. Similarly, the tetrahedron is a pyramid with four faces that are equilateral triangles.

Article / Updated 10-24-2022

The distinction between numbers and numerals is subtle but important. A number is an idea that expresses how much or how many. A numeral is a written symbol that expresses a number. Here are ten ways to represent numbers that differ from the Hindu-Arabic (decimal) system. Tally marks Numbers are abstractions that stand for real things. The first known numbers came into being with the rise of trading and commerce — people needed to keep track of commodities such as animals, harvested crops, or tools. At first, traders used clay or stone tokens to help simplify the job of counting. Over time, tally marks scratched either in bone or on clay took the place of tokens. Bundled tally marks As early humans grew more comfortable letting tally marks stand for real-world objects, the next development in numbers was probably tally marks scratched in bundles of 5 (fingers on one hand), 10 (fingers on both hands), or 20 (fingers and toes). Bundling provided a simple way to count larger numbers more easily. Of course, this system is much easier to read than non-bundled scratches — you can easily multiply or count by fives to get the total. Even today, people keep track of points in games using bundles such as these. Egyptian numerals Ancient Egyptian numerals are among the oldest number systems still in use today. Egyptian numerals use seven symbols. Egyptian Numerals Number Symbol 1 Stroke 10 Yoke 100 Coil of rope 1,000 Lotus 10,000 Finger 100,000 Frog 1,000,000 Man with raised hands Numbers are formed by accumulating enough of the symbols that you need. For example, 7 = 7 strokes 24 = 2 yokes, 4 strokes 1,536 = 1 lotus, 5 coils of rope, 3 yokes, 6 strokes Babylonian numerals Babylonian numerals, which came into being about 4,000 years ago, use two symbols: 1 = Y 10 = < For numbers less than 60, numbers are formed by accumulating enough of the symbols you need. For example, 6 = YYYYYY 34 = << For numbers 60 and beyond, Babylonian numerals use place value based on the number 60. 61 = Y Y (one 60 and one 1) 124 = YY YYYY (two 60s and four 1s) 611 = < (ten 60s and eleven 1s) Ancient Greek numerals Ancient Greek numerals were based on the Greek letters. The numbers from 1 to 999 were formed using the symbols shown: Roman numerals Although Roman numerals are over 2,000 years old, people still use them today, either decoratively (for example, on clocks, cornerstones, and Super Bowl memorabilia) or when numerals distinct from decimal numbers are needed (for example, in outlines). Roman numerals use seven symbols, all of which are capital letters in the Latin alphabet (which pretty much happens to be the English alphabet as well): I = 1 V = 5 X = 10 L = 50 C = 100 D = 500 M = 1,000 Mayan numerals Mayan numerals developed in South America during roughly the same period that Roman numerals developed in Europe. Mayan numerals use two symbols: dots and horizontal bars. A bar is equal to 5, and a dot is equal to 1. Numbers from 1 to 19 are formed by accumulating dots and bars. For example, 3 = 3 dots 7 = 2 dots over 1 bar 19 = 4 dots over 3 bars Numbers from 20 to 399 are formed using these same combinations, but raised up to indicate place value. For example, 21 = raised 1 dot, 1 dot (one 20 + one 1) 399 = raised 4 dots over 3 bars, 4 dots over 3 bars (nineteen 20s + three 5s + four 1s) Base-2 (binary) numbers Binary numbers use only two symbols: 0 and 1. This simplicity makes binary numbers useful as the number system that computers use for data storage and computation. Like the decimal system you're most familiar with, binary numbers use place value. Unlike the decimal system, binary place value is based not on powers of ten (1, 10, 100, 1,000, and so forth) but on powers of two (20, 21, 22, 23, 24, 25, 26, 27, 28, 29, and so on), as seen here: Binary Place Values 512s 256s 128s 64s 32s 16s 8s 4s 2s 1s Base-16 (hexadecimal) numbers The computer's first language is binary numbers. But in practice, humans find binary numbers of any significant length virtually undecipherable. Hexadecimal numbers, however, are readable to humans and still easily translated into binary numbers, so computer programmers use hexadecimal numbers as a sort of common language when interfacing with computers at the deepest level, the level of hardware and software design. The hexadecimal number system uses all ten digits 0 through 9 from the decimal system. Additionally, it uses six more symbols: A = 10 B = 11 C = 12 D = 13 E = 14 F = 15 Hexadecimal is a place-value system based on powers of 16. Hexadecimal Place Values 1,048,576s 65,536s 4,096s 256s 16s 1s As you can see, each number in the table is exactly 16 times the number to its immediate right. Prime-based numbers One wacky way to represent numbers unlike any of the others is prime-based numbers. Prime-based numbers are similar to decimal, binary, and hexadecimal numbers in that they use place value to determine the value of digits. But unlike these other number systems, prime-based numbers are based not on addition but on multiplication. Prime-Based Place Values 31s 29s 23s 19s 17s 13s 11s 7s 5s 3s 2s You can use the table to find the decimal value of a prime-based number.

Article / Updated 09-27-2022

To multiply two decimals, don’t worry about lining up the decimal points. In fact, to start out, ignore the decimal points. Here’s how the multiplication works: Perform the multiplication just as you would for whole numbers. When you’re done, count the number of digits to the right of the decimal point in each factor and add the results. Place the decimal point in your answer so that your answer has the same number of digits after the decimal point. Even if the last digit in the answer is 0, you still need to count this as a digit when placing the decimal point in a multiplication problem. After the decimal point is in place, however, you can drop trailing zeros. Sample question Multiply the following decimals: 74.2 x 0.35 = ? 25.97. Ignoring the decimal points, perform the multiplication just as you would for whole numbers: At this point, you’re ready to find out where the decimal point goes in the answer. Count the number of decimal places in the two factors (74.2 and 0.35), add these two numbers together (1 + 2 = 3), and place the decimal point in the answer so that it has three digits to the right of the decimal point: Practice questions Multiply these decimals: 0.635 x 0.42 = ? Perform the following decimal multiplication: 0.675 x 34.8 = ? Solve the following multiplication problem: 943 x 0.0012 = ? Find the solution to this decimal multiplication: 1.006 x 0.0807 = ? Following are answers to the practice questions: 0.635 x 0.42 = 0.2667. Place the first number on top of the second number, ignoring the decimal points. Complete the multiplication as you would for whole numbers: At this point, you’re ready to find out where the decimal point goes in the answer. Count the number of decimal places in the two factors, add these two numbers together (3 + 2 = 5), and place the decimal point in the answer so that it has five digits after the decimal point. After you place the decimal point (but not before!), you can drop the trailing zero. 0.675 x 34.8 = 23.49. Ignore the decimal points and simply place the first number on top of the second. Complete the multiplication as you would for whole numbers: Count the number of decimal places in the two factors, add these two numbers together (3 + 1 = 4), and place the decimal point in the answer so that it has four digits after the decimal point. Last, you can drop the trailing zeros. 943 x 0.0012 = 1.1316. Complete the multiplication as you would for whole numbers: Zero digits come after the decimal point in the first factor, and you have four after-decimal digits in the second factor, for a total of 4 (0 + 4 = 4); place the decimal point in the answer so that it has four digits after the decimal point. 1.006 x 0.0807 = 0.0811842. Complete the multiplication as you would for whole numbers: You have a total of seven digits after the decimal points in the two factors — three in the first factor and four in the second (3 + 4 = 7) — so place the decimal point in the answer so that it has seven digits after the decimal point. Notice that you need to create an extra decimal place in this case by attaching an additional nontrailing 0.

Cheat Sheet / Updated 03-17-2022

Following are nine little math demons that plague all sorts of otherwise smart, capable folks like you. The good news is that they’re not as big and scary as you may think, and they can be dispelled more easily than you may have dared believe.

Article / Updated 03-15-2022

When the fractions that you want to add have different denominators, there are a few different ways you can do it. Add fractions the easy way At some point in your life, some teacher somewhere told you these golden words of wisdom: “You can’t add two fractions with different denominators.” Your teacher was wrong! You can use the easy way when the numerators and denominators are small (say, 15 or under). Here’s the way to do it: Cross-multiply the two fractions and add the results together to get the numerator of the answer. Suppose you want to add the fractions 1/3 and 2/5. To get the numerator of the answer, cross-multiply. In other words, multiply the numerator of each fraction by the denominator of the other: 1*5 = 5 2*3 = 6 Add the results to get the numerator of the answer: 5 + 6 = 11 Multiply the two denominators together to get the denominator of the answer. To get the denominator, just multiply the denominators of the two fractions: 3*5 = 15 The denominator of the answer is 15. Write your answer as a fraction. When you add fractions, you sometimes need to reduce the answer that you get. Here’s an example: Because the numerator and the denominator are both even numbers, you know that the fraction can be reduced. So try dividing both numbers by 2: This fraction can’t be reduced further, so 37/40 is the final answer. In some cases, you may have to add more than one fraction. The method is similar, with one small tweak. Start out by multiplying the numerator of the first fraction by the denominators of all the other fractions. (1*5*7) = 35 Do the same with the second fraction and add this value to the first. 35 + (3*2*7) = 35 + 42 Do the same with the remaining fraction(s). 35 + 42 + (4*2*5) = 35 + 42 + 40 = 117 When you’re done, you have the numerator of the answer. To get the denominator, just multiply all the denominators together: You may need to reduce or change an improper fraction to a mixed number. In this example, you just need to change to a mixed number: Add fractions with the quick trick method You can’t always use this method, but you can use it when one denominator is a multiple of the other. Look at the following problem: First, solve it the easy way: Those are some big numbers, and you’re still not done because the numerator is larger than the denominator. The answer is an improper fraction. Worse yet, the numerator and denominator are both even numbers, so the answer still needs to be reduced. With certain fraction addition problems, there is a smarter way to work. The trick is to turn a problem with different denominators into a much easier problem with the same denominator. Before you add two fractions with different denominators, check the denominators to see whether one is a multiple of the other. If it is, you can use the quick trick: Increase the terms of the fraction with the smaller denominator so that it has the larger denominator. Look at the earlier problem in this new way: As you can see, 12 divides into 24 without a remainder. In this case, you want to raise the terms of 11/12 so that the denominator is 24: To fill in the question mark, the trick is to divide 24 by 12 to find out how the denominators are related; then multiply the result by 11: ? = (24 ÷ 12) 11 = 22 Rewrite the problem, substituting this increased version of the fraction, and add. Now you can rewrite the problem this way: As you can see, the numbers in this case are much smaller and easier to work with. The answer here is an improper fraction; changing it into a mixed number is easy: Add fractions the traditional way Use the traditional way only when you can’t use either of the other methods (or when you know the least common multiple [LCM] just by looking at the denominators). Here’s the traditional way to add fractions with two different denominators: Find the LCM of the two denominators. Suppose you want to add the fractions 3/4 + 7/10. First find the LCM of the two denominators, 4 and 10. Here’s how to find the LCM using the multiplication table method: Multiples of 10: 10, 20, 30, 40 Multiples of 4: 4, 8, 12, 16, 20 So the LCM of 4 and 10 is 20. Increase the terms of each fraction so that the denominator of each equals the LCM. Increase each fraction to higher terms so that the denominator of each is 20. Substitute these two new fractions for the original ones and add. At this point, you have two fractions that have the same denominator: When the answer is an improper fraction, you still need to change it to a mixed number:

Cheat Sheet / Updated 02-24-2022

To successfully study pre-algebra, understand that a specific order of operations needs to be applied. Also recognize some basic math principles, such as the ability to recognize and understand mathematical inequalities, place value, absolute value, and negation.

Article / Updated 07-13-2021

Even if fractions look different, they can actually represent the same amount; in other words, one of the fractions will have reduced terms compared to the other. You may need to reduce the terms of fractions to work with them in an equation. Reducing fractions to their lowest terms involves division. But because you can’t always divide, reducing takes some finesse. Here you will learn the formal way to reduce fractions, which works in all cases. Then you will learn a more informal way that you can use after you’re more comfortable. Method 1: Reduce fractions the formal way Reducing fractions the formal way relies on an understanding of how to break down a number into its prime factors. Here’s how to reduce a fraction: Break down both the numerator (top number) and denominator (bottom number) into their prime factors. For example, suppose you want to reduce the fraction 12/30. Break down both 12 and 30 into their prime factors: Cross out any common factors. In this example, you cross out a 2 and a 3, because they’re common factors — that is, they appear in both the numerator and denominator: Multiply the remaining numbers to get the reduced numerator and denominator. This shows you that the fraction 12/30 reduces to 2/5: As another example, here’s how you reduce the fraction 32/100: This time, cross out two 2s from both the top and the bottom as common factors. The remaining 2s on top, and the 5s on the bottom, aren’t common factors. So the fraction 32/100 reduces to 8/25. Method 2: Reduce fractions the informal way Here’s an easier way to reduce fractions after you get comfortable with the concept: If the numerator (top number) and denominator (bottom number) are both divisible by 2 — that is, if they’re both even — divide both by 2. For example, suppose you want to reduce the fraction 24/60. The numerator and the denominator are both even, so divide them both by 2: Repeat Step 1 until the numerator or denominator (or both) is no longer divisible by 2. In the resulting fraction, both numbers are still even, so repeat the first step again: Repeat Step 1 using the number 3, and then 5, and then 7, continuing testing prime numbers until you’re sure that the numerator and denominator have no common factors. Now, the numerator and the denominator are both divisible by 3, so divide both by 3: Neither the numerator nor the denominator is divisible by 3, so this step is complete. At this point, you can move on to test for divisibility by 5, 7, and so on, but you really don’t need to. The numerator is 2, and it obviously isn’t divisible by any larger number, so you know that the fraction 24/60 reduces to 2/5.

Article / Updated 07-13-2021

The volume of an object is how much space the object takes up — or, if you were to drop the object into a full tub of water, how much water would overflow. Capacity is how much space an object has inside — or, how much water you can fit inside the object. This distinction between volume and capacity is subtle — you can measure both in cm3, although confusingly you can also measure capacity in milliliters, each of which is the same size as 1 cm3. A liter contains 1,000 millilitres, and a cubic meter contains 1,000 liters. Incidentally, a cubic centimeter is the volume of a cube which has edges that are one centimeter long — about the size of a normal die. For the numeracy curriculum, you may need to work out the volume of a cuboid or shoebox. You normally know the width, height and depth of the box. To work out the volume, you simply times the three numbers together. A classic problem in numeracy exams involves working out how many small boxes fit into a bigger box. This kind of packing problem has real-life applications (how many DVDs can you fit into a box? Will this crate hold all the copies of Basic Maths For Dummies you want to send to your friends around the world?) and is quite straightforward. In an exam, you normally know the orientation — or which way round you need to pack the little boxes into the big box. Follow these steps to work out how to fit little boxes into a bigger box: Work out how many boxes you can fit along the front of the box. Divide the width of the big box by the width of one small box and write down the result. If you get a whole number answer, great! If not, round down, because even if your answer is 5.99, you can’t squeeze a sixth little box into the crate. Work out how many boxes you can fit along the side of the box. Divide the depth of the big box by the depth of the little box and write down the answer. Round down if you don’t have a whole number. Work out how many boxes you can fit going up the box. Divide the height of the big box by the height of the small box and write down the number. Round down if you need to. Times the three numbers together. That’s your answer! Here’s a typical question to follow as an example: A crate is 4 meters wide, 12 meters long and 3 meters deep. You want to fill it with boxes that are 2 meters wide, 3 meters long and 1 meter deep. How many boxes will fit in the crate? You can fit two boxes along the width of the crate. You can fit four boxes along the length of the crate. You can fit three boxes along the depth of the crate. You need to times those numbers together. 2 x 4 x 3 = 8 x 3 = 24. You can fit 24 boxes into the crate.

Article / Updated 07-12-2021

A lot of percent problems turn out to be easy to solve when you give them a little thought. In many cases, just remember the connection between percents and fractions and you’re halfway home. Solve simple percent problems Some percents are easy to figure. Here are a few. Finding 100% of a number: Remember that 100% means the whole thing, so 100% of any number is simply the number itself: 100% of 5 is 5 100% of 91 is 91 100% of 732 is 732 Finding 50% of a number: Remember that 50% means half, so to find 50% of a number, just divide it by 2: 50% of 20 is 10 50% of 88 is 44 Finding 25% of a number: Remember that 25% equals 1/4, so to find 25% of a number, divide it by 4: 25% of 40 is 10 25% of 88 is 22 Finding 20% of a number: Finding 20% of a number is handy if you like the service you’ve received in a restaurant, because a good tip is 20% of the check. Because 20% equals 1/5, you can find 20% of a number by dividing it by 5. But you can use an easier way: To find 20% of a number, move the decimal point one place to the left and double the result: 20% of 80 = 8 2 = 16 20% of 300 = 30 2 = 60 20% of 41 = 4.1 2 = 8.2 Finding 10% of a number: Finding 10% of any number is the same as finding 1/10 of that number. To do this, just move the decimal point one place to the left: 10% of 30 is 3 10% of 41 is 4.1 10% of 7 is 0.7 Finding 200%, 300%, and so on of a number: Working with percents that are multiples of 100 is easy. Just drop the two 0s and multiply by the number that’s left: 200% of 7 = 2 7 = 14 300% of 10 = 3 10 = 30 1,000% of 45 = 10 45 = 450 Make tough-looking percent problems easy Here’s a trick that makes certain tough-looking percent problems so easy that you can do them in your head. Simply move the percent sign from one number to the other and flip the order of the numbers. Suppose someone wants you to figure out the following: 88% of 50 Finding 88% of anything isn’t an activity that anybody looks forward to. But an easy way of solving the problem is to switch it around: 88% of 50 = 50% of 88 This move is perfectly valid, and it makes the problem a lot easier. As you learned above, 50% of 88 is simply half of 88: 88% of 50 = 50% of 88 = 44 As another example, suppose you want to find 7% of 200 Again, finding 7% is tricky, but finding 200% is simple, so switch the problem around: 7% of 200 = 200% of 7 Above, you learned that to find 200% of any number, you just multiply that number by 2: 7% of 200 = 200% of 7 = 2 7 = 14 Solve more-difficult percent problems You can solve a lot of percent problems using the tricks shown above. But what about this problem? 35% of 80 = ? Ouch — this time, the numbers you’re working with aren’t so friendly. When the numbers in a percent problem become a little more difficult, the tricks no longer work, so you want to know how to solve all percent problems. Here’s how to find any percent of any number: Change the word of to a multiplication sign and the percent to a decimal. Changing the word of to a multiplication sign is a simple example of turning words into numbers. This change turns something unfamiliar into a form that you know how to work with. So, to find 35% of 80, you would rewrite it as: 35% of 80 = 0.35 80 Solve the problem using decimal multiplication. Here’s what the example looks like: So 35% of 80 is 28. As another example, suppose you want to find 12% of 31. Again, start by changing the percent to a decimal and the word of to a multiplication sign: 12% of 31 = 0.12 31 Now you can solve the problem with decimal multiplication: So 12% of 31 is 3.72.