Pre-Algebra Articles

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.

Article / Updated 07-09-2021

The greatest common factor (GCF) of a set of numbers is the largest number that is a factor of all those numbers. For example, the GCF of the numbers 4 and 6 is 2 because 2 is the greatest number that’s a factor of both 4 and 6. Here you will learn two ways to find the GCF. Method 1: Use a list of factors to find the GCF This method for finding the GCF is quicker when you’re dealing with smaller numbers. To find the GCF of a set of numbers, list all the factors of each number. The greatest factor appearing on every list is the GCF. For example, to find the GCF of 6 and 15, first list all the factors of each number. Factors of 6: 1, 2, 3, 6 Factors of 15: 1, 3, 5, 15 Because 3 is the greatest factor that appears on both lists, 3 is the GCF of 6 and 15. As another example, suppose you want to find the GCF of 9, 20, and 25. Start by listing the factors of each: Factors of 9: 1, 3, 9 Factors of 20: 1, 2, 4, 5, 10, 20 Factors of 25: 1, 5, 25 In this case, the only factor that appears on all three lists is 1, so 1 is the GCF of 9, 20, and 25. Method 2: Use prime factorization to find the GCF You can use prime factorization to find the GCF of a set of numbers. This often works better for large numbers, where generating lists of all factors can be time-consuming. Here’s how to find the GCF of a set of numbers using prime factorization: List the prime factors of each number. Circle every common prime factor — that is, every prime factor that’s a factor of every number in the set. Multiply all the circled numbers. The result is the GCF. For example, suppose you want to find the GCF of 28, 42, and 70. Step 1 says to list the prime factors of each number. Step 2 says to circle every prime factor that’s common to all three numbers (as shown in the following figure). As you can see, the numbers 2 and 7 are common factors of all three numbers. Multiply these circled numbers together: 2 · 7 = 14 Thus, the GCF of 28, 42, and 70 is 14. Knowing how to find the GCF of a set of numbers is important when you begin reducing fractions to lowest terms.

Article / Updated 04-25-2016

Some numbers can be placed in rectangular patterns. Mathematicians probably should call numbers like these “rectangular numbers,” but instead they chose the term composite numbers. For example, 12 is a composite number because you can place 12 objects in rectangles of two different shapes, as shown in the following figure. Arranging numbers in visual patterns like this tells you something about how multiplication works. In this case, by counting the sides of both rectangles, you find out the following: 3 4 = 12 2 6 = 12 Similarly, other numbers such as 8 and 15 can also be arranged in rectangles, as shown here. As you can see, both of these numbers are quite happy being placed in boxes with at least two rows and two columns. And these visual patterns show this: 2 4 = 8 3 5 = 15 The word composite means that these numbers are composed of smaller numbers. For example, the number 15 is composed of 3 and 5 — that is, when you multiply these two smaller numbers, you get 15. Here are all the composite numbers between 1 and 16: 4 6 8 9 10 12 14 15 16 Notice that all the square numbers also count as composite numbers because you can arrange them in boxes with at least two rows and two columns. Additionally, lots of other non-square numbers are also composite numbers.

Article / Updated 04-25-2016

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 04-25-2016

We use three different types of average in maths: the mean, the mode and the median, each of which describes a different ‘normal’ value. The mean is what you get if you share everything equally, the mode is the most common value, and the median is the value in the middle of a set of data. Here are some more in-depth definitions: Median: In a sense, the median is what you normally mean when you say ‘the average man in the street’. The median is the middle-of-the road number – half of the people are above the median and half are below the median. (In America, it’s literally the middle of the road: Americans call the central reservation of a highway the ‘median’.) Try remembering ‘medium’ clothes are neither large nor small, but somewhere in between. Goldilocks was a median kind of girl. Mode: The mode is the most common result. ‘Mode’ is another word for fashion, so think of it as the most fashionable answer – ‘Everyone’s learning maths this year!’ Mean: The mean is what you get by adding up all of the numbers and dividing by how many numbers were in the list. Most people think of the mean when they use the word ‘average’ in a mathematical sense. In some ways the mean is the fairest average –you get the mean if the numbers are all piled together and then distributed equally. But the mean is also the hardest average to work out. You use the different averages in different situations, depending on what you want to communicate with your sums. Find the median To find the median of a set of numbers, you arrange the numbers into order and then find the number exactly in the middle: If the numbers aren’t in order, sort them out. You can arrange them either going up or down. Circle the number at each end of the list. Keep circling numbers two at a time (one from each end) until you have only one or two uncircled numbers. If only one number is left, that’s the median. You’re done! If two numbers are left, find the mean. Add up the two numbers and divide by two. The answer is the median. Find the mode If you have a list of numbers in order, figuring out which number shows up most often is pretty easy. You simply count the numbers – whichever number you have most of is the mode. If you have the list 1, 1, 3, 5, 5, 5, 6, 6, 7, 8, 9, 9, 10, you count each number in turn and find you have two 1s, one 3, three 5s, two 6s, one 7, one 8, two 9s and a 10. The number 5 comes up more frequently than any of the others, so the mode of these data is 5. If the data aren’t in a list, I suggest you set up a tally chart to help you count the numbers. Finding the mode in a table of numbers is very easy: whoever made the table has already done the tally chart for you and counted up the 1s. All you do is find the biggest number in the ‘count’ or ‘frequency’ column. The number labelling that row is the mode. Working out the mean of a list of numbers Here’s how to work out the mean of a set of numbers: Write out a list of all the numbers. Add up all the numbers. Count how many numbers are in the list. Divide the total from Step 2 by the total in Step 3. The answer is the mean. Check your answer makes sense. The mean should be somewhere between the highest and lowest numbers in your list. Adding up a long list of numbers is a chore. In real life you may use a calculator or a spreadsheet. But in an exam you may not have access to either of those helpful devices. You can add up longs lists of numbers by working through the list, adding a pair of numbers at a time, and writing the result in the next row. If you have a number left over at the end of the row, just copy that number into the next row and keep going.