Mark Zegarelli

Cheat Sheet / Updated 05-07-2024

Making sure you have a handle on algebra, geometry, and trigonometry is a solid start for success on the math section of the ACT. However, to boost your confidence — and your score — even higher, you should master some helpful test-taking strategies, as well as make sure you know how to translate word problems into equations and use sketches to figure out what a tricky-sounding question is really asking.

Article / Updated 03-20-2024

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.

Article / Updated 03-20-2024

Fractions, decimals, and percents are the three most common ways to give a mathematical description of parts of a whole object. Fractions are common in baking and carpentry when you're using English measurement units (such as cups, gallons, feet, and inches). Decimals are used with dollars and cents, the metric system, and in scientific notation. Percents are used in business when figuring profit and interest rates, as well as in statistics. Use the following table as a handy guide when you need to make basic conversions among the three. Fraction Decimal Percent 1/100 0.01 1% 1/20 0.05 5% 1/10 0.1 10% 1/5 0.2 20% 1/4 0.25 25% 3/10 0.3 30% 2/5 0.4 40% 1/2 0.5 50% 3/5 0.6 60% 7/10 0.7 70% 3/4 0.75 75% 4/5 0.8 80% 9/10 0.9 90% 1 1.0 100% 2 2.0 200% 10 10.0 1,000%

Article / Updated 03-20-2024

The English system of measurements is most commonly used in the United States. In contrast, the metric system is used throughout most of the rest of the world. Converting measurements between the English and metric systems is a common everyday reason to know math. This article gives you some precise metric-to-English conversions, as well as some easy-to-remember conversions that are good enough for most situations. Metric-to-English Conversion Table Metric-to-English Conversions Metric Units in Plain English 1 meter ≈ 3.28 feet A meter is about 3 feet (1 yard). 1 kilometer ≈ 0.62 miles A kilometer is about 1/2 mile. 1 liter ≈ 0.26 gallons A liter is about 1 quart (1/4 gallon). 1 kilogram ≈ 2.20 pounds A kilo is about 2 pounds. 0°C = 32°F 0°C is cold. 10°C = 50°F 10°C is cool. 20°C = 68°F 20°C is warm. 30°C = 86° 30°C is hot. Here's an easy temperature conversion to remember: 16°C = 61°F.

Article / Updated 03-20-2024

Exponents, radicals, and absolute value are mathematical operations that go beyond addition, subtraction, multiplication, and division. They are useful in more advanced math, such as algebra, but they also have real-world applications, especially in geometry and measurement. Exponents (powers) are repeated multiplication: When you raise a number to the power of an exponent, you multiply that number by itself the number of times indicated by the exponent. For example: 72 = 7 × 7 = 49 25 = 2 × 2 × 2 × 2 × 2 = 32 Square roots (radicals) are the inverse of exponent 2 — that is, the number that, when multiplied by itself, gives you the indicated value. Absolute value is the positive value of a number — that is, the value of a negative number when you drop the minus sign. For example: Absolute value is used to describe numbers that are always positive, such as the distance between two points or the area inside a polygon.

Cheat Sheet / Updated 10-05-2023

Although there's no shortcut to success on the math sections of the SAT, you can study and prepare in order to get the best SAT score you possibly can. Knowing what will be on the test (and what won't be) is key so you know what to brush up on. Also, some basic strategy goes a long way toward helping you get the best score you can. Finally, mapping out a time-management plan to answer (and skip!) the right questions can really boost your score.

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.