Unlock the power of precise math with our comprehensive guide to fractions and decimals. Elevate your skills for accurate measurements and savvy material estimates.
Whole numbers are the basic building blocks of mathematics, ranging from 1 to infinity. Mastering operations with whole numbers is essential for everyday applications like budgeting, inventory management, and construction measurements. Here are some examples:
Multiplication Tables
For example, the 5 times table is: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50. Memorizing this table will help you quickly calculate things like the cost of 7 items at $5 each.
Basic Operations
Examples of basic whole number operations include:
Addition: 12 + 8 = 20
Subtraction: 45 - 17 = 28
Multiplication: 6 x 9 = 54
Division: 84 ÷ 7 = 12
Practice Problems
Here are a couple examples of whole number practice problems:
A store sells 18 items per day. How many items will they sell in 2 weeks?
A construction crew needs 124 bricks to finish a project. If they have 78 bricks already, how many more do they need to buy?
Common Fractions
Concepts
Learn how the numerator and denominator relate, and how to compare and order fractions.
For example, the fraction 3/4 has a numerator of 3 and a denominator of 4. This means it represents 3 parts out of 4 total parts. Fractions with larger denominators represent smaller portions, so 1/8 is less than 1/4.
Operations
Master adding, subtracting, multiplying, and dividing common fractions.
To add or subtract, find a common denominator. For example, to add 1/2 and 1/4, find the least common denominator of 4, so the fractions become 2/4 and 1/4. Then you can add the numerators: 2/4 + 1/4 = 3/4.
To multiply, multiply the numerators and denominators. For example, 1/2 x 1/3 = 1/6.
To divide, invert the divisor fraction and multiply. For example, 1/2 ÷ 1/4 = 1/2 x 4/1 = 2/1 = 2.
Conversions
Convert between mixed numbers, improper fractions, and whole numbers.
To convert a mixed number like 2 1/3 to an improper fraction, multiply the whole number by the denominator and add the numerator: 2 x 3 + 1 = 7/3.
To convert an improper fraction like 7/3 to a mixed number, divide the numerator by the denominator: 7 ÷ 3 = 2 with a remainder of 1, so the mixed number is 2 1/3.
Practice Problems
Work through practice problems with real-world scenarios to reinforce your understanding. Analyze the information, determine the right operations, and solve the problems. Regular practice will build your skills.
For example:
A recipe calls for 3/4 cup of flour and 1/2 cup of sugar. How much total dry ingredients are needed?
A carpenter needs to cut a 2 1/2 foot board into three equal pieces. How long will each piece be?
If 1/6 of a tank of gas was used, and the tank holds 12 gallons, how many gallons of gas were used?
Decimal Fractions
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Place Value
Decimal numbers represent values less than one. Each digit to the right of the decimal point represents a fractional place value. The first digit is the tenths place, the second is the hundredths, the third is the thousandths, and so on.
For example, the number 0.375 has 3 in the hundredths place, 7 in the thousandths place, and 5 in the ten-thousandths place.
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Rounding
Rounding decimal numbers to a specific place value can simplify calculations and provide reasonable estimates. To round, look at the digit in the place right after the one you want to round to. If it's 5 or greater, round up. If it's 4 or less, round down.
For example, 4.678 rounded to the nearest tenth is 4.7, and 12.343 rounded to the nearest whole number is 12.
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Estimating
Estimating decimal calculations is important to check the reasonableness of your final answer.
For instance, if you need to multiply 3.456 by 2.789, you could estimate the result by rounding the numbers to 3.5 and 2.8, which gives you an estimate of 9.8.
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Basic Operations
The basic arithmetic operations of addition, subtraction, multiplication, and division apply to decimal numbers just like whole numbers. Line up the decimal points when adding/subtracting. Multiply the digits and count the total decimal places. For division, the decimal places in the quotient equal the difference between the dividend and divisor.
For example, to add 2.45 and 1.78, line up the decimal points and add the digits: 2.45 + 1.78 = 4.23. To multiply 1.2 by 3.4, multiply the digits (1.2 x 3.4 = 4.08) and count the total decimal places (2), so the result is 4.08.
Squares and Square Roots
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Perfect Squares
A perfect square is a number that can be expressed as the product of two equal integers. For example, 4, 9, 16, and 25 are perfect squares because they are 2 x 2, 3 x 3, 4 x 4, and 5 x 5 respectively. Other examples of perfect squares include 36 (6 x 6), 49 (7 x 7), and 81 (9 x 9). Identifying perfect squares is a key skill for working with square roots.
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Square Root Calculations
To find the square root of a number, we need to determine the value that, when multiplied by itself, equals the original number. For example, the square root of 16 is 4, because 4 x 4 = 16. Another example is the square root of 81, which is 9, because 9 x 9 = 81. While a calculator can quickly find square roots, it's also important to learn how to estimate and calculate square roots by hand using methods like the long division algorithm.
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Real-World Applications
Knowledge of squares and square roots has many practical applications, such as in construction (e.g. calculating the length of diagonals in a square or rectangle), engineering (e.g. determining the size of structural supports), and more. For example, if you need to build a deck that is 12 feet by 12 feet, you would need to know the diagonal length, which is the square root of 12^2 + 12^2, or about 16.97 feet. Understanding the relationships between these concepts is key to solving a variety of math problems.
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Inverse Relationship
There is an inverse relationship between squares and square roots. This means that if you know the square of a number, you can find its square root, and vice versa. For instance, if you know that 9 is the square of 3, then you can say that the square root of 9 is 3. Recognizing and applying this inverse relationship is helpful for solving equations and other math problems involving squares and square roots.
Percents and Percentages
Understanding Percent Basics
Percents are a way to express a part of a whole as a fraction out of 100. For example, 25% means 25 parts out of 100, or 1/4 of the whole.
To convert a percent to a decimal, divide the percent by 100. For example, 25% = 0.25.
To convert a decimal to a percent, multiply by 100. For example, 0.25 = 25%.
Percent Calculations
To find the percent of a number, multiply the number by the percent. For example, 15% of 80 is 0.15 x 80 = 12.
To calculate a percent increase or decrease, use the formula: Percent change = (New value - Original value) / Original value x 100.
To solve percent proportion problems, set up an equation with the percent as a fraction. For example, to find 30% of 50, the equation would be: 30/100 = x/50, where x = 15.
Real-World Percent Applications
Percents are used in many everyday situations, such as calculating tips, interest rates, sales tax, and discounts. Understanding percent concepts is a valuable life skill.
For example, if an item is regularly $50 and is on sale for 20% off, the discounted price would be $50 - (0.20 x $50) = $40.
Converting Percent to Decimal
To convert a percent to a decimal, simply divide the percent by 100. For example, 25% can be written as 0.25 in decimal form.
This is useful when performing calculations, as decimal numbers are often easier to work with than percents.
Determining the Rate, Part, and Whole
In a percent problem, there are three key pieces of information: the rate (the percent), the part (the amount represented by the percent), and the whole (the total amount). If you know any two of these, you can solve for the third.
For example, if you know that 15% of 200 is 30, you can use this to find the whole amount (200) or the part amount (30).
Converting Percents to Decimals
To convert a percent to a decimal, divide the percent by 100. For example, 15% can be written as 0.15 in decimal form.
This is a useful skill for performing percent calculations, as decimal numbers are often easier to work with than percents.
Percent Practice Problems
Work through a variety of percent problems to build your skills in calculating, converting, and applying percent concepts. This could include finding percentages of numbers, calculating percent increases and decreases, and solving percent proportion problems.
Practicing with different types of percent problems will help you become more comfortable and confident with this important math skill.
Master Feet & Inches Calculations
Easily convert, add, subtract, multiply, and divide measurements in feet and inches. Learn practical applications for construction, carpentry, and more.
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Converting Feet and Inches
Convert measurements between feet/inches and total inches. For example, convert 6 feet 3 inches to 75 inches, or 5 feet 9 inches to 69 inches.
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Adding and Subtracting Feet and Inches
Add measurements like 4 feet 6 inches + 2 feet 9 inches to get 7 feet 3 inches. Subtract measurements like 12 feet 7 inches - 8 feet 4 inches to get 4 feet 3 inches.
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Multiplying and Dividing Feet and Inches
Multiply feet measurements, like 3 feet x 2 to get 6 feet. Divide feet measurements, like 15 feet ÷ 3 to get 5 feet.
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Using a Construction Master Calculator
Quickly perform feet and inches calculations using a specialized construction calculator. This can simplify complex measurements for construction, carpentry, and home projects.
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Converting to Decimal Feet
Convert feet and inches measurements to decimal feet. For example, 6 feet 3 inches can be written as 6.25 feet.
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Feet and Inches Practice Problems
Work through a variety of practice problems to build your skills in converting, adding, subtracting, multiplying, and dividing feet and inches measurements.
Convert between degrees/minutes/seconds and decimal degrees, either by hand or using a construction calculator. For example, 45 degrees 30 minutes = 45.5 degrees.
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Degrees, Minutes, Seconds Practice
Work through sample problems like converting 135 degrees 27 minutes 15 seconds to decimal degrees (135.454167), or adding 22 degrees 15 minutes and 37 degrees 45 minutes.