Number Relations: GED Test Prep (page 6)
A good grasp of the building blocks of math will be essential for your success on the GED Mathematics Exam. This article covers the basics of mathematical operations and their sequence, variables, integers, fractions, decimals, and square and cube roots.
Basic problem solving in mathematics is rooted in whole number math facts—mainly addition facts and multiplication tables. If you are unsure of any of these facts, now is the time to review. Make sure to memorize any parts of this review that you find troublesome. Your ability to work with numbers depends on how quickly and accurately you can do simple mathematical computations.
Addition and Subtraction
Addition is used to combine amounts. The answer in an addition problem is called the sum, or the total. It is helpful to stack the numbers in a column when adding. Be sure to line up the place-value columns and work from right to left, starting with the ones column.
Add 40 + 129 + 24.
- Align the numbers you want to add on the ones column. Because it is necessary to work from right to left, begin to add starting with the ones column. The ones column equals 13, so write the 3 in the ones column and regroup or "carry" the 1 to the tens column:
- Add the tens column, including the regrouped 1.
- Then add the hundreds column. Because there is only one value, write the 1 in the answer.
Subtraction is used to find the difference between amounts. Write the greater number on top, and align the amounts on the ones column. You may also need to regroup as you subtract.
If Kasima is 45 and Deja is 36, how many years older is Kasima?
- Find the difference in their ages by subtracting. Start with the ones column. Because 5 is less than the number being subtracted (6), regroup or "borrow" a ten from the tens column. Add the regrouped amount to the ones column. Now subtract 15 – 6 in the ones column.
- Regrouping 1 ten from the tens column left 3 tens. Subtract 3 – 3, and write the result in the tens column of your answer. Kasima is 9 years older than Deja. Check: 9 + 36 = 45.
Multiplication and Division
In multiplication, you combine the same amount multiple times. For example, instead of adding 30 three times, 30 + 30 + 30, you could simply multiply 30 by 3. If a problem asks you to find the product of two or more numbers, you should multiply.
Find the product of 34 and 54.
- Line up the place values as you write the problem in columns. Multiply the ones place of the top number by the ones place of the bottom number: 4 × 4 = 16.Write the 6 in the ones place in the first partial product. Regroup the 10.
- Multiply the tens place in the top number by 4: 4 × 3 = 12. Then, add the regrouped amount 12 + 1 = 13.Write the 3 in the tens column and the 1 in the hundreds column of the partial product.
- Now multiply by the tens place of 54.Write a placeholder 0 in the ones place in the second partial product, because you're really multiplying the top number by 50. Then multiply the top number by 5: 5 × 4 = 20.Write 0 in the partial product and regroup the 2.Multiply 5 × 3 = 15. Add the regrouped 2: 15 + 2 = 17.
- Add the partial products to find the total product: 136 + 1,700 = 1,836.
In division, the answer is called the quotient. The number you are dividing by is called the divisor and the number being divided is the dividend. The operation of division is finding how many equal parts an amount can be divided into.
At a bake sale, 3 children sold their baked goods for a total of $54. If they share the money equally, how much money should each child receive?
- Divide the total amount ($54) by the number of ways the money is to be split (3).Work from left to right. How many times does 3 go into 5? Write the answer, 1, directly above the 5 in the dividend. Because 3 × 1 = 3, write 3 under the 5 and subtract 5 – 3 = 2.
- Continue dividing. Bring down the 4 from the ones place in the dividend. How many times does 3 go into 24? Write the answer, 8, directly above the 4 in the dividend. Because 3 × 8 = 24, write 24 below the other 24 and subtract 24 – 24 = 0.
- If you get a number other than 0 after your last subtraction, this number is your remainder.
9 divided by 4.
The answer is 2 R1.
Sequence of Mathematical Operations
There is an order for doing a sequence of mathematical operations, which is illustrated by the acronym PEMDAS, which can be remembered by using the first letter of each of the words in the phrase: Please Excuse My Dear Aunt Sally. Here is what it means mathematically:
P: Parentheses. Perform all operations within parentheses first.
E: Exponents. Evaluate exponents.
M/D: Multiply/Divide. Work from left to right in your expression.
A/S: Add/Subtract. Work from left to right in your expression.
Squares and Cube Roots
The square of a number is the product of a number and itself. For example, in the expression 32 = 3 × 3 = 9, the number 9 is the square of the number 3. If we reverse the process, we can say that the number 3 is the square root of the number 9. The symbol for square root is and it is called the radical. The number inside of the radical is called the radicand.
52 = 25; therefore, = 5
Because 25 is the square of 5, it is also true that 5 is the square root of 25.
The square root of a number might not be a whole number. For example, the square root of 7 is 2.645751311… It is not possible to find a whole number that can be multiplied by itself to equal 7. A whole number is a perfect square if its square root is also a whole number.
- Examples of perfect squares:
- 1, 4, 9, 16, 25, 36, 49, 64, 81, 100…
Odd and Even Numbers
An even number is a number that can be divided by the number 2: 2, 4, 6, 8, 10, 12, 14… An odd number cannot be divided by the number 2: 1, 3, 5, 7, 9, 11, 13… The even and odd numbers listed are also examples of consecutive even numbers and consecutive odd numbers because they differ by two.
Here are some helpful rules for how even and odd numbers behave when added or multiplied:
Prime and Composite Numbers
A positive integer greater than the number 1 is either prime or composite, but not both. A factor is an integer that divides evenly into a number.
- A prime number has only itself and the number 1 as factors.
- A composite number is a number that has more than two factors.
- The number 1 is neither prime nor composite.
Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23…
Examples: 4, 6, 8, 9, 10, 12, 14, 15, 16…
Number Lines and Signed Numbers
You have probably dealt with number lines before. The concept of the number line is simple: Less than is to the left and greater than is to the right.
The absolute value of a number or expression is always positive because it is the distance of a number from zero on a number line.
|–1| = 1
|2 – 4| = |– 2| = 2
Working with Integers
An integer is a positive or negative whole number. Here are some rules for working with integers:
Multiplying and Dividing
- (+) × (+) = + (+) ÷ (+) = +
- (+) × (–) = – (+) ÷ (–) = –
- (–) × (–) = + (–) ÷ (–) = +
A simple way to remember these rules: If the signs are the same when multiplying or dividing, the answer will be positive, and if the signs are different, the answer will be negative.
Adding the same sign results in a sum of the same sign:
- (+) + (+) = +
- (–) + (–) = –
When adding numbers of different signs, follow this two-step process:
- Subtract the absolute values of the numbers.
- Keep the sign of the larger number.
–2 + 3 =
- Subtract the absolute values of the numbers: 3 – 2 = 1
- The sign of the larger number (3) was originally positive, so the answer is positive 1.
8 + –11 =
- Subtract the absolute values of the numbers: 11 – 8 = 3
- The sign of the larger number (11) was originally negative, so the answer is –3.
When subtracting integers, change all subtraction to addition and change the sign of the number being subtracted to its opposite. Then, follow the rules for addition.
(+10) – (+12) = (+10) + (–12) = –2
(–5) – (–7) = (–5) + (+7) = +2
The most important thing to remember about decimals is that the first place value to the right begins with tenths. The place values are as follows:
In expanded form, this number can also be expressed as:
- 1,268.3457 = (1 × 1,000) + (2 × 100) + (6 × 10) + (8 × 1) + (3 × .1) + (4 × .01) + (5 × .001) + (7 × .0001)
Comparing decimals is actually quite simple. Just line up the decimal points and fill in any zeros needed to have an equal number of digits.
Compare .5 and .005
Line up decimal points .500
and add zeros .005
Then, ignore the decimal point and ask, which is bigger: 500 or 5?
500 is definitely bigger than 5,
so .5 is larger than .005.
In a mathematical sentence, a variable is a letter that represents a number. Consider this sentence: x + 4 = 10. It's easy to figure out that x represents 6.However, problems with variables on the GED will become much more complex than that, so you must learn the rules and procedures. Before you learn to solve equations with variables, you need to learn how they operate in formulas. The next section on fractions will give you some examples.
To do well when working with fractions, it is necessary to understand some basic concepts. Here are some math rules for fractions using variables:
Multiplying fractions is one of the easiest operations to perform. To multiply fractions, simply multiply the numerators and the denominators, writing each product in the respective place over or under the fraction bar.
Dividing fractions is the same thing as multiplying fractions by their reciprocals. To find the reciprocal of any number, switch its numerator and denominator.
For example, the reciprocals of the following numbers are:
When dividing fractions, simply multiply the dividend by the divisor's reciprocal to get the answer.
Adding and Subtracting Fractions
To add or subtract fractions with like denominators, just add or subtract the numerators and leave the denominator as it is.
To add or subtract fractions with unlike denominators, you must find the least common denominator, or LCD.
For example, for the denominators 8 and 12, the LCD is 24 because 8 × 3 = 24, and 12 × 2 = 24. In other words, the LCD is the smallest number divisible by each of the denominators.
Once you know the LCD, convert each fraction to its new form by multiplying both the numerator and denominator by the necessary number to get the LCD, and then add or subtract the new numerators.
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