B ASIC 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. Operations Addition and Subtraction Addition is used when you need 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 to work from right to left. CHAPTER Number Operations and Number Sense A GOOD grasp of the building blocks of math will be essential for your success on the GED Mathematics Test. This chapter covers the basics of mathematical operations and their sequence, variables, inte- gers, fractions, decimals, and square and cube roots. 42 405 Example Add 40 + 129 + 24. 1. Align the numbers you want to add. Since it is necessary to work from right to left, begin with the ones column. Since the ones column equals 13, write the 3 in the ones column and regroup or “carry” the 1 to the tens column: 1 40 129 +24 3 2. Add the tens column, including the regrouped 1. 1 40 129 +24 93 3. Then add the hundreds column. Since there is only one value, write the 1 in the answer. 1 40 129 +24 193 Subtraction is used when you want to find the dif- ference 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. Example If Kasima is 45 and Deja is 36, how many years older is Kasima? 1. Find the difference in their ages by subtracting. Start with the ones column. Since 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. 1 4 5 − 36 9 2. 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. 1 4 3 5 −36 09 Multiplication and Division In multiplication, you combine the same amount multi- ple 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. Example Find the product of 34 and 54. 1. Line up the place values as you write the prob- lem 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 ten. 1 34 × 54 6 2. 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. 1 34 × 54 136 3. Now multiply by the tens place of 54. Write a placeholder 0 in the ones place in the second partial product, since 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. – NUMBER OPERATIONS AND NUMBER SENSE– 406 34 × 54 136 170 —place holder 4. Add the partial products to find the total prod- uct: 136 + 1,700 = 1,836. 34 × 54 136 1700 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. Example At a bake sale, three children sold their baked goods for a total of $54. If they share the money equally, how much money should each child receive? 1. 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. Since 3 × 1 = 3, write 3 under the 5 and subtract 5 − 3 = 2. 18 3ͤ54 ෆ −3 24 −24 0 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. Since 3 × 8 = 24, write 24 below the other 24 and subtract 24 − 24 = 0. 3. If you get a number other than zero after your last subtraction, this number is your remainder. Example 9 divided by 4. 2 4ͤ9 ෆ −8 1—remainder The answer is 2 R1. Sequence of Mathematical Operations There is an order for doing a sequence of mathematical operations. That order is illustrated by the following 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. Example ᎏ (5 + 4 3) 2 ᎏ + 27 = Add 5 to 3 within parentheses. ᎏ (8 4 ) 2 ᎏ + 27 = Next, evaluate the exponential expression. ᎏ 6 4 4 ᎏ + 27 = Perform division. 16 + 27 = 43 Perform addition. Squares and Cube Roots The square of a number is the product of a number and itself. For example, in the expression 3 2 = 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 ͙ ෆ and it is called the radical. The number inside of the radical is called the radicand. 0 – NUMBER OPERATIONS AND NUMBER SENSE– 407 Example 5 2 = 25 therefore ͙25 ෆ = 5 Since 25 is the square of 5, it is also true that 5 is the square root of 25. Perfect Squares The square root of a number might not be a whole num- ber. 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 per- fect square if its square root is also a whole number. Examples of perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 . . . Numbers and Signs Odd and Even Numbers An even number is a number that can be divided by the number 2 with a whole number: 2, 4, 6, 8, 10, 12, 14 . . . An odd number cannot be divided by the number 2 as a result: 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: even + even = even and even ؋ even = even odd + odd = even and odd ؋ odd = odd odd + even = odd and even ؋ odd = even Prime and Composite Numbers A positive integer that is 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. Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23 . . . ■ A composite number is a number that has more than two factors. Examples: 4, 6, 8, 9, 10, 12, 14, 15, 16 . . . ■ The number 1 is neither prime nor composite. Number Lines and Signed Numbers You have surely dealt with number lines in your distin- guished career as a math student. The concept of the number line is simple: Less than is to the left and greater than is to the right . . . Absolute Value 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. Example ԽϪ1Խϭ1 Խ2 Ϫ 4ԽϭԽϪ2Խϭ2 Working with Integers An integer is a positive or negative whole number. Here are some rules for working with integers: Multiplying and Dividing (+) × (+) = + (+) Ϭ (+) = + (+) × (−) = − (+) Ϭ (−) = − (−) × (−) = + (−) Ϭ (−) = + A simple rule for remembering the above is that 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 Adding the same sign results in a sum of the same sign: (+) + (+) = + and (−) + (−) = − When adding numbers of different signs, follow this two-step process: 1. Subtract the absolute values of the numbers. 2. Keep the sign of the larger number. –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 Greater Than Less Than – NUMBER OPERATIONS AND NUMBER SENSE– 408 Example −2 + 3 = 1. Subtract the absolute values of the numbers: 3 − 2 = 1 2. The sign of the larger number (3) was originally positive, so the answer is positive 1. Example 8 + −11 = 1. Subtract the absolute values of the numbers: 11 − 8 = 3 2. The sign of the larger number (11) was origi- nally negative, so the answer is −3. Subtracting When subtracting integers, change all subtraction to addition and change the sign of the number being sub- tracted to its opposite. Then, follow the rules for addition. Examples (+10) − (+12) = (+10) + (−12) = −2 (−5) − (−7) = (−5) + (+7) = +2 Decimals The most important thing to remember about decimals is that the first place value to the right is tenths. The place values are as follows: In expanded form, this number can 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 Comparing decimals is actually quite simple. Just line up the decimal points and fill in any zeroes needed to have an equal number of digits. Example Compare .5 and .005 Line up decimal points .500 and add zeroes .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. Variables In a mathematical sentence, a variable is a letter that rep- resents 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 com- plex than that, and there are many rules and procedures that need to be learned. Before you learn to solve equa- tions with variables, you need to learn how they operate in formulas. The next section on fractions will give you some examples. Fractions To do well when working with fractions, it is necessary to understand some basic concepts. On the next page are some math rules for fractions using variables. 1 T H O U S A N D S 2 H U N D R E D S 6 T E N S 8 O N E S • D E C I M A L 3 T E N T H S 4 H U N D R E D T H S 5 T H O U S A N D T H S 7 T E N T H O U S A N D T H S POINT – NUMBER OPERATIONS AND NUMBER SENSE– 409 Multiplying Fractions ᎏ a b ᎏ × ᎏ d c ᎏ = ᎏ a b × × c d ᎏ Multiplying fractions is one of the easiest operations to perform. To multiply fractions, simply multiply the numerators and the denominators, writing each in the respective place over or under the fraction bar. Example ᎏ 4 5 ᎏ × ᎏ 6 7 ᎏ = ᎏ 2 3 4 5 ᎏ Dividing Fractions ᎏᎏ a b ᎏ ÷ ᎏ d c ᎏ = ᎏ a b ᎏ × ᎏ d c ᎏ = ᎏ a b × × d c ᎏ Dividing fractions is the same thing as multiplying frac- tions by their reciprocals. To find the reciprocal of any number, switch its numerator and denominator. For example, the reciprocals of the following numbers are: ᎏ 1 3 ᎏ = ᎏ 3 1 ᎏ = 3 x = ᎏ 1 x ᎏᎏ 4 5 ᎏ = ᎏ 5 4 ᎏ 5 = ᎏ 1 5 ᎏ When dividing fractions, simply multiply the divi- dend by the divisor’s reciprocal to get the answer. Example ᎏ 1 2 2 1 ᎏ ÷ ᎏ 3 4 ᎏ = ᎏ 1 2 2 1 ᎏ × ᎏ 4 3 ᎏ = ᎏ 4 6 8 3 ᎏ = ᎏ 1 2 6 1 ᎏ Adding and Subtracting Fractions ᎏ a b ᎏ × ᎏ d c ᎏ = ᎏ a b × × c d ᎏ ᎏ a b ᎏ + ᎏ d c ᎏ = ᎏ ad b + d bc ᎏ ■ To add or subtract fractions with like denomina- tors, just add or subtract the numerators and leave the denominator as it is. Example ᎏ 1 7 ᎏ + ᎏ 5 7 ᎏ = ᎏ 6 7 ᎏ and ᎏ 5 8 ᎏ − ᎏ 2 8 ᎏ = ᎏ 3 8 ᎏ ■ To add or subtract fractions with unlike denomi- nators, you must find the least common denom- inator, or LCD. For example, for the denominators 8 and 12, 24 would be the LCD 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 numera- tor and denominator by the necessary number to get the LCD, and then add or subtract the new numerators. Example ᎏ 1 3 ᎏ + ᎏ 2 5 ᎏ = ᎏ 5 5 ( ( 1 3 ) ) ᎏ + ᎏ 3 3 ( ( 2 5 ) ) ᎏ = ᎏ 1 5 5 ᎏ + ᎏ 1 6 5 ᎏ = ᎏ 1 1 1 5 ᎏ – NUMBER OPERATIONS AND NUMBER SENSE– 410 . 8 = 24, write 24 below the other 24 and subtract 24 − 24 = 0. 3. If you get a number other than zero after your last subtraction, this number is your remainder. Example 9 divided by 4. 2 4 9 ෆ −8 1—remainder The. 3 = 2. 18 3ͤ 54 ෆ −3 24 − 24 0 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 column: 1 40 129 + 24 3 2. Add the tens column, including the regrouped 1. 1 40 129 + 24 93 3. Then add the hundreds column. Since there is only one value, write the 1 in the answer. 1 40 129 + 24 193 Subtraction