NOTE: Gateway problems 1 & 2 on adding and subtracting fractions canboth be done using the same set of steps.


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1 Sample Gateway Problems:.. Working with Fractions and the Order of Operations Without Using a Calculator NOTE: Gateway problems 1 & 2 on adding and subtracting fractions canboth be done using the same set of steps. Adding fractions and subtracting fractions both require finding a least common denominator (LCD), which is most easily done by factoring the denominator (bottom number) of each fraction into a product of prime numbers (a number that can be divided only by itself and 1.) 1 2 Sample Problem #1: Adding fractions Step 1: Factor the two denominators into prime factors, then write each fraction with its denominator in factored form: 10 2 and, so Step 2: Find the least common denominator (LCD): LCD2 Sample Problem #1 (continued) Step : Multiply the numerator (top)and denominator of each fraction by the factor(s) needed to turn each denominator into the LCD. LCD Step 4: Multiply each numerator out, leaving the denominators in factored form, then add the two numerators and put them over the common denominator (note that 2 2 by the commutative property) Step : Now factor the numerator, then cancel any common factors that appear in both numerator and denominator. Once you multiply out any remaining factors, the result is your simplified answer. 2 / / 4 1
2 Full Solution to Sample Problem #1: Sample Problem #2: Subtracting fractions and, so , and LCD / / 2 14 Step 1: Factor the two denominators into prime factors, then write each fraction with its denominator in factored form: 14 2 and, so Step 2: Find the least common denominator (LCD): LCD2 6 Sample Problem #2 (continued) Step : Multiply the numerator and denominator of each fraction by the factor(s) needed to turn each denominator into the LCD: form: LCD Step 4: Multiply out the numerators, leaving the denominators in factored form, then add the two numerators and put them over the common denominator Step : Now factor the numerator, then cancel any common factors that appear in both numerator and denominator. Once you multiply out any remaining factors, the result is your simplified answer. 21 / / Full Solution to Sample Problem #2: 14 2 and, so  22, and LCD / /
3 NOTE: Gateway problems & on multiplying and dividing fractions canboth be done using the similar steps. Neither multiplying fractions nordividing fractionsrequires finding an LCD. These kinds of problems can be most easily done by factoring both the numerator(top number) and denominatorof both fractions into a product of prime numbers, and then canceling anycommon factors (numbers that appear on both the top and the bottom.) 9 Sample Problem #: Multiplying fractions Step 1: Factor both the numerators and denominators into prime factors, then write each fraction in factored form: First fraction: 9 1 and 0 2 Second fraction: 1 and So you can write 9 1 as NOTE: You do NOT need an LCD when multiplying fractions. Sample Problem # (continued) Step 2: Now just cancel any common factors that appear in both numerator and denominator. Once you multiply out any remaining factors, the result is your simplified answer. 1 / / 9. 2 / 2 1 / NOTE: It is much easier to factor first and then cancel, rather than multiplying out the numerators and denominators and then trying to simplify the answer (especially if you aren t using a calculator!) If you multiplied first, you d have gotten 8, which would be nasty to simplify by hand Full Solution to Sample Problem #: / / / 2 1 /
4 . Math TLC (Math 010 and Math 110) Sample Problem #: Dividing fractions Step 1: Multiply the first fraction by the reciprocal of the second fraction (i.e. flip the second fraction upside down and change to.) Step 2: Factor both the numerators and denominators into prime factors, then write each fraction in factored form: First fraction: 4 and 1 1 (prime) Second fraction: and 21 So you can write 4 26 as Sample Problem # (continued) NOTE: You do NOTneed an LCD when dividingfractions. Step : Now just cancel any common factors that appear in both numerator and denominator. Once you multiply out any remaining factors, the result is your simplified answer. / 2 1 / / / NOTE: Once again, it is much easier to factor first and then cancel, rather than multiplying out the numerators and denominators and then trying to simplify the answer (especially if you aren t using a calculator!) If you multiplied first, you d have gotten 110, which would be pretty hard to simplify by hand Full Solution to Sample Problem #: / 2 1/ / / NOTE: Gateway problems 4 & 6 using mixed numbers both start with the same step. A mixed number consists of an integer part and afraction part. We want to covert the mixed numberinto an improper fraction, This is done by multiplying theinteger part by the denominatorof the fraction part, thenadding that product to the numeratorof the fraction and putting that sum over the original denominator. 16 4
5 Sample Problem #4: Multiplying mixed numbers Step 1: Convert the mixed number 2 into an improper fraction: (Note that ) So 2 6 becomes, which we can then solve the same way we did problem #. 1 Sample Problem #4 (continued) 1 6 Step 2: Factor both the numerators and denominators into prime factors, then write each fraction in factored form: First fraction: 1 and are both prime Second fraction: 6 2 and is prime So you can write 1 6 as 1 2. Step : Now just cancel any common factors that appear in both numerator and denominator. Once you multiply out any remaining factors, the result is your simplified answer. 1 2 / / 18 Full Solution to Sample Problem #4: Sample Problem #6: Dividing mixed numbers / 12 4 / Step 1: Convert the mixed numbers into improper fractions:
6 0 2 0 Sample Problem #6(continued) Step 2: Factor both the numerators and denominators into prime factors, then write each fraction in factored form: First fraction: 0 2 and is prime Second fraction: 2 is prime and2 So you can write 0 2 as Step : Now just cancel any common factors that appear in Both numerator and denominator. Once you multiply out any remaining factors, the result is your simplified answer. 2 / / / / 21 Full Solution to Sample Problem #6: / / / / NOTE: Gateway problems & 8 both require using the order of operations. Order of operations: 1) First, calculate expressions within grouping symbols (parentheses, brackets, braces,absolute values, fraction bars). If there are nested sets of grouping symbols, start with the innermost ones first and work your way out. 2) Exponential expressions left to right ) Multiplication and division left to right 4) Addition and subtraction left to right Order of operations memory device: Please excuse my dear Aunt Sally 1. Please (Parentheses) 2. Excuse (Exponents). My Dear (Multiply and Divide) 4. Aunt Sally (Add and Subtract) 2 or just remember PEMDAS 24 6
7 Sample Problem # : Order of Operations Now calculate the bottom expression: 2(6+2) + 4 Step 1: Parentheses: 2(6+2) + 4 2(8) + 4 Step 2: Exponents: There aren t any in this part. Strategy: Calculate out the entire top expression and then the entire bottom expression, using the order of operations on each part. Then simplify the resulting fraction, if necessary. TOP EXPRESSION: 2 4 4( + 2) Step 1: Parentheses: 2 4 4( + 2) 2 4 4(9) Step 2: Exponents: 2 4 4(9) (9) 16 4(9) (because ) Step : Multiply/Divide: 16 4(9) Step : Multiply/Divide: 2(8) Step 4:Add/Subtract: Now put the top over the bottom and simplify the resulting fraction: TOP 2 4 4( + 2) BOTTOM 2(6+2) Step 4: Add/Subtract: Full Solution to Sample Problem #: Sample Problem # 8: Order of Operations 2 4 4( + 2) 2 4 4(9) 16 4(9) (6+2) + 4 2(8) Strategy: Deal with the expressions inside the grouping symbols (parentheses, brackets) first, starting with the innermost set ( + 6). STEP 1: (inside the parentheses) [1 + ( + 6) 10] [1 + () 10] STEP 2: (inside the brackets; multiply first, then add and subtract) [1 + () 10] [ ] [ ] [ ] [210] [22] 2 STEP : Do the final multiplication: [22]
8 Full Solution to Sample Problem #8: [1 + ( + 6)  10] [1 + ()  10] [ ] [210] [22]
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