Exercise 12 Page 288

Express each subtraction in the expression as the addition of a negative number. Then, use the Associative Property of Addition to simplify the expression.

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Practice makes perfect

We want to simplify the following expression. 14/15+13/15- 12/15+11/15- 10/15+...- 4/15+3/15- 2/15+1/15 Recall that when we subtract a number, the result is the same as adding its additive inverse. Let's rewrite the expression using this information.

Now we are adding fractions with the same denominator! The numerators are natural numbers from 1 to 14 and some fractions are positive and some fractions are negative.

Every positive fraction other than 1415 has an odd numerator and every negative fraction has an even numerator. Let's list the numerators from 14 to 1. 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1 There are 8 positive fractions and 6 negative ones. We can use the Associative Property of Addition to group each negative fraction with a corresponding positive fraction. For example, we can group and simplify the second and the third fractions in the following way.

13/15+(- 12/15)

MoveNegFracToNum

Put minus sign in numerator

13/15+(- 12/15)

AddFrac

Add fractions

13+(-12)/15

AddNeg

a+(- b)=a-b

1/15

Let's group all the fractions except the first and the last fractions in a similar way.

14/15+1/15+[11/15+(- 10/15)] +[9/15+(- 8/15)] +[7/15+(- 6/15)] +[5/15+(- 4/15)] +[3/15+(- 2/15)] +1/15

MoveNegFracToNum

Put minus sign in numerator

14/15+1/15+[11/15+(- 10/15)] +[9/15+(- 8/15)] +[7/15+(- 6/15)] +[5/15+(- 4/15)] +[3/15+(- 2/15)] +1/15

AddFrac

Add fractions

14/15+1/15+11+(- 10)/15 +9+(- 8)/15 +7+(- 6)/15 +5+(- 4)/15 +3+(- 2)/15 +1/15

AddNeg

a+(- b)=a-b

14/15+1/15+1/15 +1/15 +1/15 +1/15 +1/15 +1/15

Notice that each of the grouped addends simplifies to the fraction 115. Let's add the fractions that we got using the Associative Property of Equality.

14/15+1/15+1/15+1/15+1/15+1/15+1/15

AssociativePropAdd

Associative Property of Addition

(14/15+1/15)+(1/15+1/15+1/15+1/15+1/15)

AddFrac

Add fractions

14+1/15+1+1+1+1+1+1/15

AddTerms

Add terms

15/15+6/15

QuotOne

a/a=1