Operations with Rational Numbers (***WITH EXPLANATIONS***)

3

|-3| Read as "the absolute value of negative 3."

6

|6| Read as "the absolute value of 6." *Remember, the absolute value of a number is its distance from zero on a number line. Absolute value is never negative. Absolute value can be positive or 0.

−0.25 ÷ (−1_9/24) = −1/4 ÷ (-1_9/24) = -1/4 ÷ (-33/24) = -1/4 × (-24/33) = -1/1 × (-6/33) = 6/33 = 2/11 −0.25 ÷ (−1_9/24) = −0.25 ÷ (−1.375) = 0.181818... *Either change both to decimals or change both to fractions. *a/b form *Multiply by the reciprocal (Keep-Change-Flip) *Cross simplify *Multiply across *Same signs positive

−0.25 ÷ (−1_9/24)

-1.75 − 2_1/2 = -1.75 + (-2.5) = -4.25 or -1.75 − 2_1/2 = -1_3/4 + (-2_2/4) = -3_5/4 = -4_1/4 *Either change both to decimals or change both to fractions. Then, add the opposite (Keep-Change-Change).

−1.75 − 2_1/2

−3_2/5 ÷ (−7/10) = −17/5 × (-10/7) = -17/1 × (-2/7) = -34/7 = -4_6/7 *a/b form *Multiply by the reciprocal (Keep-Change-Flip) *Cross simplify *Multiply across *Different signs negative

−3_2/5 ÷ (-7/10)

1.25 + (−3_1/2) = 1.25 + (-3.5) = -2.25 or 1.25 + (−3_1/2) = 1_1/4 + (-3_2/4) = -2_1/4 *Either change both to decimals or change both to fractions. Then, follow the real numbers addition rules.

1.25 + (−3_1/2)

1.25 + 0.75 = 2 *SAME SIGNS ADD

1.25 + 0.75

10 - (-9) = 10 + 9 = 19 When subtracting rational numbers, you ADD THE OPPOSITE (Keep-Change-Change). Once you change the problem to addition, then you simply just follow the addition rules. More examples: -8 - (-9) = -8 + 9 = 1 -17 - 27 = -17 + (-27) = -44 11 - (-11) = 11 + 11 = 22 18 - 21 = 18 + (-21) = -3 26 - 11 = 26 + (-11) = 15

10 - (-9)

15 - 23 = 15 + (-23) = -8 When subtracting rational numbers, you ADD THE OPPOSITE (Keep-Change-Change). Once you change the problem to addition, then you simply just follow the addition rules. More examples: -9 - (-3) = -9 + 3 = -6 -20 - 12 = -20 + (-12) = -32 6 - (-6) = 6 + 6 = 12 11 - 16 = 11 + (-16) = -5 95 - 25 = 95 + (-25) = 70

15 - 23

2/3 ÷ (-3/4) = 2/3 × (-4/3) = -8/9 *a/b form *Multiply by the reciprocal (Keep-Change-Flip) *Multiply across *Different signs negative

2/3 ÷ (-3/4)

28 ÷ (-4) = -7 When multiplying or dividing rational numbers, you multiply or divide as normal. If the signs are the same, the product or quotient will be positive. If the signs are different, the product of quotient will be negative. *SAME SIGNS POSITIVE *DIFFERENT SIGNS NEGATIVE More examples: -11 × (-3) = 33 -6 × 1 = -6 7(-9) = -63 32 ÷ (-8) = -4 -81 ÷ (-9) = 9 -35/5 = -7

28 ÷ (-4)

3 + (-11) = -8 When adding rational numbers, you SUBTRACT the numbers when they have DIFFERENT SIGNS. You take the sign of the number with the greater absolute value. *DIFFERENT SIGNS SUBTRACT Ask yourself these two questions to help: (1) What do I have more of (+ or -)? *This determines the sign of your answer. (2) How many more? *This determines the amount of your answer. More examples: -20 + 25 = 5 -2 + 1 = -1 5 + (-3) = 2 11 + (-15) = -4

3 + (-11)

3(0) = 0 The is the Zero Property of Multiplication. Anything times zero is zero. More examples: -7 × 0 = 0 0(-1) = 0 a × 0 = 0

3(0)

4.75 - (-3.5) = 4.75 + 3.5 = 8.25 *ADD THE OPPOSITE. *THEN, FOLLOW THE ADDITION RULES.

4.75 - (-3.5)

7 + (-7) = 0 This represent the Additive Inverse Property. When you add a value and its opposite (or additive inverse), you will always get zero. Opposites cancel each other out.

7 + (-7)

9.7 - 10.3 = 9.7 + (-10.3) = -0.6 *ADD THE OPPOSITE. *THEN, FOLLOW THE ADDITION RULES.

9.7 - 10.3

1

-(-1) Read as "the opposite of the opposite of 1" or "the opposite of negative 1."

-1.75 - (-0.5) = -1.75 + 0.5 = -1.25 *ADD THE OPPOSITE. *THEN, FOLLOW THE ADDITION RULES.

-1.75 - (-0.5)

-1/2 + 1/3 = -3/6 + 2/6 = -1/6 *FIRST, GET EQUIVALENT FRACTIONS WITH COMMON DENOMINATORS. *DIFFERENT SIGNS SUBTRACT

-1/2 + 1/3

-1/6 + (-5/6) = -6/6 or -1 *SAME SIGNS ADD

-1/6 + (-5/6)

-10 - 21 = -10 + (-21) = -31 When subtracting rational numbers, you ADD THE OPPOSITE (Keep-Change-Change). Once you change the problem to addition, then you simply just follow the addition rules. More examples: -8 - (-2) = -8 + 2 = -6 -9 - 2 = -9 + (-2) = -11 12 - (-12) = 12 + 12 = 24 15 - 25 = 15 + (-25) = -10 20 - 14 = 20 + (-14) = 6

-10 - 21

-14/-7 = 2 When multiplying or dividing rational numbers, you multiply or divide as normal. If the signs are the same, the product or quotient will be positive. If the signs are different, the product of quotient will be negative. *SAME SIGNS POSITIVE *DIFFERENT SIGNS NEGATIVE More examples: -5 × (-3) = 15 -8 × 1 = -8 6(-3) = -18 54 ÷ (-6) = -9 -100 ÷ (-4) = 25 -144/12 = -12

-14/-7

-19.2 × (-4_1/2) = -19.2 × (-4.5) = -86.4 or -19.2 × (-4_1/2) = -19_1/5 × (-4_1/2) = -96/5 × (-9/2) = -48/5 × (-9/1) = -432/5 = -86_2/5 *a/b form *Cross simplify *Multiply across *Same signs positive

-19.2 × (-4_1/2)

-1_5/7 - 2/7 = -1_5/7 + (-2/7) = -1_7/7 or -2 *ADD THE OPPOSITE (Keep -Change-Change)

-1_5/7 - 2/7

-2.25 - 0.75 = -2.25 + (-0.75) = -3 *ADD THE OPPOSITE. *THEN, FOLLOW THE ADDITION RULES.

-2.25 - 0.75

-2/3 × (-9/10) = -1/1 × (-3/5) = 3/5 *a/b form *Cross simplify *Multiply across *Same signs positive

-2/3 × (-9/10)

-3 × 6 = -18 When multiplying or dividing rational numbers, you multiply or divide as normal. If the signs are the same, the product or quotient will be positive. If the signs are different, the product of quotient will be negative. *SAME SIGNS POSITIVE *DIFFERENT SIGNS NEGATIVE More examples: -5 × (-4) = 20 -1 × 1 = -1 6(-9) = -54 24 ÷ (-3) = -8 -36 ÷ (-4) = 9 -30/5 = -6

-3 × 6

-3 ÷ 0 is undefined or -3/0 is undefined Zero divided by a non-zero rational number is undefined. This is not okay or "N/O."

-3 ÷ 0 or -3/0

-3.2 + (-0.6) = -3.8 *SAME SIGNS ADD

-3.2 + (-0.6)

-3/4 - (-2/4) = -3/4 + 2/4 = -1/4 *ADD THE OPPOSITE (Keep -Change-Change)

-3/4 - (-2/4)

-4 + (-9) = -13 When adding rational numbers, you ADD the numbers when they have SAME SIGNS. You keep the sign of the addends. *SAME SIGNS ADD More examples: -3 + (-5) = -8 2 + 5 = 7

-4 + (-9)

-5.25 + 2.5 = -2.75 *DIFFERENT SIGNS SUBTRACT

-5.25 + 2.5

-6 - (-3) = -6 + 3 = -3 When subtracting rational numbers, you ADD THE OPPOSITE (Keep-Change-Change). Once you change the problem to addition, then you simply just follow the addition rules. More examples: -5 - (-1) = -5 + 1 = -4 -6 - 7 = -6 + (-7) = -13 8 - (-8) = 8 + 8 = 16 25 - 35 = 25 + (-35) = -10 19 - 15 = 19 + (-15) = 4

-6 - (-3)

-7/8 × 2_1/3 = -7/8 × 7/3 = -49/24 = -2_1/4 *a/b form *Multiply across *Different signs negative

-7/8 × 2_1/3

-9(-5) = 45 When multiplying or dividing rational numbers, you multiply or divide as normal. If the signs are the same, the product or quotient will be positive. If the signs are different, the product of quotient will be negative. *SAME SIGNS POSITIVE *DIFFERENT SIGNS NEGATIVE More examples: -6 × (-4) = 24 -7 × 1 = -7 8(-9) = -72 25 ÷ (-5) = -5 -42 ÷ (-6) = 7 -56/7 = -8

-9(-5)

-4

-[-(-4)] Read as "the opposite of the opposite of the opposite of 4" or "the opposite of the opposite of negative 4."

-2

-|-2| Read as "the opposite of the absolute value of negative 2."

0 ÷ 2 = 0 or 0/2 = 0 A non-zero rational number divided by zero is always zero. This is "O/K."

0 ÷ 2 or 0/2

0.5 + (-1.75) = -1.25 *DIFFERENT SIGNS SUBTRACT

0.5 + (-1.75)

0.5 × (−2/5) = 0.5 × (-0.4) = -0.2 or 0.5 × (−2/5) = 1/2 × (-2/5) = 1/1 × (-1/5) = -1/5 *Either change both to decimals or change both to fractions. Then, cross simplify and multiply across. *Different signs negative

0.5 × (−2/5)