We will learnmixed calculations with the four operations, combining addition, subtraction, multiplication, division, and powers; techniques using thedistributive law; and applications of positive and negative numbers in daily life（such as averages）. This completes the unit on “positive and negative numbers.”
Order of Operations
Perform calculations in the following order.
① Calculate insideparenthesesfirst
② Calculatepowers
③ Calculatemultiplication and division
④ Calculateaddition and subtraction
Remember: “parentheses → powers → multiplication and division → addition and subtraction.”
Calculate the following：8 + (−3)² × (−2) − 12 ÷ (−4)
Answer：
Power：(−3)² = +9
Multiplication and division：+9 × (−2) = −18、 12 ÷ (−4) = −3
Substituting gives 8 + (−18) − (−3) ＝ 8 − 18 + 3 ＝ −7
Techniques Using the Distributive Law
Thedistributive lawholds for multiplication and addition（subtraction）.
a×(b+c) ＝ a×b + a×c
It can also be used in the reverse direction（factoring out a common factor）.
a×b + a×c ＝ a×(b+c)
It is often used to make calculations easier.
Example：13×98 ＝ 13×(100−2) ＝ 1300 − 26 ＝ 1274
Applications of Positive and Negative Numbers（Averages）
Calculations become easier when a reference value is chosen and differences from it are represented as positive or negative numbers.
For example, take 20℃ as the reference temperature for a day, and suppose the temperatures over 5 days have differences of +2, −1, +3, 0, −4（℃）.
Average difference from the reference：(2 + (−1) + 3 + 0 + (−4)) ÷ 5 = 0 ÷ 5 = 0
Actual average temperature：20 + 0 ＝ 20℃。
Thus, it can be found using “reference value +（average difference）.”
Five students’ heights are represented relative to 150cm as +3, −2, +5, −4, and +3. Find the average height of the five students.
Answer：Sum of differences：3+(−2)+5+(−4)+3 ＝ 5。Average difference：5÷5 = 1。
Average height ＝ 150 + 1 ＝ 151cm。
- Order of operations：parentheses → powers → multiplication and division → addition and subtraction
- Using the distributive law a×(b+c)=ab+ac skillfully makes calculations easier
- Choosing a reference value and representing differences as positive or negative numbers makes calculations such as averages easier
- (Average) = (Reference value) + (Average difference)
Practice Problems
- Calculate the following：(−4) × {6 + (−9)} ÷ (−3)
- Use the distributive law to calculate efficiently：(−25) × 7 + (−25) × 3
- A test score was recorded relative to 70 points, and the differences for four students were +8, −6, +3, and −1. Find the average score of the four students.
Answers and Explanations
- Answer：−4
Explanation：Inside the parentheses：6+(−9)=−3。The expression is (−4)×(−3)÷(−3) = (+12)÷(−3) = −4。 - Answer：−250
Explanation：(−25)×7 + (−25)×3 ＝ (−25)×(7+3) ＝ (−25)×10 ＝ −250。Factor out the common factor −25。 - Answer：71 points
Explanation：Sum of differences：8+(−6)+3+(−1) ＝ 4。Average difference：4÷4=1。Average score＝70+1＝71 points。