highschool/Grade 9/Math/MFM1P1/Final_Exam_Study_Sheet.md

Study Sheet

Rounding and Decimals

Decimals

• Terms:
  • Given the number 9123.456:
    • The tenth is the 4.
    • The hundredth is 5.
    • The thousandths is 6.
    • The ones is 3.
    • The tens is 5.
    • The hundreds is 1.
    • The thousands is 9.
    • Remember, tens and tenths may sound the same, but they are DIFFERENT!
• To round to a tenth, hundredth, and thousandths
  • Tenths
    • If the hundredth is 5 or higher, round up, else, round down.
    • Example:
      • Round 12.53223 to the tenths
      • The answer is 12.5, as the hundredths, or 3 is smaller than 5.
  • Hundredth
    • If the thousandth is 5 or higher, round up, else, round down.
    • Example:
      • Round 12.53521 to the hundredth
      • The answer is 12.4, as the thousandths, or 5 is bigger or equal to 5.
  • Thousandth
    • If the number of the thousandth is 5 or higher, round up, else, round down.
      
    • Example:
      • Round 12.5356 to the thousandths
      • The answer is 12.536, as the number after the thousandths, or 6 is bigger than 5.
• To round to a ones, tens, hundreds, and thousands
  • Ones
    • If the tenths is 5 or higher, round up, else, round down.
    • Example:
      • Round 123.5333 to the ones
      • The answer is 124, as the tenths, or 5 is bigger than or equal to 5.
  • Tens
    • If the ones is 5 or higher, round up, else, round down.
    • Example:
      • Round 123.5777 to the tens
      • The answer is 120, as the ones, or 3 is smaller than or equal to 5.
  • Hundreds
    • If the tens is 5 or higher, round up, else, round down.
    • Example:
      • Round 177.34343 to the hundreds
      • The answer is 200, as the tens, or 7 is bigger than 5.
  • Thousands
    • If the hundreds is 5 or higher, round up, else round down.
    • Example:
      • Round 566.777` to the thousands
      • The answer is 1000, as the hundreds, or 5 is bigger or equal to 5.

Integers

Multiplication and Division

• Pretend a and b are random positive numbers

  Type	Outcome
  a × b	Positive number
  a × (-b)	Negative number
  (-a) × b	Negative number
  (-a) × (-b)	Positive number
  a ÷ b	Positive number
  a ÷ (-b)	Negative number
  (-a) ÷ b	Negaitve number
  (-a) ÷ (-b)	Positive number

• Treat as normal divion and multiplacation, and just add the negative sign infront of the number according to the rules above.

• Practice

  • 8 × -7
    • Answer: -56
  • 2 × 4
    • Answer: 8
  • -7 × -7
    • Answer: 1
  • -10 × 4
    • Answer: -40
  • 8 ÷ 4
    • Answer: 2
  • -16 ÷ -8
    • Answer: 2
  • -4 ÷ 1
    • Answer -4
  • 9 ÷ -3
    • Answer: -3

Addition and Division

• Pretend a and b are random postive numbers

  Type	Equivalent
  a+b	a+b
  b+a	b+a
  a+(-b)	a-b
  (-a)+b	b-a
  a-b	a-b
  b-a	b-a
  a-(-b)	a+b
  (-a)-b	a-b

Order Or Operation

• BEDMAS
• Follow BEDMAS for order of operations if there are more than one operation

Letter	Meaning
B / P	Bracket / Parentheses
E	Exponent
D	Divison
M	Multiplication
A	Addition
S	Subtraction

• 
• Follow order of operation, inorder to do know which operation to do first.
• Example: Given \(`(2+4) \times 5 - 9 \div 3`\)
  • First do everything in brackets: \(`(6) \times 5 - 9 \div 3`\)
  • Then do multiplication/division: \(`30 - 3`\)
  • Then finally, do subtaction/addition: \(`27`\)
  • The answer is 27.

Fractions / Rational Numbers

• The number on the top is called the numerator.
• The number on the bottom is called the denominator.
• A fraction in its most simple form is when the numerator and denominator cannot be both divided by the same number.

Additions / Subtractions With Fractions

• Example: \(`\frac{3}{5} + \frac{4}{3}`\)
• Find common denominator, which is 15, as 5 and 3 both are factors of 15.
  • You can do this easily with a table, just count by the number you are using, for example:

  • Counting by 5s	Counting by 3s
    5	3
    10	6
    15	9
    20	12
    25	15

  • As you can see, both columns contain the number 15, so 15 is the common denominator.
  • Now, after we find the denominator, we must convert the fraction so that it has the common denominator. To do this, we must multiply the denominator by a number, so that it equals the common denominator. For the first fracion \(`\frac{3}{5}`\), the denominator is 5, to get to 15, we must multiply it by 3. Now, whatever we do on the bottom, me MUST do it on the top too, so we also multiply the numerator by 3 as well, the new fraction is now \(`\frac{3 \times 3}{5 \times 3} = \frac{9}{15}`\).
  • We now do the same thing to the other fraction: \(`\frac{4 \times 5}{3 \times 5} = \frac{20}{15}`\)
  • Now that the denominators are the same and the fractions are converted, we can just simply add the numerators together while keeping the denominator the same. The result is \(`\frac{9 + 20}{15} = \frac{29}{15}`\).
  • The same steps applied to subtracion, with the only difference of subtacting the numerators rather than adding them.

Multiplaction With Fractions

• To multiply a fracion, simply multiply the numerators together, and the denominators together.
• Example: \(`\frac{3}{6} \times \frac{7}{4}`\)
  • Answer: \(`\frac{3 \times 7}{6 \times 4} = \frac{21}{24}`\)

Division With Fractions

• To divide 2 fractions, flip the second fraction upside down and multiply them togehter.
• Or, in advanced terms, mulitply the first fraction by the reciporocal of the second fraction.
• Given an example: \(`\frac{4}{2} \div \frac{6}{9}`\)
  • First, flip the second fraction upside down: \(`\frac{4}{2} \div \frac{9}{6}`\)
  • Then change the division to a multiply: \(`\frac{4}{2} \times \frac{9}{6}`\)
  • Then multiply the 2 fractions \(`\frac{4 \times 9}{2 \times 6} = \frac{36}{12}`\)