Middle School Mathematics: Combinatorics, Conversions, and Signed Operations

Fundamental Counting Principle and Combinatorics

  • Definition of the Fundamental Counting Principle (Rule of Product):

    • If a compound event or selection consists of multiple successive choices where the first choice has n1n_1 possible outcomes, the second choice has n2n_2 possible outcomes, and the kk-th choice has nkn_k possible outcomes, the total number of distinct combinations is given by the product:
    • Total Combinations=n1×n2×⋯×nk\text{Total Combinations} = n_1 \times n_2 \times \dots \times n_k

  • Cafeteria Breakfast Package Problem:

    • Problem Context: A school cafeteria offers a breakfast package consisting of one sandwich (torta), one beverage (bebida), and one dessert (postre).
    • Selection Categories and Available Options:
    • Tortas (3 options): Pollo (Chicken), Jamón (Ham), and Queso (Cheese).
    • Bebidas (4 options): Jamaica (Hibiscus tea), Horchata (Rice milk drink), Limón (Lemonade), and Coca (Coca-Cola).
    • Postres (2 options): Galleta (Cookie) and Flan.
    • Question: How many different packages consisting of a torta, a beverage, and a dessert can be formed?
    • Step-by-step Multiplication:
    • 3 tortas×4 bebidas×2 postres=243 \text{ tortas} \times 4 \text{ bebidas} \times 2 \text{ postres} = 24
    • Conclusion: There are 2424 distinct breakfast package options available.

  • Custom Ice Cream Configuration Problem:

    • Problem Context: Constructing an ice cream serving by choosing one base, one ice cream flavor, and one topping.
    • Selection Categories and Available Options:
    • Base (2 options): Cono (Cone) or Canasta (Waffle bowl/basket).
    • Ice Cream Flavor / Sabor del helado (4 options): Vainilla (Vanilla), Chocolate, Fresa (Strawberry), and Menta (Mint).
    • Topping (3 options): Chispas de colores (Rainbow sprinkles), Rallado de coco (Shredded coconut), and Pistache (Pistachio).
    • Question: How many different ice cream combinations can be assembled by selecting one base, one flavor, and one topping?
    • Step-by-step Multiplication:
    • 2 bases×4 sabores×3 toppings=242 \text{ bases} \times 4 \text{ sabores} \times 3 \text{ toppings} = 24
    • Conclusion: There are 2424 distinct ice cream configurations that can be formed.

Conversion Between Decimal Numbers and Fractions

  • Mathematical Fundamentals of Conversion:

    • Decimals to Fractions: A terminating decimal is expressed as a fraction by writing the decimal digits in the numerator over a power of 1010 corresponding to the number of decimal places (e.g., tenths as 1010, hundredths as 100100, thousandths as 10001000), followed by reducing to lowest terms using the greatest common divisor (gcd⁡\gcd).
    • Fractions to Decimals: A fraction ab\frac{a}{b} is converted to a decimal by carrying out the arithmetic division of the numerator aa by the denominator bb.

  • Exercise 1: Decimal to Simplified Fraction Conversion:

    • Problem Statement: What is the representation in simplified fraction form of the decimal number 0.750.75?
    • Provided Multiple-Choice Options:
    • A) 7510\frac{75}{10}
    • B) 34\frac{3}{4}
    • C) 76\frac{7}{6}
    • D) 14\frac{1}{4}
    • Worked Solution and Reduction:
    • Identify decimal place value: 0.750.75 has two decimal places, representing hundredths: 0.75=751000.75 = \frac{75}{100}
    • First simplification step (divide numerator and denominator by 55): 75÷5100÷5=1520\frac{75 \div 5}{100 \div 5} = \frac{15}{20}
    • Second simplification step (divide numerator and denominator by 55): 15÷520÷5=34\frac{15 \div 5}{20 \div 5} = \frac{3}{4}
    • Alternative direct method: gcd⁡(75,100)=25\gcd(75, 100) = 25, yielding 75÷25100÷25=34\frac{75 \div 25}{100 \div 25} = \frac{3}{4}
    • Correct Answer: Option B (34\frac{3}{4}

  • Exercise 2: Converting Fraction 45\frac{4}{5} to a Decimal:

    • Problem Statement: If we convert the fraction 45\frac{4}{5} to a decimal number, what do we obtain?
    • Provided Multiple-Choice Options:
    • A) 0.40.4
    • B) 0.450.45
    • C) 0.80.8
    • Long Division Algorithm:
    • Set up the division: 5)4.0‾5 \overline{) 4.0}
    • Since 55 does not divide into 44, place 0.0. in the quotient.
    • Append a zero to the dividend to divide 4040 by 55: 40÷5=840 \div 5 = 8
    • Product and remainder: 5×8=405 \times 8 = 40, with a remainder of 40−40=040 - 40 = 0
    • 45=0.8\frac{4}{5} = 0.8
    • Correct Answer: Option C (0.80.8)

  • Exercise 3: Converting Fraction 120\frac{1}{20} to a Decimal:

    • Problem Statement: If we convert the fraction 120\frac{1}{20} to a decimal number, what is the result?
    • Provided Multiple-Choice Options:
    • A) 0.50.5
    • B) 0.050.05
    • C) 0.0050.005
    • D) 0.020.02
    • Long Division Algorithm:
    • Set up the division: 20)1.00‾20 \overline{) 1.00}
    • 2020 divides into 11 zero times (0.0.)
    • 2020 divides into 1010 zero times (0.00.0)
    • 2020 divides into 100100 exactly 55 times (20×5=10020 \times 5 = 100
    • Remainder: 100−100=0100 - 100 = 0
    • The resulting decimal is 0.050.05
    • Correct Answer: Option B (0.050.05)

Number Line Representation (Ubicación en la Recta Numérica)

  • Conceptual Framework:
    • Real numbers are situated along a one-dimensional coordinate axis centered at an origin of zero (00).
    • Positive values are located to the right of zero (>0> 0).
    • Negative values are located to the left of zero (<0< 0).
    • Rational values represented either as fractions (ab\frac{a}{b}) or decimals correspond to exact spatial coordinates between whole integers.

Addition and Subtraction with Signed Numbers (Sumas y Restas con Signos)

  • Governing Rules for Signed Addition and Subtraction:

    • Same Signs: Add the absolute values of the terms together and retain the shared sign ((+)+(+)=(+)(+) + (+) = (+) and (−)+(−)=(−)(-) + (-) = (-)).
    • Different Signs: Calculate the absolute difference between the larger magnitude and the smaller magnitude, and apply the sign of the term with the larger absolute value.

  • Exercise 1: Addition of Signed Decimals with Unlike Signs:

    • Problem Statement: Solve (−4.5)+(+2.1)=(-4.5) + (+2.1) =
    • Provided Multiple-Choice Options:
    • A) −6.6-6.6
    • B) +2.4+2.4
    • C) −2.4-2.4
    • Step-by-step Calculation:
    • Identify absolute values: ∣−4.5∣=4.5|-4.5| = 4.5 and ∣+2.1∣=2.1|+2.1| = 2.1
    • Since the signs are opposite, subtract the smaller magnitude from the larger magnitude:       4.5−2.1=2.44.5 - 2.1 = 2.4
    • Determine the dominant sign: Since 4.5>2.14.5 > 2.1 and the term 4.54.5 is negative, the resulting sign must be negative:       (−4.5)+(+2.1)=−2.4(-4.5) + (+2.1) = -2.4
    • Correct Answer: Option C (−2.4-2.4)

  • Exercise 2: Addition of Signed Fractions with Like Denominators:

    • Problem Statement: Perform the addition of the following fractions with equal denominators:     (−47)+(−67)=\left(-\frac{4}{7}\right) + \left(-\frac{6}{7}\right) =
    • Provided Multiple-Choice Options:
    • A) 47\frac{4}{7}
    • B) −107-\frac{10}{7}
    • C) −47-\frac{4}{7}
    • D) 107\frac{10}{7}
    • Step-by-step Calculation:
    • Maintain the common denominator of 77.
    • Combine the numerators following the rule for numbers with identical signs:       (−4)+(−6)7=−107=−107\frac{(-4) + (-6)}{7} = \frac{-10}{7} = -\frac{10}{7}
    • Correct Answer: Option B (−107-\frac{10}{7}

Multiplication and Division with Signed Numbers (Multiplicación y División con Signos)

  • Law of Signs (Ley de los Signos):

    • Same Signs (Product or Quotient is Positive):
    • (+)×(+)=(+)(+) \times (+) = (+)
    • (−)×(−)=(+)(-) \times (-) = (+)
    • (+)÷(+)=(+)(+) \div (+) = (+)
    • (−)÷(−)=(+)(-) \div (-) = (+)
    • Different Signs (Product or Quotient is Negative):
    • (+)×(−)=(−)(+) \times (-) = (-)
    • (−)×(+)=(−)(-) \times (+) = (-)
    • (+)÷(−)=(−)(+) \div (-) = (-)
    • (−)÷(+)=(−)(-) \div (+) = (-)

  • Exercise 1: Multiplication of Signed Decimals:

    • Problem Statement: Apply the law of signs to solve the following operation:     (−0.6)×(+1.2)=(-0.6) \times (+1.2) =
    • Provided Multiple-Choice Options:
    • A) +0.72+0.72
    • B) −0.72-0.72
    • C) +7.2+7.2
    • D) +0.18+0.18
    • Step-by-step Calculation:
    • Multiply the numerical values ignoring signs and decimals: 12×6=7212 \times 6 = 72
    • Count the total number of decimal places: 1.21.2 has 11 decimal place and 0.60.6 has 11 decimal place, making a total of 1+1=21 + 1 = 2 decimal places from the right:       1.2×0.6=0.721.2 \times 0.6 = 0.72
    • Apply the sign rule: A negative multiplied by a positive yields a negative result ((−)×(+)=(−)(-) \times (+) = (-)(−0.6)×(+1.2)=−0.72(-0.6) \times (+1.2) = -0.72
    • Correct Answer: Option B (−0.72-0.72)

  • Exercise 2: Division of Signed Decimals:

    • Problem Statement: Solve the following division:     (+4.5)÷(−0.5)=(+4.5) \div (-0.5) =
    • Division Setup and Simplification:
    • Set up the quotient as a fraction: +4.5−0.5\frac{+4.5}{-0.5}
    • Eliminate decimals by multiplying both numerator and denominator by 1010:       4.5×100.5×10=455=9\frac{4.5 \times 10}{0.5 \times 10} = \frac{45}{5} = 9
    • In long division notation: 0.5)4.5‾0.5 \overline{) 4.5} shifts the decimal place by one position to become 5)45‾5 \overline{) 45}, yielding a numerical quotient of 99.
    • Apply the sign rule: A positive divided by a negative yields a negative result ((+)÷(−)=(−)(+) \div (-) = (-)(+4.5)÷(−0.5)=−9(+4.5) \div (-0.5) = -9
    • Final Value: −9-9