Adv. Math notes

Keep, Change, Flip (KCF)

The Keep, Change, Flip (KCF) method is a fundamental strategy used to divide fractions.

  • Keep: Keep the first fraction unchanged.

  • Change: Change the division operator (÷\div) to a multiplication operator (×\times).

  • Flip: Flip the second fraction to its reciprocal (swap the numerator and denominator).

Example

To solve 34÷25\frac{3}{4} \div \frac{2}{5}:

  1. Keep: 34\frac{3}{4}

  2. Change: ÷\div becomes ×\times

  3. Flip: 25\frac{2}{5} becomes 52\frac{5}{2}

  4. Multiply: 34×52=3×54×2=158\frac{3}{4} \times \frac{5}{2} = \frac{3 \times 5}{4 \times 2} = \frac{15}{8}

The Zip Zap Method

The "Zip Zap" (or butterfly) method is used to combine fractions with non-common denominators without needing to explicitly compute the least common denominator first.

  • Zip (First Cross-Product): Multiply the numerator of the first fraction by the denominator of the second fraction.

  • Zap (Second Cross-Product): Multiply the denominator of the first fraction by the numerator of the second fraction.

  • Combine Numerators: Add or subtract the two cross-products depending on the operator in the expression.

  • Multiply Denominators: Multiply the two original denominators together to form the common denominator.

General Formula

ab+cd=a×d+b×cb×d\frac{a}{b} + \frac{c}{d} = \frac{a \times d + b \times c}{b \times d}

Example

Simplify 13+25\frac{1}{3} + \frac{2}{5}:

  1. Zip: 1×5=51 \times 5 = 5

  2. Zap: 3×2=63 \times 2 = 6

  3. New Numerator: 5+6=115 + 6 = 11

  4. New Denominator: 3×5=153 \times 5 = 15

  5. Result: 1115\frac{11}{15}

Multiplying and Dividing Positive and Negative Integers

Working with positive and negative (cohesive) integers follows clear sign rules for multiplication and division.

Rule Summary

  • Same Signs Produce a Positive Result:

    • (+)×(+)=(+)(+) \times (+) = (+)

    • (−)×(−)=(+)(-) \times (-) = (+)

    • (+)÷(+)=(+)(+) \div (+) = (+)

    • (−)÷(−)=(+)(-) \div (-) = (+)

  • Different Signs Produce a Negative Result:

    • (+)×(−)=(−)(+) \times (-) = (-)

    • (−)×(+)=(−)(-) \times (+) = (-)

    • (+)÷(−)=(−)(+) \div (-) = (-)

    • (−)÷(+)=(−)(-) \div (+) = (-)

Worked Examples

  • Same signs (Multiplication): (−4)×(−6)=24(-4) \times (-6) = 24

  • Different signs (Multiplication): (−5)×3=−15(-5) \times 3 = -15

  • Same signs (Division): (−20)÷(−4)=5(-20) \div (-4) = 5

  • Different signs (Division): 36÷(−6)=−636 \div (-6) = -6