Addition, subtraction, multiplication, and division of fractions

LVL: FREE

MODULE: Fractions, Proportions, and Percentages

✖️ 1. Finding common denominators for addition and subtraction using LCM

🔗 Finding Common Denominators Using LCM

  • To add or subtract fractions, the denominators must match.
  • Find the LCM (Least Common Multiple) of both denominators.
  • Convert each fraction by multiplying top and bottom by the same number.
  • The LCM becomes the new common denominator.
  • Only add or subtract the numerators after conversion.

Example: 14+16\frac{1}{4} + \frac{1}{6} → LCM of 4 and 6 is 12 → 312+212=512\frac{3}{12} + \frac{2}{12} = \frac{5}{12}

💡 Same bottom, then add the top!

1. Finding common denominators for addition and subtraction using LCM

Finding Common Denominators Using LCM

To add or subtract fractions, they must share a common denominator. The least common multiple (LCM) of the denominators provides the smallest such shared base, minimizing arithmetic complexity.

Intuition: Think of fractions as pieces of different-sized pies. To combine them, we must first cut all pies into equally-sized slices.

Core Rules:

  • Identify the LCM of all denominators involved
  • Convert each fraction by multiplying numerator and denominator by the factor that transforms the original denominator into the LCM
  • Only numerators are added or subtracted; the common denominator remains unchanged
  • Simplify the result by dividing numerator and denominator by their greatest common divisor (GCD)

Consequence: Using the LCM instead of arbitrary common multiples keeps numbers smaller and calculations cleaner.

Example: For 16+14\frac{1}{6} + \frac{1}{4}, the LCM of 6 and 4 is 12. Convert: 212+312=512\frac{2}{12} + \frac{3}{12} = \frac{5}{12}.

Progress0 / 3
Level 2
EXEC: ALGORITHM

Add the following complex numbers: (4+3i)+(2+5i)(4 + 3i) + (2 + 5i).

Deep reasoning
Ultra

✖️ 2. Multiplication rules and the mechanics of cross-canceling

✖️ Multiplication Rules and Cross-Canceling

  • Multiply fractions by multiplying numerators together and denominators together.
  • No common denominator needed for multiplication.
  • Cross-cancel before multiplying to simplify work.
  • Cross-cancel means divide any numerator and any denominator by their GCF.
  • Cross-canceling prevents large numbers and reduces final simplification.

Example: 49×38\frac{4}{9} \times \frac{3}{8} → Cancel 4 and 8 by 4, cancel 3 and 9 by 3 → 13×12=16\frac{1}{3} \times \frac{1}{2} = \frac{1}{6}

💡 Straight across, but cancel first to save time!

2. Multiplication rules and the mechanics of cross-canceling

Multiplication Rules and Cross-Canceling

Multiplying fractions requires multiplying numerators together and denominators together: ab×cd=a⋅cb⋅d\frac{a}{b} \times \frac{c}{d} = \frac{a \cdot c}{b \cdot d}. No common denominator is needed.

Intuition: Taking a fraction of a fraction scales both dimensions proportionally.

Core Rules:

  • Multiply straight across: numerator with numerator, denominator with denominator
  • Cross-canceling simplifies before multiplication by dividing any numerator and any denominator by their common factors
  • Cross-canceling reduces intermediate products, preventing unnecessarily large numbers
  • Always simplify the final result

Consequence: Cross-canceling is optional but highly efficient, especially with large numbers.

Example: For 815×512\frac{8}{15} \times \frac{5}{12}, cancel 5 with 15 (giving 3) and 8 with 12 (giving 2 and 3): 23×13=29\frac{2}{3} \times \frac{1}{3} = \frac{2}{9}.

Progress0 / 3
Level 2
EXEC: ALGORITHM

Evaluate ini^n when n=14n = 14.

Deep reasoning
Ultra

✖️ 3. Division as multiplication by the reciprocal (Keep-Change-Flip)

🔄 Division as Multiplication by Reciprocal (Keep-Change-Flip)

  • To divide fractions, use Keep-Change-Flip.
  • Keep the first fraction unchanged.
  • Change division to multiplication.
  • Flip the second fraction (swap numerator and denominator).
  • Then multiply normally using the multiplication rule.

Example: 23÷45\frac{2}{3} \div \frac{4}{5} → Keep 23\frac{2}{3}, change to ×\times, flip to 54\frac{5}{4} → 23×54=1012=56\frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}

💡 Dividing by a fraction? Flip it and multiply!

3. Division as multiplication by the reciprocal (Keep-Change-Flip)

Division as Multiplication by the Reciprocal

Dividing by a fraction is equivalent to multiplying by its reciprocal. The Keep-Change-Flip method formalizes this: keep the first fraction, change division to multiplication, flip the second fraction.

Intuition: Dividing by 12\frac{1}{2} asks "how many halves fit?" which equals multiplying by 2.

Core Rules:

  • Keep the dividend (first fraction) unchanged
  • Change the operation from division to multiplication
  • Flip the divisor (second fraction) to its reciprocal by swapping numerator and denominator
  • Proceed with standard fraction multiplication

Consequence: Division never requires special handling beyond converting to multiplication.

Example: For 34÷25\frac{3}{4} \div \frac{2}{5}, apply Keep-Change-Flip: 34×52=158\frac{3}{4} \times \frac{5}{2} = \frac{15}{8}.

Progress0 / 3
Level 2
EXEC: ALGORITHM

Multiply the complex numbers: (3+2i)(4+i)(3 + 2i)(4 + i).

Deep reasoning
Ultra

✖️ 4. Common mistakes in fraction operations and how to detect them

⚠️ Common Mistakes and Detection

  • Never add or subtract denominators (only numerators change).
  • Forgetting to find a common denominator before adding causes wrong answers.
  • Flipping the wrong fraction in division (always flip the second one).
  • Multiplying denominators when adding (only do this for multiplication).
  • Check: if adding two fractions less than 1 gives a result greater than 2, you made an error.

Example of error: 12+13≠25\frac{1}{2} + \frac{1}{3} \neq \frac{2}{5} (wrong: added denominators). Correct: 36+26=56\frac{3}{6} + \frac{2}{6} = \frac{5}{6}

💡 Denominators stay put when adding; only tops combine!

4. Common mistakes in fraction operations and how to detect them

Common Mistakes and Detection Strategies

Fraction errors typically arise from misapplying operation rules. Recognizing patterns helps prevent systematic mistakes.

Intuition: Each operation has distinct rules; mixing them produces nonsensical results that fail basic sanity checks.

Core Rules:

  • Never add or subtract denominators directly; only numerators combine after finding a common denominator
  • Never flip the first fraction in division; only the divisor (second fraction) becomes its reciprocal
  • Cross-canceling applies only to multiplication and division, never to addition or subtraction
  • Check if your answer's magnitude makes sense (e.g., adding two positive fractions less than 1 cannot exceed 2)

Consequence: Dimensional analysis and estimation catch most errors before final answers.

Example: The error 12+13=25\frac{1}{2} + \frac{1}{3} = \frac{2}{5} fails because 25<12\frac{2}{5} < \frac{1}{2}, which is impossible when adding positive quantities.

Progress0 / 3
Level 2
EXEC: ALGORITHM

Calculate the product of the complex number z=3+4iz = 3 + 4i and its conjugate. Enter the final real number.

Deep reasoning
Ultra

✖️ 5. Applications: Calculating equivalent resistance of parallel circuits in physics

⚡ Equivalent Resistance in Parallel Circuits

  • In parallel circuits, total resistance uses the formula 1Rtotal=1R1+1R2\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2}.
  • You must add fractions with resistor values as denominators.
  • Find the common denominator of the resistances.
  • After adding, take the reciprocal to find total resistance.
  • Parallel resistance is always less than the smallest individual resistor.

Example: Two resistors 6 ohms and 3 ohms → 1R=16+13=16+26=36\frac{1}{R} = \frac{1}{6} + \frac{1}{3} = \frac{1}{6} + \frac{2}{6} = \frac{3}{6} → R=63=2R = \frac{6}{3} = 2 ohms

💡 Parallel paths make current flow easier, so total resistance drops!

5. Applications: Calculating equivalent resistance of parallel circuits in physics

Equivalent Resistance in Parallel Circuits

In parallel circuits, the reciprocal of total resistance equals the sum of reciprocals of individual resistances: 1Rtotal=1R1+1R2+⋯\frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots. This requires fraction addition and reciprocal operations.

Intuition: Parallel paths allow current to split, reducing overall resistance below the smallest individual resistor.

Core Rules:

  • Write each resistance as a reciprocal fraction: 1Ri\frac{1}{R_i}
  • Add these fractions using a common denominator (typically LCM)
  • Take the reciprocal of the sum to find RtotalR_{\text{total}}
  • The result is always less than the smallest individual resistance

Consequence: Mastery of fraction operations directly enables solving real-world electrical engineering problems.

Example: For resistors 6 ohms and 3 ohms in parallel: 1R=16+13=16+26=36\frac{1}{R} = \frac{1}{6} + \frac{1}{3} = \frac{1}{6} + \frac{2}{6} = \frac{3}{6}, so R=2R = 2 ohms.

Progress0 / 3
Level 2
EXEC: ALGORITHM

In an AC circuit, two components are connected in series. Their impedances are Z1=5+3jZ_1 = 5 + 3j ohms and Z2=2−jZ_2 = 2 - j ohms.

Calculate the total impedance ZtotalZ_{total}. Write your answer in the form a+bja + bj.

Deep reasoning
Ultra

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