Addition, subtraction, multiplication, and division of fractions

LVL: FREE

MODULE: Fractions, Proportions, and Percentages

[EXEC: MICRO_CORE]

✖️ 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!

[EXEC: DEEP_COMPUTE]

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}.

TASK_1[0 / 3]
LVL_2
EXEC: ALGORITHM

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

DEEP_COMPUTE
ULTRA
[EXEC: MICRO_CORE]

✖️ 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!

[EXEC: DEEP_COMPUTE]

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}.

TASK_1[0 / 3]
LVL_2
EXEC: ALGORITHM

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

DEEP_COMPUTE
ULTRA
[EXEC: MICRO_CORE]

✖️ 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!

[EXEC: DEEP_COMPUTE]

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}.

TASK_1[0 / 3]
LVL_2
EXEC: ALGORITHM

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

DEEP_COMPUTE
ULTRA
[EXEC: MICRO_CORE]

✖️ 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!

[EXEC: DEEP_COMPUTE]

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.

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LVL_2
EXEC: ALGORITHM

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

DEEP_COMPUTE
ULTRA
[EXEC: MICRO_CORE]

✖️ 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!

[EXEC: DEEP_COMPUTE]

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.

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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_COMPUTE
ULTRA

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