Subtract 4 1/2-3 5/6

Solution:

Step 1: Transform $4\frac{1}{2}$ into an improper fraction.

• Step 1.1: Recognize that a mixed number is the sum of its integer and fractional parts: $4+\frac{1}{2}-3\frac{5}{6}$.

• Step 1.2: Combine $4$ and $\frac{1}{2}$.

  • Step 1.2.1: Express $4$ as a fraction with a denominator of $2$ by multiplying by $\frac{2}{2}$: $4\cdot \frac{2}{2}+\frac{1}{2}-3\frac{5}{6}$.

  • Step 1.2.2: Merge $4\cdot \frac{2}{2}$: $\frac{4\cdot 2}{2}+\frac{1}{2}-3\frac{5}{6}$.

  • Step 1.2.3: Add the numerators over the shared denominator: $\frac{4\cdot 2+1}{2}-3\frac{5}{6}$.

  • Step 1.2.4: Simplify the numerator:

    • Step 1.2.4.1: Calculate $4\cdot 2$: $\frac{8+1}{2}-3\frac{5}{6}$.

    • Step 1.2.4.2: Add $8$ and $1$: $\frac{9}{2}-3\frac{5}{6}$.

Step 2: Change $3\frac{5}{6}$ into an improper fraction.

• Step 2.1: Note that a mixed number is the sum of its integer and fractional parts: $\frac{9}{2}-(3+\frac{5}{6})$.

• Step 2.2: Add $3$ and $\frac{5}{6}$.

  • Step 2.2.1: Convert $3$ into a fraction with a denominator of $6$ by multiplying by $\frac{6}{6}$: $\frac{9}{2}-(3\cdot \frac{6}{6}+\frac{5}{6})$.

  • Step 2.2.2: Combine $3\cdot \frac{6}{6}$: $\frac{9}{2}-(\frac{3\cdot 6}{6}+\frac{5}{6})$.

  • Step 2.2.3: Add the numerators over the shared denominator: $\frac{9}{2}-\frac{3\cdot 6+5}{6}$.

  • Step 2.2.4: Simplify the numerator:

    • Step 2.2.4.1: Multiply $3$ by $6$: $\frac{9}{2}-\frac{18+5}{6}$.

    • Step 2.2.4.2: Add $18$ and $5$: $\frac{9}{2}-\frac{23}{6}$.

Step 3: Adjust $\frac{9}{2}$ to have a common denominator with $\frac{23}{6}$ by multiplying by $\frac{3}{3}$: $\frac{9}{2}\cdot \frac{3}{3}-\frac{23}{6}$.

Step 4: Express both fractions with a common denominator of $6$.

• Step 4.1: Multiply $\frac{9}{2}$ by $\frac{3}{3}$: $\frac{9\cdot 3}{2\cdot 3}-\frac{23}{6}$.

• Step 4.2: Multiply $2$ by $3$: $\frac{9\cdot 3}{6}-\frac{23}{6}$.

Step 5: Combine the numerators over the common denominator: $\frac{9\cdot 3-23}{6}$.

Step 6: Simplify the numerator.

• Step 6.1: Multiply $9$ by $3$: $\frac{27-23}{6}$.

• Step 6.2: Subtract $23$ from $27$: $\frac{4}{6}$.

Step 7: Reduce the fraction by eliminating common factors.

• Step 7.1: Extract the factor of $2$ from $4$: $\frac{2(2)}{6}$.

• Step 7.2: Eliminate the common factors:

  • Step 7.2.1: Extract the factor of $2$ from $6$: $\frac{2\cdot 2}{2\cdot 3}$.

  • Step 7.2.2: Cancel the common factor: $\frac{\cancel{2}\cdot 2}{\cancel{2}\cdot 3}$.

  • Step 7.2.3: Rewrite the simplified expression: $\frac{2}{3}$.

Step 8: Present the result in various forms.

• Exact Form: $\frac{2}{3}$
• Decimal Form: $0.666\dots$

Knowledge Notes:

The problem involves subtracting mixed numbers, which are numbers consisting of an integer and a fraction. The steps taken to solve this problem involve several key mathematical concepts:

1. Conversion of Mixed Numbers to Improper Fractions: A mixed number is converted to an improper fraction by multiplying the whole number by the denominator of the fractional part and then adding the numerator of the fractional part.

2. Common Denominator: To subtract fractions, they must have a common denominator. This is achieved by finding the least common multiple (LCM) of the denominators and adjusting the fractions accordingly.

3. Simplifying Fractions: After performing the subtraction, the resulting fraction is simplified by canceling out any common factors in the numerator and denominator.

4. Multiplication and Addition of Fractions: When combining fractions, it's important to multiply the numerators and denominators separately and then add or subtract the numerators while keeping the common denominator.

5. Reduction of Fractions: A fraction is reduced to its simplest form by dividing both the numerator and the denominator by their greatest common divisor (GCD).