Solution: Step 1: Transform 4 frac12 into an improper fraction. - Step 1.1: Recognize that a mixed number is the sum of its integer and fractional parts: 4 + frac12 - 3 frac56. - Step 1.2: Combine 4 and frac12. - Step 1.2.1: Express 4 as a fraction with a denominator of 2 by multiplying by frac22: 4 cdot frac22 + frac12 - 3 frac56. - Step 1.2.2: Merge 4 cdot frac22: frac4 cdot 22 + frac12 - 3 frac56. - Step 1.2.3: Add the numerators over the shared denominator: frac4 cdot 2 + 12 - 3 frac56. - Step 1.2.4: Simplify the numerator: - Step 1.2.4.1: Calculate 4 cdot 2: frac8 + 12 - 3 frac56. - Step 1.2.4.2: Add 8 and 1: frac92 - 3 frac56. Step 2: Change 3 frac56 into an improper fraction. - Step 2.1: Note that a mixed number is the sum of its integer and fractional parts: frac92 - (3 + frac56). - Step 2.2: Add 3 and frac56. - Step 2.2.1: Convert 3 into a fraction with a denominator of 6 by multiplying by frac66: frac92 - (3 cdot frac66 + frac56). - Step 2.2.2: Combine 3 cdot frac66: frac92 - (frac3 cdot 66 + frac56). - Step 2.2.3: Add the numerators over the shared denominator: frac92 - frac3 cdot 6 + 56. - Step 2.2.4: Simplify the numerator: - Step 2.2.4.1: Multiply 3 by 6: frac92 - frac18 + 56. - Step 2.2.4.2: Add 18 and 5: frac92 - frac236. Step 3: Adjust frac92 to have a common denominator with frac236 by multiplying by frac33: frac92 cdot frac33 - frac236. Step 4: Express both fractions with a common denominator of 6. - Step 4.1: Multiply frac92 by frac33: frac9 cdot 32 cdot 3 - frac236. - …

Subtract 4 1/2-3 5/6

The problem is to perform a subtraction operation between two mixed numbers. In this case, you are asked to subtract 3 5/6 from 4 1/2. A mixed number is a number consisting of an integer and a proper fraction. To solve this, one typically converts mixed numbers into improper fractions before carrying out the subtraction, adjusts them to have a common denominator, and then performs the subtraction of the numerators while keeping the denominator the same. Finally, if the resulting fraction is improper, it might need to be converted back into a mixed number.

$4 \frac{1}{2} - 3 \frac{5}{6}$

Answer

Expert–verified

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: $0.666\ldots$

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:

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