Answer :
Certainly! Let's break down the solution to the given problem, step-by-step.
The expression to solve is:
[tex]\[
\left(\frac{3}{6}+\frac{1}{4}\right)+\frac{6}{9} \cdot \frac{2}{3}
\][/tex]
### Step 1: Simplify the fractions within the addition
1. Simplify [tex]\(\frac{3}{6}\)[/tex]:
- [tex]\(\frac{3}{6}\)[/tex] simplifies to [tex]\(\frac{1}{2}\)[/tex] because both the numerator and the denominator are divisible by 3.
2. Add [tex]\(\frac{1}{2}\)[/tex] and [tex]\(\frac{1}{4}\)[/tex]:
- To add [tex]\(\frac{1}{2}\)[/tex] and [tex]\(\frac{1}{4}\)[/tex], find a common denominator. The common denominator between 2 and 4 is 4.
- Convert [tex]\(\frac{1}{2}\)[/tex] to [tex]\(\frac{2}{4}\)[/tex], since [tex]\(\frac{1 \times 2}{2 \times 2} = \frac{2}{4}\)[/tex].
- Now, add [tex]\(\frac{2}{4}\)[/tex] and [tex]\(\frac{1}{4}\)[/tex]:
[tex]\[
\frac{2}{4} + \frac{1}{4} = \frac{3}{4}
\][/tex]
### Step 2: Simplify the multiplication
1. Simplify [tex]\(\frac{6}{9}\)[/tex]:
- [tex]\(\frac{6}{9}\)[/tex] simplifies to [tex]\(\frac{2}{3}\)[/tex] because both the numerator and the denominator are divisible by 3.
2. Multiply [tex]\(\frac{2}{3}\)[/tex] and [tex]\(\frac{2}{3}\)[/tex]:
- Perform the multiplication:
[tex]\[
\frac{2}{3} \times \frac{2}{3} = \frac{4}{9}
\][/tex]
### Step 3: Add the results
Now, combine the results from the addition and multiplication:
- We have [tex]\(\frac{3}{4}\)[/tex] from the addition and [tex]\(\frac{4}{9}\)[/tex] from the multiplication.
To add these, find a common denominator. The least common denominator between 4 and 9 is 36.
1. Convert [tex]\(\frac{3}{4}\)[/tex] to have a denominator of 36:
- [tex]\(\frac{3}{4} = \frac{3 \times 9}{4 \times 9} = \frac{27}{36}\)[/tex]
2. Convert [tex]\(\frac{4}{9}\)[/tex] to have a denominator of 36:
- [tex]\(\frac{4}{9} = \frac{4 \times 4}{9 \times 4} = \frac{16}{36}\)[/tex]
3. Add the two fractions:
[tex]\[
\frac{27}{36} + \frac{16}{36} = \frac{43}{36}
\][/tex]
Thus, the solution to the expression is [tex]\(\frac{43}{36}\)[/tex], which can also be written as a mixed number: [tex]\(1 \frac{7}{36}\)[/tex].
Therefore, the answer to the multiple-choice question is [tex]\(1 \frac{7}{36}\)[/tex].
The expression to solve is:
[tex]\[
\left(\frac{3}{6}+\frac{1}{4}\right)+\frac{6}{9} \cdot \frac{2}{3}
\][/tex]
### Step 1: Simplify the fractions within the addition
1. Simplify [tex]\(\frac{3}{6}\)[/tex]:
- [tex]\(\frac{3}{6}\)[/tex] simplifies to [tex]\(\frac{1}{2}\)[/tex] because both the numerator and the denominator are divisible by 3.
2. Add [tex]\(\frac{1}{2}\)[/tex] and [tex]\(\frac{1}{4}\)[/tex]:
- To add [tex]\(\frac{1}{2}\)[/tex] and [tex]\(\frac{1}{4}\)[/tex], find a common denominator. The common denominator between 2 and 4 is 4.
- Convert [tex]\(\frac{1}{2}\)[/tex] to [tex]\(\frac{2}{4}\)[/tex], since [tex]\(\frac{1 \times 2}{2 \times 2} = \frac{2}{4}\)[/tex].
- Now, add [tex]\(\frac{2}{4}\)[/tex] and [tex]\(\frac{1}{4}\)[/tex]:
[tex]\[
\frac{2}{4} + \frac{1}{4} = \frac{3}{4}
\][/tex]
### Step 2: Simplify the multiplication
1. Simplify [tex]\(\frac{6}{9}\)[/tex]:
- [tex]\(\frac{6}{9}\)[/tex] simplifies to [tex]\(\frac{2}{3}\)[/tex] because both the numerator and the denominator are divisible by 3.
2. Multiply [tex]\(\frac{2}{3}\)[/tex] and [tex]\(\frac{2}{3}\)[/tex]:
- Perform the multiplication:
[tex]\[
\frac{2}{3} \times \frac{2}{3} = \frac{4}{9}
\][/tex]
### Step 3: Add the results
Now, combine the results from the addition and multiplication:
- We have [tex]\(\frac{3}{4}\)[/tex] from the addition and [tex]\(\frac{4}{9}\)[/tex] from the multiplication.
To add these, find a common denominator. The least common denominator between 4 and 9 is 36.
1. Convert [tex]\(\frac{3}{4}\)[/tex] to have a denominator of 36:
- [tex]\(\frac{3}{4} = \frac{3 \times 9}{4 \times 9} = \frac{27}{36}\)[/tex]
2. Convert [tex]\(\frac{4}{9}\)[/tex] to have a denominator of 36:
- [tex]\(\frac{4}{9} = \frac{4 \times 4}{9 \times 4} = \frac{16}{36}\)[/tex]
3. Add the two fractions:
[tex]\[
\frac{27}{36} + \frac{16}{36} = \frac{43}{36}
\][/tex]
Thus, the solution to the expression is [tex]\(\frac{43}{36}\)[/tex], which can also be written as a mixed number: [tex]\(1 \frac{7}{36}\)[/tex].
Therefore, the answer to the multiple-choice question is [tex]\(1 \frac{7}{36}\)[/tex].
