Comparing Fractions: Identify the greater value

To solve this problem, we will compare the given fractions to determine which is largest:

• Compare $\frac{2}{3}$ and $\frac{7}{11}$:

  • Cross multiply: $2 \times 11 = 22$ and $3 \times 7 = 21$.
    Since $22 > 21$, $\frac{2}{3}$ is larger than $\frac{7}{11}$.

• Compare $\frac{2}{3}$ and $\frac{6}{10}$:

  • Simplify $\frac{6}{10} = \frac{3}{5}$.
    Cross multiply: $2 \times 5 = 10$ and $3 \times 3 = 9$.
    Since $10 > 9$, $\frac{2}{3}$ is larger than $\frac{3}{5}$ (same as $\frac{6}{10}$).

• To confirm, compare $\frac{2}{3}$ and $\frac{3}{5}$ again directly:

  • Cross multiply: $2 \times 5 = 10$ and $3 \times 3 = 9$.
    Since $10 > 9$, $\frac{2}{3}$ is also larger than $\frac{3}{5}$.

Therefore, the largest fraction of all given choices is indeed $\frac{2}{3}$.

To solve this problem, let's evaluate each of the fraction addition expressions:

• Expression 1: $\frac{1}{2} + \frac{1}{4}$
  To add these fractions, we need a common denominator. The least common denominator of 2 and 4 is 4.
  Thus, $\frac{1}{2} = \frac{2}{4}$ and keep $\frac{1}{4}$ as it is.
  Adding the two fractions, $\frac{2}{4} + \frac{1}{4} = \frac{3}{4}$.
• Expression 2: $\frac{1}{3} + \frac{1}{3}$
  Since these have the same denominator, we can directly add them:
  $\frac{1}{3} + \frac{1}{3} = \frac{2}{3}$.
• Expression 3: $\frac{1}{2} + \frac{3}{10}$
  The least common denominator of 2 and 10 is 10.
  Convert $\frac{1}{2}$ to $\frac{5}{10}$ and add it to $\frac{3}{10}$:
  $\frac{5}{10} + \frac{3}{10} = \frac{8}{10} = \frac{4}{5}$.
• Expression 4: $\frac{5}{10} + \frac{1}{5}$
  Convert the fractions to have the same denominator. Here, the least common denominator is 10.
  Keep $\frac{5}{10}$ as it is, and convert $\frac{1}{5}$ to $\frac{2}{10}$:
  $\frac{5}{10} + \frac{2}{10} = \frac{7}{10}$.

Now, let's compare these results:
- $\frac{3}{4} = 0.75$
- $\frac{2}{3} = 0.666\ldots$
- $\frac{4}{5} = 0.8$
- $\frac{7}{10} = 0.7$

Comparing these decimals, we see that $0.8$ (from Expression 3) is the largest result.

Therefore, the expression with the highest result is $\frac{1}{2} + \frac{3}{10}$.