If $\frac{x}{5} = \frac{y}{8}$, then the value of $\frac{x+5}{y+8}$ is equal to :

Q: If $\frac{x}{5} = \frac{y}{8}$, then the value of $\frac{x+5}{y+8}$ is equal to :

To solve the problem, we start with the given equation: \[ \frac{x}{5} = \frac{y}{8} \] From this equation, we can express $x$ in terms of $y$ or vice versa. Let's express $x$ in terms of $y$: \[ x = \frac{5}{8}y \] Now, we need to find the value of $\frac{x+5}{y+8}$. Substitute $x = \frac{5}{8}y$ into the expression: \[ \frac{x+5}{y+8} = \frac{\frac{5}{8}y + 5}{y + 8} \] To simplify this expression, we first find a common denominator for the numerator: \[ \frac{5}{8}y + 5 = \frac{5}{8}y + \frac{40}{8} = \frac{5y + 40}{8} \] Now, substitute back into the expression: \[ \frac{x+5}{y+8} = \frac{\frac{5y + 40}{8}}{y + 8} \] This can be simplified by multiplying the numerator by the reciprocal of the denominator: \[ = \frac{5y + 40}{8(y + 8)} \] To further simplify, notice that: \[ 5y + 40 = 5(y + 8) \] Thus, the expression becomes: \[ \frac{5(y + 8)}{8(y + 8)} = \frac{5}{8} \] Therefore, the value of $\frac{x+5}{y+8}$ is $\frac{5}{8}$, which corresponds to Option 3. **Explanation of Incorrect Options:** 1. **Option 1: $\frac{7}{8}$** - This option is incorrect because the simplification of the expression $\frac{x+5}{y+8}$ does not yield $\frac{7}{8}$. The correct simplification results in $\frac{5}{8}$. 2. **Option 2: $\frac{3}{5}$** - This option is incorrect because the relationship between $x$ and $y$ given by $\frac{x}{5} = \frac{y}{8}$ leads to a different ratio when substituting into $\frac{x+5}{y+8}$. The correct ratio is $\frac{5}{8}$. 3. **Option 4: $\frac{8}{5}$** - This option is incorrect because it suggests a reversal of the correct ratio. The simplification process clearly shows that the correct ratio is $\frac{5}{8}$, not $\frac{8}{5}$. In conclusion, the correct answer is Option 3: $\frac{5}{8}$, as demonstrated through the logical simplification of the given expression.

Question

Accepted Answer

To solve the problem, we start with the given equation:

$$
\frac{x}{5} = \frac{y}{8}
$$

From this equation, we can express $x$ in terms of $y$ or vice versa. Let's express $x$ in terms of $y$:

$$
x = \frac{5}{8}y
$$

Now, we need to find the value of $\frac{x+5}{y+8}$. Substitute $x = \frac{5}{8}y$ into the expression:

$$
\frac{x+5}{y+8} = \frac{\frac{5}{8}y + 5}{y + 8}
$$

To simplify this expression, we first find a common denominator for the numerator:

$$
\frac{5}{8}y + 5 = \frac{5}{8}y + \frac{40}{8} = \frac{5y + 40}{8}
$$

Now, substitute back into the expression:

$$
\frac{x+5}{y+8} = \frac{\frac{5y + 40}{8}}{y + 8}
$$

This can be simplified by multiplying the numerator by the reciprocal of the denominator:

$$
= \frac{5y + 40}{8(y + 8)}
$$

To further simplify, notice that:

$$
5y + 40 = 5(y + 8)
$$

Thus, the expression becomes:

$$
\frac{5(y + 8)}{8(y + 8)} = \frac{5}{8}
$$

Therefore, the value of $\frac{x+5}{y+8}$ is $\frac{5}{8}$, which corresponds to Option 3.

**Explanation of Incorrect Options:**

1. **Option 1: $\frac{7}{8}$** - This option is incorrect because the simplification of the expression $\frac{x+5}{y+8}$ does not yield $\frac{7}{8}$. The correct simplification results in $\frac{5}{8}$.

2. **Option 2: $\frac{3}{5}$** - This option is incorrect because the relationship between $x$ and $y$ given by $\frac{x}{5} = \frac{y}{8}$ leads to a different ratio when substituting into $\frac{x+5}{y+8}$. The correct ratio is $\frac{5}{8}$.

3. **Option 4: $\frac{8}{5}$** - This option is incorrect because it suggests a reversal of the correct ratio. The simplification process clearly shows that the correct ratio is $\frac{5}{8}$, not $\frac{8}{5}$.

In conclusion, the correct answer is Option 3: $\frac{5}{8}$, as demonstrated through the logical simplification of the given expression.