If $\frac{a}{3}=\frac{b}{4}=\frac{c}{7}$, then value of $\frac{a+b+c}{c}$ is.

Question

Accepted Answer

The correct option is (3):

This problem involves the concept of ratios and proportions. When a series of ratios are equal, we can set them equal to a constant, say $k$, to express the variables in terms of this constant.

**1. Recall the Key Concept:**
If we have a proportion like $\frac{a}{x} = \frac{b}{y} = \frac{c}{z}$, we can use the property of equal ratios (also known as the $k$-method or property of equal proportions), which states that each ratio is equal to the ratio of the sum of the numerators to the sum of the denominators:
$$\frac{a}{x} = \frac{b}{y} = \frac{c}{z} = \frac{a+b+c}{x+y+z}$$

**2. Introduce the Constant $k$ (Alternative Method):**
Given the equation:
$$\frac{a}{3} = \frac{b}{4} = \frac{c}{7}$$
Let this common ratio be equal to a constant $k$:
$$\frac{a}{3} = k \implies a = 3k$$
$$\frac{b}{4} = k \implies b = 4k$$
$$\frac{c}{7} = k \implies c = 7k$$

**3. Show the Calculation Process:**
We need to find the value of the expression $\frac{a+b+c}{c}$.
Substitute the values of $a$, $b$, and $c$ in terms of $k$ into the expression:
$$a+b+c = 3k + 4k + 7k = (3+4+7)k = 14k$$
Now, substitute the sum and the value of $c$ into the target expression:
$$\frac{a+b+c}{c} = \frac{14k}{7k}$$

**4. Simplification and Final Result:**
Cancel out the common term $k$ from the numerator and denominator (since $k$ cannot be zero for the ratios to be defined in this context, assuming $a, b, c$ are not all zero):
$$\frac{14k}{7k} = \frac{14}{7} = 2$$
Thus, the value of the expression $\frac{a+b+c}{c}$ is 2.

**5. Justification of the Correct Option (Option 3):**
The calculated value is 2, which corresponds exactly to Option 3. This result is obtained by correctly expressing the variables based on the given ratio equality and substituting them into the expression to simplify.

The incorrect option is (1):
The value is 2, not $\sqrt{2}$. This option might arise from an incorrect manipulation of the ratio terms or accidental root calculation.

The incorrect option is (2):
The value is 2, not $\frac{1}{\sqrt{7}}$. This choice is likely the result of an algebraic error or choosing a denominator term with a root incorrectly.

The incorrect option is (4):
The value is 2, not 7. 7 is the value of the denominator $c$ (or $7k$), but the final expression is the ratio of the sum $(a+b+c)$ to $c$, which simplifies to 2. Choosing 7 would mean mistaking the value of $c$ for the value of the entire expression $\frac{a+b+c}{c}$.