Directions: These questions are based on the following information: Given that $a : b = 5 : 3$ and $b : c = 2 : 5$. If $c = 50$, the value of $a + b + c$ will be

### Concept & Proportional Distribution Once a unified ratio is established, finding the absolute value of one component allows you to determine the scale factor (constant $k$). This scale factor can then be applied to the sum of the ratio parts to find the total sum efficiently. ### Step-by-Step Solution 1. **Given:** From previous calculations, the unified ratio is $a : b : c = 10 : 6 : 15$. 2. Let the actual values be $a = 10k$, $b = 6k$, and $c = 15k$. 3. We are given that $c = 50$. Set up the equation: $$15k = 50$$ $$k = \frac{50}{15} = \frac{10}{3}$$ 4. The requested expression is the sum $a + b + c$: $$a + b + c = 10k + 6k + 15k = 31k$$ 5. Substitute the exact value of $k$ into the sum expression: $$31 \cdot \left(\frac{10}{3}\right) = \frac{310}{3}$$ 6. Convert the fraction to a decimal to check the ranges: $$\frac{310}{3} = 103.333...$$ 7. Analyzing the options, $103.33$ is strictly greater than $103$ and less than $105$. ### Exam Strategy & Shortcut Do not waste time calculating the individual absolute values of $a$ and $b$ before adding. Sum the ratio proportions ($10 + 6 + 15 = 31$) and multiply by the multiplier ($\frac{50}{15}$) in one single step: $31 \cdot \frac{50}{15} = 103.33$. ### Common Pitfall Converting the scale factor $k = \frac{10}{3}$ into a rounded decimal like $3.33$ too early. Doing so yields $31 \cdot 3.33 = 103.23$, which could cause hesitation when distinguishing between narrow bounds, especially if rounding errors push the value into an incorrect bucket. Always keep fractions until the final step. ### Final Answer Therefore, the correct answer is **more than 103 but less than 105**.