In two types of powdered detergent the ratio of soda and soap-dust is 2 : 19 and 1 : 11 respectively. If 7 kg of the first type is mixed with 4 kg of the second type, find the ratio of soda to soap-dust in the new detergent mixture.

### Concept & Mixture of Mixtures To find the final ratio when two different mixtures are combined, calculate the absolute weight of each individual component (soda and soap-dust) from both types, sum them up respectively, and then take the new ratio. ### Step-by-Step Solution * **Step 1: Analyze the first type of detergent.** * Quantity used = $7$ kg. * Ratio of soda to soap-dust = $2 : 19$. Total parts = $21$. * Soda from first type = $\frac{2}{21} \times 7 = \frac{2}{3}$ kg. * Soap-dust from first type = $\frac{19}{21} \times 7 = \frac{19}{3}$ kg. * **Step 2: Analyze the second type of detergent.** * Quantity used = $4$ kg. * Ratio of soda to soap-dust = $1 : 11$. Total parts = $12$. * Soda from second type = $\frac{1}{12} \times 4 = \frac{1}{3}$ kg. * Soap-dust from second type = $\frac{11}{12} \times 4 = \frac{11}{3}$ kg. * **Step 3: Combine the components.** * Total soda = $\frac{2}{3} + \frac{1}{3} = \frac{3}{3} = 1$ kg. * Total soap-dust = $\frac{19}{3} + \frac{11}{3} = \frac{30}{3} = 10$ kg. * **Step 4: Find the final ratio.** * Ratio of total soda to total soap-dust = $1 : 10$. ### Exam Strategy & Shortcut Because the fractions result in the same denominator ($3$) for both types, you can simply add the numerators directly. Notice how $(2/21 \times 7)$ and $(1/12 \times 4)$ both neatly cancel down to denominators of $3$. This allows for rapid mental addition: $(2+1)/3 = 1$ for soda, and $(19+11)/3 = 10$ for soap, immediately giving $1:10$. ### Common Pitfall Attempting to average the ratios directly without accounting for the different total weights ($7$ kg and $4$ kg) mixed together. Ratios cannot be directly added; absolute quantities must be derived first. ### Final Answer Therefore, the correct answer is **1 : 10**.