In a plastic bag, there are (a + 5) red rubber bands, (2b - 5) blue rubber bands, and (2.5a - 5) black rubber bands. The number of red rubber bands is 40% less than the number of blue rubber bands, which is 25% more than the number of black rubber bands, In a separate box, there are (a + b) white rubber bands, (3b - a) orange rubber bands, and (b - a) yellow rubber bands, Calculate the probability of randomly selecting 2 white rubber bands, from the box, given that the selection is done without replacement.

### Concept & Algebraic Probability First, we use basic linear equations to find the values of unknowns $a$ and $b$ based on percentage relationships. Next, we apply classical probability rules for sequential events without replacement. $$ P(A \text{ and } B) = P(A) \times P(B|A) $$ ### Step-by-Step Solution 1. **Define Quantities in the Bag:** * Red = $a + 5$ * Blue = $2b - 5$ * Black = $2.5a - 5$ 2. **Formulate Equations based on conditions:** Condition 1: Red is $40\%$ less than Blue (meaning Red is $60\%$ of Blue). $a + 5 = 0.6(2b - 5)$ --- (Eq 1) Condition 2: Blue is $25\%$ more than Black (meaning Blue is $125\%$ of Black). $2b - 5 = 1.25(2.5a - 5)$ --- (Eq 2) 3. **Solve for $a$ and $b$:** Substitute $(2b - 5)$ from Eq 2 directly into Eq 1: $a + 5 = 0.6 \times [1.25(2.5a - 5)]$ $a + 5 = 0.75(2.5a - 5)$ $a + 5 = 1.875a - 3.75$ $0.875a = 8.75$ $a = 10$ Now, find $b$ using Eq 2: $2b - 5 = 1.25(2.5 \times 10 - 5)$ $2b - 5 = 1.25(25 - 5)$ $2b - 5 = 1.25 \times 20 = 25$ $2b = 30 \implies b = 15$ 4. **Calculate Box Contents:** * White = $a + b = 10 + 15 = 25$ * Orange = $3b - a = 3(15) - 10 = 45 - 10 = 35$ * Yellow = $b - a = 15 - 10 = 5$ Total in box = $25 + 35 + 5 = 65$ rubber bands. 5. **Calculate Probability:** We need the probability of drawing 2 white rubber bands without replacement. First draw (White): $25 / 65 = 5 / 13$ Second draw (White): $24 / 64 = 3 / 8$ $P(\text{Both White}) = \frac{5}{13} \times \frac{3}{8} = \frac{15}{104}$ ### Exam Strategy & Shortcut Whenever percentages involve straightforward multiplier relations ($40\%$ less = $0.6 \times$, $25\%$ more = $1.25 \times$), chain the equations directly. For instance, $\text{Red} = 0.6 \times \text{Blue}$ and $\text{Blue} = 1.25 \times \text{Black}$, meaning $\text{Red} = 0.6 \times 1.25 \times \text{Black} = 0.75 \times \text{Black}$. This bypasses solving complex simultaneous equations by creating one variable equation instantly. ### Common Pitfall A common error is making the denominator for the second probability term the same as the first. Since selection is "without replacement", you must deduct 1 from both the total count (denominator) and the successful count (numerator). ### Final Answer Therefore, the correct answer is **15/104**.