Initial Stock Composition
Let $T$ represent the total stock (in kg) at the beginning.
Initially, product A constituted 80% of the total stock.
- Initial quantity of Product A: $A_{initial} = 0.80 \times T$
- Initial quantity of Product B: $B_{initial} = (1 - 0.80) \times T = 0.20 \times T$
Stock Changes and Final Ratio
Following the removal of specific quantities:
- Product A removed: 20 kg
- Product B removed: 8 kg
The quantities remaining in the stock are:
- Final quantity of Product A: $A_{final} = (0.80T - 20)$ kg
- Final quantity of Product B: $B_{final} = (0.20T - 8)$ kg
The ratio of the final quantities of Product A to Product B is given as 9 : 2.
We can express this relationship as an equation:
$ \frac{A_{final}}{B_{final}} = \frac{9}{2} $Substituting the expressions for $A_{final}$ and $B_{final}$:
$ \frac{0.80T - 20}{0.20T - 8} = \frac{9}{2} $Solving for the Initial Total Stock
To find $T$, we solve the equation:
Cross-multiply the terms:
$ 2 \times (0.80T - 20) = 9 \times (0.20T - 8) $ $ 1.60T - 40 = 1.80T - 72 $Rearrange the equation to isolate $T$:
$ 72 - 40 = 1.80T - 1.60T $ $ 32 = 0.20T $Calculate the value of $T$:
$ T = \frac{32}{0.20} $ $ T = \frac{320}{2} $ $ T = 160 $The total stock in the beginning was 160 kg.
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