Track / Stage 5 / 109 of 400 #709 of 2000

Problem 709

AIME late

Algebra Difficulty 5.2 Find the answer

13.198. A certain substance absorbs moisture, thereby increasing its mass. To absorb 1400 kg of moisture, 300 kg more of the uncrushed substance is required than of the crushed substance. What percentage of the mass of the substance is the mass of the absorbed moisture in the case of crushed substance and in the case of uncrushed substance, if in the latter case this percentage is 105 less than in the former?

Your answer

Official solution

Solution.

Let's take $x$ kg of crushed substance, $(x+300)$ kg of uncrushed.

$\begin{aligned} & x \text { kg is } 100 \% \\ & 1400 \text{ kg } \quad-\quad y_{1} \% \\ & x+300 \text { kg is } 100 \% \\ & 1400 \text { kg } \quad-\quad y_{2} \% . \\ & \left\{\begin{array}{l} y_{1}=\frac{140000}{x} \\ y_{2}=\frac{140000}{x+300} \\ y_{1}=y_{2}+105 \end{array}\right. \\ & y_{2}=175 \% \text {. } \end{aligned}$

Answer: 280 and $175 \%$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.