Question

There are 100 persons and 100 plates
Rich one eat in 5 plates
Men with Service eat in 3 plates
And 3 poor eat in 1 plate then how many rich service and poor people are there

Answer 1

Answer on Question #77409 – Math – Algebra

Question

There are 100 persons and 100 plates

Rich one eat in 5 plates

Men with Service eat in 3 plates

And 3 poor eat in 1 plate then how many rich, service and poor people are there

Solution

$$R + S + P = 100$$ $$5R + 3S + \frac{P}{3} = 100$$

where $R$ is the number of rich men, $S$ is the number of service men, $P$ is the number of poor men.

$$R = 100 - S - P$$ $$5(100 - S - P) + 3S + \frac{P}{3} = 100$$ $$500 - 2S - \frac{14P}{3} = 100$$ $$S + \frac{7P}{3} = 200$$ $$S = 200 - \frac{7P}{3}$$ $$0 \leq S \leq \frac{100}{3} \Rightarrow 0 \leq S \leq 33$$ $$0 \leq R \leq \frac{100}{5} = 20$$ $$0 \leq P \leq 100$$ $$0 \leq 200 - \frac{7P}{3} \leq \frac{100}{3} \Rightarrow \frac{500}{7} \leq P \leq \frac{600}{7}$$ $$71 \leq P \leq 85$$

Since $P$ is a multiple of 3, then

$$72 \leq P \leq 84$$

So we have

$$P = 72; \quad S = 200 - \frac{7 \cdot 72}{3} = 32; \quad R = 100 - 72 - 32 = -4 < 0$$ $$P = 75; \quad S = 200 - \frac{7 \cdot 75}{3} = 25; \quad R = 100 - 75 - 25 = 0$$ $$P = 78; \quad S = 200 - \frac{7 \cdot 78}{3} = 18; \quad R = 100 - 78 - 18 = 4$$ $$P = 81; \quad S = 200 - \frac{7 \cdot 81}{3} = 11; \quad R = 100 - 81 - 11 = 8$$ $$P = 84; \quad S = 200 - \frac{7 \cdot 84}{3} = 4; \quad R = 100 - 84 - 4 = 12$$

Answer:

$$P = 75; \quad S = 25; \quad R = 0$$ $$P = 78; \quad S = 18; \quad R = 4$$ $$P = 81; \quad S = 11; \quad R = 8$$ $$P = 84; \quad S = 4; \quad R = 12$$

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