Algebra word problem solving

Question

In an island, which had a total population of 55009, a war was fought between ‘Benos’ and ‘Malos’ the only tribes residing in the island. During the war every ‘Benos’ fought with a different number of ‘malos’. One of them fought with exactly 140 ‘Malos’, a second one fought with exactly 141 ‘Malos’, a third one fought with exactly 142 ‘Malos’, a fourth one with with exactly 143 ‘Malos’ and so on till one of them fought with every ‘Malos’ residing in the island. Find the number of ‘Malos’ residing in the island.

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My attempt :-

1st Bano will fight against 140 Manos 2nd Bano will fight against 141 Manos . . . nth Bano will fight with (139+n) Manos

so total Banos = n

and total Manos will be the sum of Manos will turn out to be 140+141+142.... = n/2 * (140+139+n) =n/2 * (279+n)

so now, n + n/2 * (279+n) = 55009

n is not turning out to be an integer from solving this. What did I do wrong in here ?

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I am not able to understand the official solution either, it simply does the following :-

No. of Manos = 139 + Number of Banos

139 + Banos + Banos = 55009

Banos = 27435 => Manos = 55009-27435 = 27574

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Answer 1

Let's say there were $b$ Benos and $m$ Malos.

• Beno no. $1$ fought $140$ Malos.
• Beno no. $2$ fought $141$ Malos.
• Beno no. $3$ fought $142$ Malos.
• ...
• for each $k$, Beno no. $k$ fought $139+k$ Malos.
• ...
• Beno no. $b$ (the last one) fought $139+b$ Malos.

But, in the last line, we also know that the last Beno fought all $m$ Malos. Thus,

$$m=139+b$$

On the other hand, we know that

$$b+m=55009$$

(the total number of people on the island). Now this is a system of two equations with two unknowns $b$ and $m$, and it needs to be solved somehow. One way is: substitute $m=139+b$ in the second equation to get:

$$b+(139+b)=55009$$ $$2b+139=55009$$ $$2b=54870$$ $$b=27435$$ $$m=139+b=139+27435=27574$$

Answer 2

I think you misunderstood the question. The question means that $n$th Banos fight with $(n+139)$th Manos, and you did this part correctly, the place that you are wrong with is Manos can fight again untill there is a specific $n_0$ where all Manos are in the fight. As all Manos are in the fight, and the island only contains $55009$ people, $$\mathrm{Manos}=n_0+139$$ and $$\mathrm{Benos}=n_0$$ As there won't left any people that haven't fought in this fight, the term of Benos is $\mathrm{no.(Benos)}$ and term of Manos is $\mathrm{no.(Benos)}+139$ Therfore the equation: $$2\mathrm{no.(Benos)}+139=55009$$ and lead to the solution.