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There are 3 persons named Ram,Shyam and Hari (Yeah,Indian names). In an 100 m race Ram defeats Shyam by 15 m. In another race of 100 m, Hari defeats Shyam by 20 m. If Ram and Hari take part in a 150 m race,who will win and by how much distance?

I attempted the problem and got an answer that Hari will win by $8 \frac{14}{17}$ m. Is that correct? Also can you please mention the steps how to solve it?

Archit Pasayat
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1 Answers1

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We can resolve it by comparing the ratio of distance between the runners

$R:=$ Ram

$S:=$ Shyam

$H:=$ Hari

From the first run, we can see that Ram runs 100 meters while Shyam 85 meters in the same time, so the ratio is $$(I):\frac{R}{S} = \frac{100}{85}$$

From the second run, we get the ratio $$(II):\frac{H}{S} = \frac{100}{80}$$

Using $I$ and $II$, we can create a system of equations: $$ \left\{\begin{matrix} R = \frac{100S}{85}\\ H = \frac{100S}{80} \end{matrix}\right.$$

Now, dividing both sides, we get: $$\frac{R}{H} = \frac{\frac{100S}{85}}{\frac{100S}{80}} \Leftrightarrow H = \frac{100}{80}\frac{85}{100}R \Leftrightarrow H= \frac{17R}{16}$$

Now, se can see that Hari is $\frac{17}{16}$ times faster than Ram. If Hari runs 100 meters, we get: $$H: 100 = \frac{17R}{16} \Leftrightarrow R = \frac{1600}{17} \approx 94.1176471 \text{ meters}$$

Marco Antonio
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