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Question

# Sachin, Sehwag and Dhoni together scored 228 runs. Their individual scores are in G.P. Sehwag and Dhoni together scored 12 runs more than Sachin. Find their individual scores.

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Solution

## Let the runs made by Sachin, Sehwag and Dhoni be $\frac{a}{r}$, a and ar, respectively. Thus, we have: $\frac{a}{r}$ + a + ar = 228 …(1) Sehwag and Dhoni together scored 12 runs more than Sachin. Thus, we have: a + ar = $\frac{a}{r}$ + 12 $⇒$ a + ar – $\frac{a}{r}$ = 12 …(2) On subtracting (2) from (1), we get: $\frac{a}{r}$+ a + ar – a – ar + $\frac{a}{r}$ = 228 – 12 = 216 $⇒$$\frac{2a}{r}$ = 216 $⇒$ a = 108r On putting a = 108r in (1), we get: $\frac{108r}{r}+108r+108{r}^{2}=228\phantom{\rule{0ex}{0ex}}⇒108\left[1+r+{r}^{2}\right]=228\phantom{\rule{0ex}{0ex}}⇒1+r+{r}^{2}=\frac{19}{9}\phantom{\rule{0ex}{0ex}}⇒r+{r}^{2}=\frac{10}{9}$ On using the quadratic formula, we get: $r=\frac{-1±\sqrt{1-4×1×\left(\frac{-10}{9}\right)}}{2\left(1\right)}\phantom{\rule{0ex}{0ex}}=\frac{-1±\sqrt{1+\frac{40}{9}}}{2}\phantom{\rule{0ex}{0ex}}=\frac{-1±\sqrt{\frac{49}{9}}}{2}\phantom{\rule{0ex}{0ex}}=\frac{-1±\frac{7}{3}}{2}\phantom{\rule{0ex}{0ex}}=\frac{-3±7}{6}\phantom{\rule{0ex}{0ex}}=\frac{-10}{6}\mathrm{and}\frac{4}{6}\phantom{\rule{0ex}{0ex}}=\frac{-5}{3}\mathrm{and}\frac{2}{3}$ For r = $\frac{-5}{3}$, we have: a = 108 × $\frac{-5}{3}$ = $-$180 < 0 The value cannot be negative. For r =$\frac{2}{3}$, we have: a = 108 × $\frac{2}{3}$ = 72 Thus, we have: $\frac{a}{r}=\frac{72}{\frac{2}{3}}=\frac{72×3}{2}=108\phantom{\rule{0ex}{0ex}}a=72\phantom{\rule{0ex}{0ex}}ar=72×\frac{2}{3}=48$ Therefore, Sachin scored 108 runs, Sehwag scored 72 runs and Dhoni scored 48 runs.

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