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.

Answer 1

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
$\Rightarrow$ 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
$\Rightarrow$$\frac{2a}{r}$ = 216
$\Rightarrow$ a = 108r
On putting a = 108r in (1), we get:

$$\begin{gathered} \frac{108r}{r}+108r+108r^2=228 \\ \Rightarrow 108\left[1+r+r^2\right]=228 \\ \Rightarrow 1+r+r^2=\frac{19}{9} \\ \Rightarrow r+r^2=\frac{10}{9} \end{gathered}$$

On using the quadratic formula, we get:

$$\begin{gathered} r=\frac{-1\pm \sqrt{1-4\times 1\times \left({\displaystyle \frac{-10}{9}}\right)}}{2\left(1\right)} \\ =\frac{-1\pm \sqrt{1+{\displaystyle \frac{40}{9}}}}{2} \\ =\frac{-1\pm \sqrt{{\displaystyle \frac{49}{9}}}}{2} \\ =\frac{-1\pm {\displaystyle \frac{7}{3}}}{2} \\ =\frac{-3\pm 7}{6} \\ =\frac{-10}{6}\mathrm{and}\frac{4}{6} \\ =\frac{-5}{3}\mathrm{and}\frac{2}{3} \end{gathered}$$

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:

$$\begin{gathered} \frac{a}{r}=\frac{72}{{\displaystyle \frac{2}{3}}}=\frac{72\times 3}{2}=108 \\ a=72 \\ ar=72\times \frac{2}{3}=48 \end{gathered}$$

Therefore, Sachin scored 108 runs, Sehwag scored 72 runs and Dhoni scored 48 runs.