Logical Reasoning Discussion

**Common Information**

**A**bdul, **B**ikram and **C**hetan are three professional traders who trade in shares of a company XYZ Ltd.

**A**bdul follows the strategy of buying at the opening of the day at 10 am and selling the whole lot at the close of the day at 3 pm.

**B**ikram follows the strategy of buying at hourly intervals: 10 am , 11 am, 12 noon, 1 pm and 2 pm, and selling the whole lot at the close of the day. Further, he buys an equal number of shares in each purchase.

**C**hetan follows a similar pattern as **B**ikram but his strategy is somewhat different. **C**hetan’s total investment amount is divided equally among his purchases.

The profit or loss made by each investor is the difference between the sale value at the close of the day less the investment in purchase.

The “return” for each investor is defined as the ratio of the profit or loss to the investment amount expressed as a percentage.

Q. |
One day, two other traders,
At the close of the day the following was observed: **A**bdul lost money in the transactions.- Both
**D**ane and**E**mily made profits. - There was an increase in share price during the closing hour compared to the price at 2 pm.
- Share price at 12 noon was lower than the opening price.
Which of the following is necessarily false? |

✔ A. |
Share price was at its lowest at 2 pm |

✖ B. |
Share price was at its lowest at 11 am |

✖ C. |
Share price at 1 pm was higher than the share price at 2 pm |

✖ D. |
Share price at 1 pm was higher than the share price at 12 noon |

✖ E. |
None of the above |

**Solution:**

Option(**A**) is correct

Let $x_1, x_2, ... , x_6$ be the share prices at 10 am, 11 am, 12 noon, 1 pm, 2 pm and 3 pm respectively.

Now, since Abdul lost money in the transaction,

$x_1 > x_6$

Also, it is given that,

$x_1 > x_3$, and $x_6 > x_5$

Combining the above, we have,

$x_1 > x_6 > x_5$

and $x_1 > x_3$

Also, let the money Emily invests at 10 am be Rs. $P$. Then,

Her investment $= \text{Rs. }P$

and the number of shares she buys, $=\dfrac{P}{x_1}$

So, after selling these shares at 12 noon, she will get,

$=\text{Rs. } \dfrac{P}{x_1} \times x_3$

Now, she invests this money at 1 pm, and the number of shares she buys,

$=\text{Rs. } \dfrac{Px_3}{x_1x_4}$

So after selling these shares at 3pm, she gets

$=\text{Rs. } \dfrac{P x_3}{x_1 x_4} \times x_6$

So, her returns,

$=\dfrac{\dfrac{Px_3 x_6}{x_1 x_4}-P}{P}$

$=\left(\dfrac{x_3 x_6}{x_1 x_4}\right)-1$

Since she made profit, her returns > 0;

$\Rightarrow \left(\dfrac{x_3 x_6}{x_1 x_4}\right)-1>0$

$\Rightarrow \left(\dfrac{x_3 x_6}{x_1 x_4}\right)>1$

$\therefore \dfrac{x_3}{x_4}$ has to be $>1$; i.e. $x_3 > x_4$

$\therefore$ The share price at 12 noon is greater than that at 1 pm.

**Hence, option D is definitely false.**

Also, since in the first half, Emily invests at 10 am and sells at 12 noon, and we know that the share price at 10 am was greater than at 12 noon; hence she must have suffered a loss during this transaction. However, she makes a net profit in the end. So, she must have made profit during the second part of the transaction; i.e. the share price at 1 pm must have been less than that at 3 pm.

i.e. $x_4 < x_6$,

Also, let Dane buy n shares at 10 am, 11 am and 12 noon.

Hence, her investment $= n(x_1 + x_2 + x_3)$

And she sells these at 1 pm, 2 pm and 3 pm for

$=n(x_4 + x_5 + x_6)$

$\therefore$ Her returns,

$=\dfrac{n(x_4 + x_5 + x_6) - n(x_1 + x_2 + x_3)}{n(x_1 + x_2 + x_3)}$

$=\left(\dfrac{x_4 + x_5 + x_6}{x_1 + x_2 + x_3}\right)-1$

Since she made profit, her returns are greater than 0;

$\Rightarrow \left(\dfrac{x_4 + x_5 + x_6}{x_1 + x_2 + x_3}\right)-1 >0$

$\Rightarrow \left(\dfrac{x_4 + x_5 + x_6}{x_1 + x_2 + x_3}\right)>1$

Hence, $(x_4 + x_5 + x_6) > (x_1 + x_2 + x_3)$

Since, $x_1 > x_6$ and $x_3 > x_4$, hence $x_5 > x_2$

So far, we have,

$x_1 > x_6 > x_5 > x_2$, $x_4 < x_6$ and $x_1 > x_3 > x_4$

Now from Dane’s investment, we know that,

$(x_4+ x_5+ x_6) – (x_1+ x_2+x_3) > 0$ -------- (i)

Keeping in mind the relationships between the share prices, we have

$x_6 = x_1 – b$

$x_4 = x_1 – b – c$

$x_3 = x_1 – b – c + a$

$x_5 = x_1 – d$, where $a$, $b$, $c$ and $d$ are all positive.

Substituting the above in equation i, we have,

$(x_1 – b – c + x_1 – d + x_1 – b) – (x_1 + x_2 + x_1 – b – c + a) > 0$

$\therefore x_1 – x_2 > b + d + a$ (which is $> 0$, since all the variables are positive)

$\Rightarrow x_1 > x_2$

$\Rightarrow x_2 < x_1 – b – a – d$

$\therefore x_2$ is definitely less than $x_6$ and $x_5$.

Although we don’t know when the share price is at its lowest, we do know that $x_5 > x_2$.

$\therefore x_5$, i.e. the share price at 2 pm is not the lowest.

**Hence, option A is also definitely false.**

Thus, there are two options which are correct for this question. This is an ambiguity therefore we are referring **option A** as the correct choice but **option D** is also a correct choice.