Time and Distance - Q1 - To and Fro Motion in a Straight Line

Q. Ram and Gopi run towards each other starting from K and L respectively with respective speeds of 2 kmph and 3 kmph. After meeting each other, to reach L, if Ram takes 7 hours less than the square of the magnitude of the time (in hours) taken by Gopi to reach K, find distance b/w K and L.

A.

Let M be the Meeting Point of Ram and Gopi.
Let KM = x and ML = y.

Let time taken to meet be 't' hours.
Therefore,
Distance traveled by Ram in time t = KM = x = 2t km
Distance traveled by Gopi in time t = ML = y = 3t km

Hence,
$$\frac{x}{y}=\frac{2}{3}$$
Hence, Let x = 2k and y = 3k.
After meeting at time t,
let t_R be the time taken by Ram to reach L, and let t_G be the time taken by Gopi to reach K

$$t_R=\frac{\text{Distance traveled by Ram}}{\text{Speed of Ram}}=\frac{y}{2}$$

$$t_G=\frac{\text{Distance traveled by Gopi}}{\text{Speed of Gopi}}=\frac{x}{3}$$

From "to reach L, if Ram takes 7 hours less than the square of the magnitude of the time (in hours) taken by Gopi to reach K",

$$t_R={\left(t_G\right)}^2-7$$ $$\begin{array}{l}\to \frac{y}{2}={\left(\frac{x}{3}\right)}^2-7\\ \to 2x^2-9y-126=0\end{array}$$

Substituting for x and y as 2k and 3k,
$$\begin{array}{l}\to {\left(2k\right)}^2-9\left(3k\right)-126=0\\ \to 4k^2-27k-126=0\end{array}$$

Solving for k,
$$\begin{array}{l}k=\frac{27\pm \sqrt{{27}^2-\left[\left(4\right)\times \left(8\right)\times \left(-126\right)\right]}}{2\times 8}\\ \\ \to k=\frac{27\pm \sqrt{729+4032}}{16}=\frac{27\pm 69}{16}\\ \\ \to k=6\text{ OR }-\left(\raisebox{1ex}{$21$}\!\left/ \!\raisebox{-1ex}{$8$}\right.\right)\end{array}$$

Therefore,
$$\begin{array}{l}x=2k=12\\ y=3k=18\end{array}$$

Distance between K & L,
$$\text{KL}$$

Therefore the distance between K & L is 30 kms.

Profit and Loss - Q1 - Linc and Cello Pens

Q: A retailer bought 3850 Linc pens and 1848 cello pens at the same price. He sells pens in such a way that he can buy 650 Linc pens with the sale of 481 Linc pens and he can buy only 408 Cello pens with the sale of 629 Cello pens. What is the overall profit percentage of the retailer?

A.

Let Selling Price of 1 Linc Pen = $S_L$
Let Cost Price of 1 Linc Pen = $C_L$
Let Selling Price of 1 Cello Pen = $S_C$
Let Cost Price of 1 Cello Pen = $C_C$

From "he can buy 650 Linc pens with the sale of 481 Linc pens",
$$650C_L=481S_L$$
$$\to \frac{S_L}{C_L}=\frac{650}{481}=\frac{13\times 50}{13\times 37}=\frac{50}{37}$$
Hence, Profit Percentage on 1 Linc Pen = $$\%P_L=\frac{S_L-C_L}{C_L}=\frac{50-37}{37}=\frac{13}{37}$$

From "he can buy only 408 Cello pens with the sale of 629 Cello pens",
$$408C_L=629S_L$$
$$\to \frac{S_C}{C_C}=\frac{408}{629}=\frac{17\times 24}{17\times 37}=\frac{24}{37}$$
Hence, Profit Percentage on 1 Cello Pen =
$$\%P_C=\frac{S_C-C_C}{C_C}=\frac{24-37}{37}=-\frac{13}{37}$$

Cost of 3850 Linc Pens = $3850C_L$
Cost of 1848 Cello Pens = $1848C_C$

From "A retailer bought 3850 Linc pens and 1848 cello pens at the same price",
$$3850C_L=1848C_C$$

Overall Profit Percentage is the cost-weighted average of the individual Profit Percentages. Hence,
$$\%P=\frac{\left[3850C_L\times \frac{13}{37}\right]+\left[1848C_C\times \left(-\frac{13}{37}\right)\right]}{3850C_L+1848C_C}$$
Since, $3850C_L=1848C_C$
$$\%P=\frac{\left[1848C_C\times \frac{13}{37}\right]+\left[1848C_C\times \left(-\frac{13}{37}\right)\right]}{1848C_C+1848C_C}=0$$

Hence, overall the retailer makes No Profit, No Loss.