5.4 Real World Problems

5.4 Real World Problems

Example 14

A tradesman buys a number of articles for Rs. 300. Four of them are broken in transit but by selling the remaining at a profit of Rs. 1.25 each, he gains Rs. 350 altogether. How many articles did he buy?

Solution

Suppose he buys $x$ articles for Rs. 300 $\therefore$ Each cost = Rs. $\frac{300}{x}$ He sold $(x - 4)$ articles for Rs. 350 Each sold for Rs. $\frac{350}{x - 4}$ Now, $\frac{350}{x - 4} - \frac{300}{x} = 1.25$ $\frac{350}{x - 4} - \frac{300}{x} = \frac{125}{100} = \frac{5}{4}$ Multiplying both sides by $4x(x - 4)$, we get $350(4x) - 300(x - 4)(4) = 5x(x - 4)$ $1400x - 1200x + 4800 = 5x^2 - 20x$ $5x^2 - 20x + 1200x - 1400x - 4800 = 0$ $5x^2 - 220x - 4800 = 0$ (Dividing by 5) $x^2 - 44x - 960 = 0$ $x^2 - 60x + 16x - 960 = 0$ $x(x - 60) + 16(x - 60) = 0$ $(x - 60)(x + 16) = 0$ $x - 60 = 0, \quad x + 16 = 0$ $\therefore x = 60, \quad -16$ Negative value is rejected. Number of articles purchased = 60

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Challenge!
Amna bought 50 kg of fruits consisting of mangoes and grapes. She paid twice as much per kg for the mango as she did for the grapes. If Amna bought Rs. 18000 worth of mangoes and Rs. 6000 worth grapes, then how many kg's of each fruit did she buy?

Example 15

A train is scheduled to cover a distance of 120 km at a certain average speed, owing to a service checkup this average speed is reduced by 5 km/h, and in consequence the journey takes 20 minutes more than the scheduled time. What was the scheduled speed?

Solution

Let $x$ km/h be scheduled speed; then scheduled time for 120 km = $\frac{120}{x}$ hrs 20 minute = $\frac{20}{60}$ hrs = $\frac{1}{3}$ hrs But $(x - 5)$ km/h is actual speed: therefore, actual time for 120 km = $\frac{120}{x - 5}$ $\therefore \frac{120}{x - 5} - \frac{120}{x} = \frac{1}{3}$ Multiply each term by $3x(x - 5)$ $360x - 360(x - 5) = x(x - 5)$ $x^2 - 5x - 1800 = 0$ $(x - 45)(x + 40) = 0$ $x - 45 = 0, \quad x + 40 = 0$ $\therefore x = 45, \quad x = -40$ Negative value is rejected. The scheduled speed is 45 km/h.