Elements of Algebra

196 ELEMENTS SECT. IV. as a daughter. Required how much each of them is to receive. Suppose the share of each daughter to be $x$ Then each son's is consequently - - - $2x$ And the widow's - - - - - - - - $4x$ The whole inheritance, therefore, is $3x + 4x + 4x$; or $11x = 11000$, and, consequently, $x = 1000$. Each daughter, therefore, is to receive 1000 crowns; So that the three receive in all 3000 Each son receives 2000; So that the two sons receive - - 4000 And the mother receives - - - 4000 Sum 11000 crowns. 589. *Question* 6. A father intends by his will, that his three sons should share his property in the following manner: the eldest is to receive 1000 crowns less than half the whole fortune; the second is to receive 800 crowns less than the third of the whole; and the third is to have 600 crowns less than the fourth of the whole. Required the sum of the whole fortune, and the portion of each son. Let the fortune be expressed by $x$: The share of the first son is $\frac{1}{2}x - 1000$ That of the second - - - - $\frac{1}{3}x - 800$ That of the third - - - - $\frac{1}{4}x - 600$ So that the three sons receive in all $\frac{1}{2}x + \frac{1}{3}x + \frac{1}{4}x - 2400$, and this sum must be equal to $x$. We have, therefore, the equation $\frac{13}{12}x - 2400 = x$; and subtracting $x$, there remains $\frac{1}{12}x - 2400 = 0$; then adding 2400, we have $\frac{1}{12}x = 2400$; and, lastly, multiplying by 12, we obtain $x = 28800$. The fortune, therefore, consists of 28800 crowns; of which The eldest son receives 13400 crowns The second - - - - - 8800 And the youngest - - - - 6600 28800 crowns. 590. *Question* 7. A father leaves four sons, who share his property in the following manner: the first takes the half of the fortune, minus 3000*l*.; the second takes the third, minus 1000*l*.; the third takes exactly the fourth of the property; and the fourth takes 600*l*. and the fifth part of the property. What was the whole fortune, and how much did each son receive?