Elements of Algebra

CHAP. III. OF ALGEBRA. 197 Let the whole fortune be represented by $x$: Then the eldest son will have $\frac{1}{2}x - 3000$ The second - - - - $\frac{1}{3}x - 1000$ The third - - - - - $\frac{1}{4}x$ The youngest - - - - $\frac{1}{5}x + 600$. And the four will have received in all $\frac{1}{2}x + \frac{1}{3}x + \frac{1}{4}x + \frac{1}{5}x - 3400$, which must be equal to $x$. Whence results the equation $\frac{77}{60}x - 3400 = x$; then subtracting $x$, we have $\frac{17}{60}x - 3400 = 0$; adding 3400, we obtain $\frac{17}{60}x = 3400$; then dividing by 17, we have $\frac{1}{60}x = 200$; and multiplying by 60, gives $x = 12000$. The fortune therefore consisted of 12000. The first son received 3000 The second - - - - 3000 The third - - - - - 3000 And the fourth - - - 3000 591. Question 8. To find a number such, that if we add to it its half, the sum exceeds 60 by as much as the number itself is less than 65. Let the number be represented by $x$: Then $x + \frac{1}{2}x - 60 = 65 - x$, or $\frac{3}{2}x - 60 = 65 - x$. Now, by adding $x$, we have $\frac{5}{2}x - 60 = 65$; adding 60, we have $\frac{5}{2}x = 125$; dividing by 5, gives $\frac{1}{2}x = 25$; and multiplying by 2, we have $x = 50$. Consequently, the number sought is 50. 592. Question 9. To divide 32 into two such parts, that if the less be divided by 6, and the greater by 5, the two quotients taken together may make 6. Let the less of the two parts sought be $x$; then the greater will be $32 - x$. The first, divided by 6, gives $\frac{x}{6}$; and the second, divided by 5, gives $\frac{32-x}{5}$. Now $\frac{x}{6} + \frac{32-x}{5} = 6$: so that multiplying by 5, we have $\frac{5}{6}x + 32 - x = 30$, or $-\frac{1}{6}x + 32 = 30$; adding $\frac{1}{6}x$, we have $32 = 30 + \frac{1}{6}x$; subtracting 30, there remains $2 = \frac{1}{6}x$; and lastly, multiplying by 6, we have $x = 12$. So that the less part is 12, and the greater part is 20. 593. Question 10. To find such a number, that if multiplied by 5, the product shall be as much less than 40 as the number itself is less than 12. Let the number be $x$; which is less than 12 by $12 - x$; then taking the number $x$ five times, we have $5x$, which is $5x + 12 - x = 40$ $4x = 28$ $x = 7$