Elements of Algebra

302 ELEMENTS PART II. 6. Question 3. To divide 100 into two such parts, that dividing the first by 5, there may remain 2; and dividing the second by 7, the remainder may be 4. Since the first part, divided by 5, leaves the remainder 2, let us suppose it to be $5x + 2$; and, for a similar reason, we may represent the second part by $7y + 4$: we thus have $5x + 7y + 6 = 100$, or $5x = 94 - 7y = 90 + 4 - 5y - 2y$; whence we obtain $x = 18 - y + \frac{4-2y}{5}$. Hence it follows, that $4 - 2y$, or $2y - 4$, or the half $y - 2$, must be divisible by 5. For this reason, let us make $y - 2 = 5z$, or $y = 5z + 2$, and, as $5x + 7y = 94$, we shall have $x = 16 - 7z$; whence we conclude, that $7z$ must be less than 16, and $z$ less than $\frac{16}{7}$, that is to say, $z$ cannot exceed 2. The question proposed, therefore, admits of three answers: 1. $z = 0$ gives $x = 16$, and $y = 2$; whence the two parts are $82 + 18$. 2. $z = 1$ gives $x = 9$, and $y = 7$; and the two parts are $47 + 53$. 3. $z = 2$ gives $x = 2$, and $y = 12$; and the two parts are $12 + 88$. 7. Question 4. Two women have together 100 eggs: one says to the other; 'When I count my eggs by eights, there is an overplus of 7.' The second replies: 'If I count mine by tens, I find the same overplus of 7.' How many eggs had each? As the number of eggs belonging to the first woman, divided by 8, leaves the remainder 7; and the number of eggs belonging to the second, divided by 10, gives the same remainder 7; we may express the first number by $8x + 7$, and the second by $10y + 7$; so that $8x + 10y + 14 = 100$, or $8x = 86 - 10y$, or $4x = 43 - 5y = 40 + 3 - 4y - y$. Consequently, if we make $y - 3 = 4z$, so that $y = 4z + 3$, we shall have $x = 10 - 4z - 3 - z = 7 - 5z$; whence it follows, that $5z$ must be less than 7, or $z$ less than 2; that is to say, we have the two following answers: 1. $z = 0$ gives $x = 7$, and $y = 3$; so that the first woman had 63 eggs, and the second 37. 2. $z = 1$ gives $x = 2$, and $y = 7$; therefore the first woman had 23 eggs, and the second had 77. 8. Question 5. A company of men and women spent 1000 sous at a tavern. The men paid each 19 sous, and each woman 13. How many men and women were there?