Now, 4s. 6d. is equal to 4 shillings; 3s. 6d. is equal to 3 shillings; 86×4, or 387, shillings is the value of the wheat; 2×3, or 7x is the value of the rye, in shillings; 3y is the value of the barley in shillings; and 136 × 4, or 544 shillings, is the value of the mixture. The two Equations will then evidently be from which we shall find x=14, and y=36 bushels. 41. A composition of copper and tin, containing 100 cubic inches, weighs 505 ounces. How many ounces of each metal does it contain, supposing the weight of a cubic inch of copper to be 51 ounces, and of a cubic inch of tin 4 ounces? Let x represent the number of ounces of copper, and y the number of ounces of tin; then we shall have 4x 21' x+y=505. Now, x-51, or represents the number of cubic inches of copper; From these Equations x will be found equal to 420 ounces, and y equal to 85 ounces. 42. A general, having lost a battle, found that he had only one half of his army plus 3600 men left fit for action; of his men plus 600 being wounded, and the rest, who were of his whole army, either slain, taken prisoners, or missing. Of how many men did the army consist? Let x represent the number; then +600 represents the number wounded; represents the number slain, taken prisoners, or missing; and Now, as only of his men plus 3600 were left, we shall have from which we shall find x=24000 men. 43. Two pipes, one of them running 5 hours, and the other 4, filled a cistern containing 330 gallons; and the same two pipes, the first running 2 hours, and the second 3, filled another cistern containing 195 gallons. How many gallons did each pipe discharge per hour? Let x represent the number of gallons discharged by the first, and y the number discharged by the second, per hour; then 5x is the number discharged by the 1st in 5 hours; and The first Equation will then be 5x+4y=330; and in a similar manner, we shall have 2x+3y=195; from which Equations, we shall find x=30, and y=45 gallons. 44. After A and B had been employed on a piece of work for 14 days, they called in C, by whose aid it was completed in 28 days. Had C worked with them from the beginning, the work would have been completed in 21 days. In how many days would C alone have accomplished the work? Let x represent the number of days; then 1 X 14 х 1 is the part C did in 1 day; and, since he was employed 14 days, X- -14 28x , is the part performed by A and B in 28 days; is the part A and B did in 1 day. Hence, we have the Equation. 45. Some smugglers discovered a cave which would exactly hold their cargo, viz, 13 bales of cotton, and 33 casks of wine. A revenue cutter coming in sight while they were unloading, they sailed away with 9 casks, and 5 bales, leaving the cave two-thirds full. How many bales or casks would it contain? Let x represent the number of bales, and y the number of casks, 1 then represents the part of the cave which one bale would occupy, and 1 У represents the part which one cask would occupy. The first Equation will therefore be and, as there were left in the cave 13-5, or 8 bales, and 33-9, or 24 casks, the second Equation will be which will give y=3x. 2y-6x=0; Substituting 3r for y, in the third Equation, and dividing both sides of the resulting Equation by a, we shall have which gives x=24. 30+33=3x; The value of y is now easily found to be 72. 46. A gentleman left a sum of money to be divided among four servants, so that the share of the first was the sum of the shares of the other three; the share of the second was of the sum of the other three; and the share of the third of the sum of the other three; and it was also found that the share of the first exceeded that of the last by $14. What was the whole sum? and the share of each? Let x, y, z, and w, represent the several shares; By the conditions of the problem, we shall then have These Equations will give x=$40, y=$30, z=24, and w=$26. The whole sum will then be 40+30+24+26=$120. PROBLEMS In Proportion, Percentage, Interest, &c. 1. Divide $950 between two persons, so that their shares shall be to each other as 3 to 5. Let x represent the share of the 1st, and y the share of the 2d; then we shall have x+y=950; and xy::3:5. By converting this proportion into an Equation, we find 5x=3y, (149). From this and the first Equation we shall find x=$356, and y= $5933. 2. Find the Formulas for dividing any given sum s between two persons so that the shares shall be to each other as any two numbers a and b. Let x and y represent the respective shares; then x+y=s; and xy::a:b. This proportion converted into an Equation, gives bx=ay, (149). as From the two Equations we shall find x= bs and y= a+b' a+b 3. Divide the sum of $3000 between A, B, and C, in the propor tions of 1, 2, and 3. Let x represent A's share; then 12 x 2x, B's share; and 2:3 :: 2x : 3x, C's share (150). Then we shall have the Equation x+2x+3x=3000; from which we shall find x=$500, A's share; hence 2×500=$1000, B's share; and 3 × 500 $1500, C's share. 4. Divide the sum of $7600 between three persons, in the propor tions of,, and }. to Let x, y, and z, represent the several shares; then The three Equations will give x=4000, y=2000, and z=1600. 5. A bankrupt is indebted to A $400, and to B $700. He is able pay both $900. What sum should each of the two creditors receive? Let x and y represent the respective shares; then we shall have x+y=900; and |