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; 7x x × 31, or 2 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 86+x+y=136; and 7.x 387+ +3y=544; 2 from which we shall find x=14, and y=36 bushels. and 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 x+y=505. 4x Now, x-51, or represents the number of cubic inches of copper; 21' The second Equation will then be 4x 4y + = =100. 21 From these Equations x will be found equal to 420 ounces, and y equal to 85 ounces. 4y y÷41, or represents the number of cubic inches of tin. 17' 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 X 8100 -600, represents the number re maining. Now, as only of his men plus 3600 were left, we shall have 27x x +3600; 40 -600 1 )。 2 from which we shall find x=24000 men. 14 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 a 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 4y is the number discharged by the 2d in 4 hours. 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. 27x 40 or 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 is the part C did in 1 day; and, since he was employed 14 days, X 14 is the part of the work performed by C; X -600: = 2-14 28x Hence, we have the Equation 1 x-14 1 28x x 21' This Equation will give x=42 days. , is the part performed by A and B in 28 days; is the part A and B did in 1 day. the part the three could do in 1 day. 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 represents the part which one cask would occupy. y The first Equation will therefore be 13 33 =1, the entire cave; + and, as there were left in the cave 13-5, or 8 bales, and 33-9, or 24 casks, the second Equation will be 8 24 + 2 y 3 Clearing these Equations of fractions, we have 13y+33x=xy; 24y+72x=2xy. Multiplying the third Equation by 2, x 26y+66x=2xy; subtracting the fourth Equation from this, we have 2y-6x=0; which will give y=3x. Substituting 3.x for y, in the third Equation, and dividing both sides of the resulting Equation by x, we shall have 30+33=3x; which gives x=24. The value of y is now easily found to be 72. 46. A gentleman left a sum of money to be divided among four serthe sum of the shares of the vants, so that the share of the first was of the sum of the other three; of the sum of the other three; and it was the first exceeded that of the last by $14. and the share of each? Let x, y, z, and w, represent the several shares; x=w+14. 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::35. By converting this proportion into an Equation, we find 5x=3y, (149). From this and the first Equation we shall find x= =$3561, 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 This proportion converted into an Equation, gives From the two Equations we shall find x= Let x represent A's share; then 3. Divide the sum of $3000 between A, B, and C, in the propor tions of 1, 2, and 3. 1: 2 ::x : 2x, B's share; and 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 x 500-$1500, C's share. x+y+2=7600; xy; and y: 2. The two proportions converted into Equations, will give 4. Divide the sum of $7600 between three persons, in the propor tions of,, and }. Let x, y, and z, represent the several shares; then . y as and y= a+b' 2 Z bs a+b ; and 4 The three Equations will give x=4000, y=2000, and 2= = 1600. 5. A bankrupt is indebted to A $400, and to B $700. He is able to 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 |