KEY TO DODD'S ALGEBRA. PROBLEMS In Simple Equations of one Unknown Quantity. 1. What number is that to the double of which if 13 be added, the sum will be 75? Let x represent the number; then 2x represents double the number; and by the conditions of the problem, the Equation will be 2x+13=75. This Equation will give x=31. 2. Find a number such that if it be multiplied by 5, and 24 be subtracted from the product, the remainder will be 36. Let x represent the number; then 5x represents 5 times the number; and by the conditions of the problem, the Equation will be 5 –24=36 ; which will give x=12. 3. What number is that to of which if 25 be added, the sum obtained will be equal to the number itself minus 39? 4 3x represents three times the number. We shall then have the Equation represents one fourth of the number; and =77; from which the value of x will be found equal to 28. 5. Find what number added to the sum of one half, one third, and one fourth of itself will be equal to 4 added to twice the number. Let x represent the number; then X 4 3x of it be subtracted from three represents one half of the number; represents one third of the number; and X represents one fourth of the number; 4 and, by the conditions of the problem, the Equation will be +x=2x+4; X Xx x 2 + + from which we shall find x=48. 6. Divide the number 165 into two such parts, that the less equal to of the greater. Let x represent the less part; then may be and the Equation will be x= This Equation will give x=15, the less part; hence 165-15=150 is the greater. 7. Divide the number 100 into two such parts that six times the less may be equal to twice the greater. 165-x Let x represent the less part; then 100-x represents the greater; 6x represents 6 times the less; and 200-2x represents twice the greater. We shall then have the Equation 6x=200-2x ; which will give 2=25, the less part; hence 100-25=75 is the greater. 8. It is required to divide 75 into two such parts that 3 times the greater may exceed 7 times the less by 15. Let x represent the less part; then 75-x represents the greater; 225-3x represents 3 times the greater; and The Equation will then be 225-3x=7x+15. This Equation will give x=21, the less part; hence 75-21-54 is the greater. 3 after adding $100. 9. What sum of money is that to which if $100 be added, of the amount will be $400 ? Let x represent the sum; then 2 2x+200 (x+100), or 3 The Equation will then be which will give x=$500. represents 2x+200 3 =400; of the amount obtained 10. A prize of $100 is to be divided between two persons-the share of the first being of that of the other. What are the shares? Let x represent the share of the second; then 100-x represents the share of the first; and the Equation will be - 100-x= from which will be found $564, the share of the second; hence 100-561 $433, the share of the first. 11. A post has of its length in the mud, of it in the water, and 15 feet above the water. What is the length of the post? 3 Let x represent the length, in feet; then 7x 9 represents the part in the mud; represents the part in the water; and x 4 15 feet is the part above the water. The Equation will then be X 3 =36 feet. + +15=x; from which x will be found 12. Find a number such that if it be divided by 12, the divisor, dividend, and quotient together shall make 64. 13. In a mixture of wine and cider, of the whole plus 25 gallons was wine, and part minus 5 gallons was cider. What was the whole number of gallons in the mixture? |