SECTION VIII. Examples of the Solution of Problems producing pure 1. WHAT two numbers are those, whose sum is to the greater as 10 to 7; and whose sum multiplied by the less produces 270? 2. There are two numbers in the proportion of 4 to 5, the difference of whose squares is 81. What are those numbers? 3. What two numbers are those, whose difference is to the greater as 2 to 9, and the difference of whose squares is 128? P 4. A Mercer bought a piece of silk for £16. 48.; and the number of shillings which he paid for a yard was to the number of yards as 4: 9. How many yards did he buy, and what was the price of a yard? Let 4 the number of shillings he paid for a yard; .. 9x the number of yards, and 36x2 = (the price of the whole =) 324; .. x2 = 9, and .. x = ±3; consequently there were 27 yards, at 128. per yard. 5. It is required to divide the number 18 into two such parts, that the squares of those parts may be in the proportion of 6. Find three numbers in the proportion of the sum of whose squares is 724. Reducing the fractions to a common denominator, the required numbers will evidently be in the proportion of 6, 8, and 9; let. 6x, 8x, and 9x, represent the numbers; then (36x2 + 64x2 + 81x2 =) 181x2 = 724 ; and consequently, the numbers are ± 12, ± 16, and ± 18. 7. It is required to divide the number 14 into two such parts that the quotient of the greater part divided by the less, may be to the quotient of the less divided by the greater as 8. What two numbers are those whose difference is to the less as 4 to 3; and their product multiplied by the less is equal to 504 ? 9. What two numbers are as 5 to 4, the sum of whose cubes is 5103? Let 5x and 4x = the numbers; •. (125æ3 + 64x3 =) 189x3 = 5103, .. x = 3, and the numbers are 15 and 12. 10. A number of boys set out to rob an orchard, each carrying as many bags as there were boys in all, and each bag capable of containing 4 times as many apples as there were boys. They filled their bags, and found the number of apples was 2916. How many boys were there? |