to produce a mixture worth 16s. per gallon, an integral number of gallons of each sort being always taken. (18) How many fractions are there with denominators 3 and 4, whose sum is 31? (19) How many fractions are there with denominators 12 and 18, whose sum is ? 25 Transform the following numbers from the Denary to the Senary (8) Transform 13256 from scale 7 to scale twelve. (9) Transform 1341120 from the senary to the duodenary scale. (10) Transform 654321 from the septenary to the duodenary scale. (11) Subtract 20404020 from 103050301 in the octenary scale. (12) Extract the square root of the result in the last Ex. (13) Divide 51117344 by 675 in the octenary scale. (14) Find the radix of the scale in which 40501 is equivalent to 5365 in the denary scale. (15) In what scale is the denary number 2704 written 20304 ? (16) Extract the square root of 1010001 in the binary scale, and reduce the result to the denary scale. (17) Apply the duodenary notation to find the square of 4ft. 2in. O'. 2". 10". (18) Apply the duodenary notation to find the cube of 16ft. 10in. EXERCISES. Zo. Transform the following quantities from scale 10 to scale 5: (1) 221-342. (2) 357-234. (3) 101.265. (4) Transform 179-25 from the denary to the senary scale. (6) Transform from the denary to the duodenary scale. 25 (7) Transform 21 from the denary to the octenary scale. (9) Transform 3197e from the duodenary to the octenary scale. 11 (11) Transform from the denary to the duodenary scale. (12) A certain number is 125 in the scale whose radix is x, 78 in the scale whose radix is y, and 49 in the scale whose radix is a+y; find the number in the scale whose radix is 10. that EXERCISES. Zp. (1) If p and q are any positive whole numbers, and p+q is even, shew p-q is also even. (2) Shew that the difference between any number and its square is always an even number. (3) Shew that the difference between any number and its cube is always divisible by 6 without remainder. (4) Shew that the product of two odd numbers will always be odd. (5) Shew that the sum of any two consecutive odd numbers will always be divisible by 4. (6) Shew that the product of any two consecutive even numbers is divisible by 8. (7) Shew that every odd square number, greater than 1, leaves a remainder 1, when divided by 8. (8) Shew that every perfect cube is of one of the forms 7m, or 7m±1. (9) Shew that upon any number, greater than 12, which is a perfect square, being divided by 12, the remainder, if there be any, is a square. (10) Shew that the difference of the squares of any two odd numbers is exactly divisible by 8. (11) Shew that the square of any number prime to 4 is of the form 4p+1. (12) Shew that the difference of the squares of any two prime numbers, each of which is greater than 3, is divisible by 24. EXERCISES. Zq. Find the value of each of the following fractions: EXERCISES. Zr. (1) Given log a find the log. of "Ja". (2) Given log 2=0·30103, and log 3=0·47712, find log (3) Given log 2, and log 3, as in last example, find log (5)" (4) Given log n, find the log. of √n. In2 (5) Given log 2, and log 3, as in Ex. (2), find log 12. (6) Given log 2, find the log. of 5. 4 (7) Given log 2, find the log. of 6.4. (8) If a*b*=c, find x. (9) If ab, find x. 10 20 (10) If Ja=b, find x. (11) If log x= log a- log b, find x. (12) If log x = n log a+m log b-p log c, find x. (13) Given log 2=0·30103, find the log. of 1620. 5 2 1 (14) Given log + log y=, and log x-log y = (15) Given log x-log y=log n, and ax+by=c, find x and y. (16) If logo=3 logoa-2, find x. y. (17) If log x+log y=log a, and 2 log x-2 log y=log b, find x and y. EXERCISES. Zs. [N.B. log 1·05-0-02119, and log 104-001703.] (1) What would £200 amount to in 7 years at 4 per cent., compound interest? (2) How much money must be invested at compound interest to amount to £500 in 12 years at 5 per cent.? (3) In how many years will a sum of money double itself placed out at 4 per cent. compound interest? (4) A freehold estate which produces a clear rental of £100 a year is sold for £2500; at what rate is interest reckoned? (5) Find the amount of £100 in 10 years at £100 per cent., compound interest. (6) If a person returns 100 guineas for the loan of £100 for 3 months, what is the rate of interest allowed? (7) A person returns £287. 10s. for the loan of £250 at the rate of 5 per cent. per annum, simple interest. For what time was the money lent? (8) Supposing interest paid half-yearly, what will £500 amount to in 8 years at 5 per cent., compound interest? (given log 1.025-001072.) (9) How many years purchase should be given for a freehold estate when money is worth 3 per cent.? |