to exercise his ingenuity in discovering short methods for working them, and is also thus better prepared for the practice of Algebra. Examples in the metric system and decimal coinage are interspersed amongst the others. These weights and measures are now legally a part of the English system, and it seems advisable from their juxtaposition that the pupil should see the advantages they possess, whereas when they are relegated to an Appendix they are generally omitted altogether. I have added a few questions from various Examination Papers, as specimens of the hardest questions set in the examinations referred to. GRAMMAR SCHOOL, GRANTHAM, January, 1867. R. D. BEASLEY. EXPLANATION OF SIGNS AND TERMS. THE sign of equality is; thus 3 tons = 60 cwt., means 3 tons are equal to 60 cwt. When two or more quantities are added together, the result is called the sum. The sign of addition is + (plus). Thus 3+ 4 + 5 = 12, means that 12 is the sum of 3, 4, 5 added together. When one quantity is subtracted from another, the former is called the subtrahend, and the result the difference, or remainder. The sign of subtraction is (minus). Thus in £7£5 = £2, £5 is the subtrahend, and £2 the difference. When a quantity is multiplied by a number, the first quantity is called the multiplicand, the number the multiplier, and the result the product. The sign of multiplication is × (read multiplied by). Thus 3 tons × 5 = 15 tons; where 3 tons is the multiplicand, 5 the multiplier, 15 tons the product. When several numbers are multiplied together, the result is also called the product. When a quantity is divided by a number, the first quantity is called the dividend, the number the divisor, and the result the quotient. The sign of division is ÷ (read divided by). Thus 8 yds. 42 yds.; where 8 yds. is the dividend, 4 the divisor, and 2 yds. the quotient. When numbers are used without reference to any particular thing they are called abstract numbers; when applied to particular things they are called concrete numbers. Thus 7, 25 are abstract numbers; £3, 14 cwt., 25 yds., 16 hrs. are concrete numbers. these cases £1, I cwt., I yd., I hr. are called concrete units. The sign. stands for the word therefore. In CHAPTER I. PRIME AND COMPOSITE NUMBERS.-GREATEST COMMON MEASURE. -LEAST COMMON MULTIPLE. 1. A number which cannot be divided by any other number less than itself, except unity, is called a prime number. Thus 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, &c., are prime numbers. All other numbers are compounded of two or more numbers multiplied together, and are called composite numbers. Thus 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, &c., are composite numbers. The numbers which multiplied together compose another number are called its factors. Thus 2 and 15, 3 and 10, 5 and 6, are factors of 30. When the factors are prime numbers, they are called the prime factors of the composite number. Thus 2, 3, 5 are the prime factors of 30. Sometimes the same prime factor is repeated: thus 2, 2, 3 are the prime factors of 12; 2, 2, 2, 3, 3, 5 of 360; that is, 2 x 2 x 3 = 12, and 2 × 2 × 2 × 3 × 3 × 5 = 360. These expressions are usually written 12 = 22.3, 360 = 23. 32. 5, where 22 signifies that the factor 2 occurs twice, 23 that it occurs three times, and so on. The dot between the 22 and the 3 is often used as a sign of multiplication. 2. It is sometimes convenient to resolve numbers into their prime factors, which may be done as follows The labour of this may often be shortened. B The student should familiarize himself with this process, so as to be able to write down at sight the factors of all composite numbers from 1 to 200. The reverse operation of finding the numbers compounded of certain prime factors is one of simple multiplication, but the labour may often be shortened by remembering that every 2 and 5 that occur make 10. Thus 2. 3. 533. 5. 100 = 1500. EXERCISE XXVIII. 3. Every number that divides another number without a remainder is called a measure of that number. Thus 2, 3, 4, 6 are measures of 12. Every number that divides each of two or more numbers without a remainder is called a common measure of those numbers. Thus 2, 3, 6 are all common measures of 18, 24, 30. The greatest number that will divide two or more numbers without a remainder is called their Greatest common measure. Thus 6 is the greatest common measure of 18, 24, 30. Greatest common measure is usually written G. C. M. 4. To find the G. C. M. of two numbers, e. g. 348, 738. 348) 738 (2 696 42) 348 (8 12) 42 (3 6) 12 (2 12 The student will easily see the rule from this example. The larger 738 is divided by 348, leaving the remainder 42. 348 is now divided by 42, leaving the remainder 12. 42 is next divided by 12, leaving the remainder 6. 12 is divided by 6, leaving no remainder. The last divisor 6 is the G. C. M. required. Reason of the process. Whatever number divides 348 and 738 must divide twice 348 or 696 and 738, and therefore 738 — 696 or 42. For suppose it does not go an exact number of times in 42, then (since it does go an exact number of times in 696) it would not go an exact number of times in 696 + 42 or 738; but by the hypothesis it does, since it is a measure of 738. Hence every number |