CHAPTER V DENOMINATE NUMBERS A denominate number is one in which a quantity is given in one denomination; as 2 feet, 3 yards, 4 pounds. A compound denominate number is one in which a quantity is given in two or more denominations of the same table of measures; as 3 yds. 2 ft. 6 in., or 3° 14' 30". REDUCTION OF DENOMINATE NUMBERS. Reduction of denominate numbers consists in changing them from one denomination to another without altering their value. CASE I.-TO REDUCE FROM A HIGHER TO A LOWER DENOMINATION. RULE.-Multiply the quantity in the highest denomination by the number of times a unit of this denomination contains one of the next lower, and to the product add the quantity, if any, in the lower denomination. Proceed in like manner with this result and continue the operation until the desired denomination is reached. CASE II.-TO REDUCE FROM A LOWER TO A HIGHER DENOM INATION. RULE.-Divide the given quantity by the number of times a unit of its denomination is contained in one of the next higher. Proceed in like manner with the quotient, and continue the operation until the desired denomination is reached. The last quotient, with the several remainders, if any, annexed in order will be the answer. Denominate numbers in the metric system are reduced by simply moving the decimal point to the right or to the left. PROBLEMS. 1. Reduce 6 rds. 4 yds. 2 ft. 9 in. to inches. Ans. 1,365 in. Ans. 5 ft. 2 in. 2. Reduce 15 hands to feet and inches. Reduce 8,589" to degrees and decimals of a degree. Ans. 2° .386. 5. Reduce 40 cm. to inches (approximately). Ans. 16 in. 6. Reduce 50 kilos. to pounds (legal value). Ans. 110.23 lbs. ADDITION OF COMPOUND DENOMINATE NUMBERS. RULE.-Write the quantities, placing numbers of the same denomination in the same column. Beginning at the right hand, add the numbers of the lowest denomination and divide the sum by the number of units of this denomination required to make ONE of the next higher. Write the REMAINDER under this denomination and carry the QUOTIENT to the next column. Add the column of each denomination in the same way. PROBLEMS. 1. 73° 42′ 35′′+8° 29′ 52" = what? Solution: 73° 42′ 35′′ = 8° 29′ 52" 82° 12′ 27′′ Ans. 35"+52" 87" =1' 27". The 27" is written in the seconds column and the 1' is carried to the next higher. 42'+29'+1' =72′ = 1° 12'. The 12' is written in the minute column and the 1° carried. 73°+8°+1° 82°. 2. 26° 31′ 28′′+31° 27′ 43′′=what? Ans. 57° 59′ 11′′. 3. 115° 57' 45"+9′ 32" what? Ans. 116° 7' 17". = 4. 10yds 2ft 10in+7yds 1ft 7in=what? Ans. 18yds 1ft 5in 5. 7m 6dm 5cm 4mm +9m 4dm 6cm 7mm = what? Ans. 17m 1dm 2cm 1mm. 6. 14cwt 3r 15lb 8oz 12dr +12cwt 2ar 13lb 12oz 14dr+18cwt 3ar 12lb 9oz 10dr=what? Ans. 2t 6ewt 19r 16lb 150z 4dr. SUBTRACTION OF COMPOUND DENOMINATE NUMBERS. RULE.-Write the subtrahend under the minuend so that numbers of the same denomination may be in the same column. Beginning at the right hand, subtract the numbers of each denomination separately and write the remainder under the numbers subtracted. If any number in the subtrahend is greater than the number above it in the minuend, borrow a unit from the next higher denomination of the minuend and reduce it to the next lower denomination, add this to the number in the minuend to be subtracted from, and then subtract. Proceed in the same way with each denomination, remembering that a number in the minuend from which a unit has been borrowed must be regarded as ONE LESS than it stands. 27"-14"13". The remainder 13" is written under the numbers subtracted. 28' is greater than 14'. Borrowing 1° from the 11° of the minuend, reducing this to minutes and adding to 14' we have 74'. 74'-28' 46'. Remembering that 1° has been borrowed from the 11° we have 10°—7° = 3°. 2. 63° 47′ 28′′-15° 28′ 13" what? Ans. 48° 19' 15". 3. 75° 36′ 19′′-18° 29′ 38"=what? Ans. 57° 6' 41". 4. 43° 27' 15"-19° 38′ 17" what? 5. 14lb 12oz gdr-6lb 14oz 11dr=what? = = Ans. 23° 48' 58". 6. 85bu 2pk 5at-58bu 3pk 2at 1pt what? Ans. 26bu 3pk 2qt 1pt = Multiplication and division of compound denominate numbers may be performed by reducing all numbers to lowest given denomination, performing the required operation, and then reducing the result to any higher denomination desired. This method of performing these operations will, it is thought, be sufficient to meet the wants of gunners in ordinary cases. CHAPTER VI RATIO AND PROPORTION A ratio is the measure of the relation of one quantity to another of the same kind and denomination. It is found by dividing the first quantity by the second, and is always an abstract number. Thus, the ratio of 8 ft. to 4 ft. is 2. The sign of ratio is, which is read "is to." Thus: the ratio of 8 to 4 is written 8: 4, read "8 is to 4," and equals 8÷4, or 2. A simple ratio is the ratio of two quantities. Each quantity is called a term of the ratio. A proportion is a comparison of equal ratios. The sign of proportion is ::, which is read "as". A simple proportion is a comparison of two equal simple ratios. Thus: 3: 6::8: 16, which is read "3 is to 6 as 8 is to 16." Each quantity in a proportion is called a term. The 1st and 4th terms are called the extremes; the 2d and 3d, the means. In any proportion the product of the extremes is equal to the product of the means. Hence either extreme is equal to the product of the means divided by the other extreme, and either mean is equal to the product of the extremes divided by the other mean. Simple proportion is employed when three terms are given to find a fourth. Two of the three terms must be of the same denomination and the other of the same as that to be found. The rule by which the fourth term is found is often called the single rule of three. |