Elements of arithmetic

104. Suppose it required to divide 49 yards into five equal parts, or, as it is called, to find the fifth part of 49 yards. If we divide 45 by 5, the quotient is 9, and the remainder is 4; that is (72), 49 is made up of 5 times 9 and 4. Let the line a b represent 49 yards:

Take 5 lines, c, d, e, f, and g, each 9 yards in length, and the line h, 4 yards in length. Then, since 49 is 5 nines and 4, c, d, e, f, g, and h, are together equal to a b. Divide h, which is 4 yards, into five equal parts, i, k, l, m, and n, and place one of these parts opposite to each of the lines, c, d, e, f, and g. It follows that the ten lines, c, d, e, f, g, i, k, l, m, n, are together equal to a b, or 49 yards. Now d and k together are of the same length as c and i together, and so are e and l, f and m, and g and n. Therefore, c and i together, repeated 5 times, will be 49 yards; that is, c and i together make up the fifth part of 49 yards.

105. c is a certain number of yards, viz. 9; but i is a new sort of quantity, to which hitherto we have never come. It is not an exact number of yards, for it arises from dividing 4 yards into 5 parts, and taking one of those parts. It is the fifth part of 4 yards, and is called a fraction of a yard. It is written thus, ⁴/₅(23), and is what we must add to 9 yards in order to make up the fifth part of 49 yards.

The same reasoning would apply to dividing 49 bushels of corn, or 49 acres of land, into 5 equal parts. We should find for the fifth part of the first, 9 bushels and the fifth part of 4 bushels; and for the second, 9 acres and the fifth part of 4 acres.

We say, then, once for all, that the fifth part of 49 is 9 and ⁴/₅, or 9 + ⁴/₅; which is usually written (9⁴/₅), or if we use signs, 49/5 = (9⁴/₅).

EXERCISES.

What is the seventeenth part of 1237?—Answer, (72-¹³/₁₇).

106. By the term fraction is understood a part of any number, or the sum of any of the equal parts into which a number is divided. Thus, ⁴⁹/₅, ⁴/₅, ²⁰/₇, are fractions. The term fraction even includes whole numbers:[14] for example, 17 is ¹⁷/₁, ³⁴/₂, ⁵¹/₃, &c.

The upper number is called the numerator, the lower number is called the denominator, and both of these are called terms of the fraction. As long as the numerator is less than the denominator, the fraction is less than a unit: thus, ⁶/₁₇ is less than a unit; for 6 divided into 6 parts gives 1 for each part, and must give less when divided into 17 parts. Similarly, the fraction is equal to a unit when the numerator and denominator are equal, and greater than a unit when the numerator is greater than the denominator.

107. By ⅔ is meant the third part of 2. This is the same as twice the third part of 1.

To prove this, let a b be two yards, and divide each of the yards a c and c b into three equal parts.

Then, because a e, e f, and f b, are all equal to one another, a e is the third part of 2. It is therefore ⅔. But a e is twice a d, and a d is the third part of one yard, or ⅓; therefore ⅔ is twice ⅓; that is, in order to get the length ⅔, it makes no difference whether we divide two yards at once into three parts, and take one of them, or whether we divide one yard into three parts, and take two of them. By the same reasoning, ⅝ may be found either by dividing 5 into 8 parts, and taking one of them, or by dividing 1 into 8 parts, and taking five of them. In future, of these two meanings I shall use that which is most convenient at the time, as it is proved that they are the same thing. This principle is the same as the following: The third part of any number may be obtained by adding together the thirds of all the units of which it consists. Thus, the third part of 2, or of two units, is made by taking one-third out of each of the units, that is,

⅔ = ⅓ × 2.

This meaning appears ambiguous when the numerator is greater than the denominator: thus, ¹⁵/₇ would mean that 1 is to be divided into 7 parts, and 15 of them are to be taken. We should here let as many units be each divided into 7 parts as will give more than 15 of those parts, and take 15 of them.

108. The value of a fraction is not altered by multiplying the numerator and denominator by the same quantity. Take the fraction ¾, multiply its numerator and denominator by 5, and it becomes ¹⁵/₂₀, which is the same thing as ¾; that is, one-twentieth part of 15 yards is the same thing as one-fourth of 3 yards: or, if our second meaning of the word fraction be used, you get the same length by dividing a yard into 20 parts and taking 15 of them, as you get by dividing it into 4 parts and taking 3 of them. To prove this,

let a b represent a yard; divide it into 4 equal parts, a c, c d, d e, and e b, and divide each of these parts into 5 equal parts. Then a e is ¾. But the second division cuts the line into 20 equal parts, of which a e contains 15. It is therefore ¹⁵/₂₀. Therefore, ¹⁵/₂₀ and ¾ are the same thing.

Again, since ¾ is made from ¹⁵/₂₀ by dividing both the numerator and denominator by 5, the value of a fraction is not altered by dividing both its numerator and denominator by the same quantity. This principle, which is of so much importance in every part of arithmetic, is often used in common language, as when we say that 14 out of 21 is 2 out of 3, &c.

109. Though the two fractions ¾ and ¹⁵/₂₀ are the same in value, and either of them may be used for the other without error, yet the first is more convenient than the second, not only because you have a clearer idea of the fourth of three yards than of the twentieth part of fifteen yards, but because the numbers in the first being smaller, are more convenient for multiplication and division. It is therefore useful, when a fraction is given, to find out whether its numerator and denominator have any common divisors or common measures. In (98) was given a rule for finding the greatest common measure of any two numbers; and it was shewn that when the two numbers are divided by their greatest common measure, the quotients have no common measure except 1. Find the greatest common measure of the terms of the fraction, and divide them by that number. The fraction is then said to be reduced to its lowest terms, and is in the state in which the best notion can be formed of its magnitude.

EXERCISES.

With each fraction is written the same reduced to its lowest terms.

110. When the terms of the fraction given are already in factors,[15] any one factor in the numerator may be divided by a number, provided some one factor in the denominator is divided by the same. This follows from (88) and (108). In the following examples the figures altered by division are accented.

111. As we can, by (108), multiply the numerator and denominator of a fraction by any number, without altering its value, we can now readily reduce two fractions to two others, which shall have the same value as the first two, and which shall have the same denominator. Take, for example, ⅔ and ⁴/₇; multiply both terms of ⅔ by 7, and both terms of ⁴/₇ by 3. It then appears that

Here are then two fractions ¹⁴/₂₁ and ¹²/₂₁, equal to ⅔ and ⁴/₇, and having the same denominator, 21; in this case, ⅔ and ⁴/₇ are said to be reduced to a common denominator.

It is required to reduce ⅒, ⅚, and ⁷/₉ to a common denominator. Multiply both terms of the first by the product of 6 and 9; of the second by the product of 10 and 9; and of the third by the product of 10 and 6. Then it appears (108) that

On looking at these last fractions, we see that all the numerators and the common denominator are divisible by 6, and (108) this division will not alter their values. On dividing the numerators and denominators of ⁵⁴/₅₄₀, ⁴⁵⁰/₅₄₀, and ⁴²⁰/₅₄₀ by 6, the resulting fractions are, ⁹/₉₀, ⁷⁵/₉₀, and ⁷⁰/₉₀. These are fractions with a common denominator, and which are the same as ⅒, ⅚, and ⁷/₉; and therefore these are a more simple answer to the question than the first fractions. Observe also that 540 is one common multiple of 10, 6, and 9, namely, 10 × 6 × 9, but that 90 is the least common multiple of 10, 6, and 9 (103). The following process, therefore, is better. To reduce the fractions ⅒, ⅚, and ⁷/₉, to others having the same value and a common denominator, begin by finding the least common multiple of 10, 6, and 9, by the rule in (103), which is 90. Observe that 10, 6, and 9 are contained in 90 9, 15, and 10 times. Multiply both terms of the first by 9, of the second by 15, and of the third by 10, and the fractions thus produced are ⁹/₉₀, ⁷⁵/₉₀, and ⁷⁰/₉₀, the same as before.

If one of the numbers be a whole number, it may be reduced to a fraction having the common denominator of the rest, by (106).

EXERCISES.

112. By reducing two fractions to a common denominator, we are able to compare them; that is, to tell which is the greater and which the less of the two. For example, take ½ and ⁷/₁₅. These fractions reduced, without alteration of their value, to a common denominator, are ¹⁵/₃₀ and ¹⁴/₃₁. Of these the first must be the greater, because (107) it may be obtained by dividing 1 into 30 equal parts and taking 15 of them, but the second is made by taking 14 of those parts.

It is evident that of two fractions which have the same denominator, the greater has the greater numerator; and also that of two fractions which have the same numerator, the greater has the less denominator. Thus, ⁸/₇ is greater than ⁸/⁹, since the first is a 7th, and the last only a 9th part of 8. Also, any numerator may be made to belong to as small a fraction as we please, by sufficiently increasing the denominator. Thus, ¹⁰/₁₀₀ is ¹/₁₀, ¹⁰/₁₀₀₀ is ¹/₁₀₀, and ¹⁰/₁₀₀₀₀₀₀ is ¹/₁₀₀₀₀₀₀ (108).

We can now also increase and diminish the first fraction by the second. For the first fraction is made up of 15 of the 30 equal parts into which 1 is divided. The second fraction is 14 of those parts. The sum of the two, therefore, must be 15 + 14, or 29 of those parts; that is, ½ + ⁷/₁₅ is ²⁹/₃₀. The difference of the two must be 15-14, or 1 of those parts; that is, ½-⁷/₁₅ = ¹/₃₀.

113. From the last two articles the following rules are obtained:

I. To compare, to add, or to subtract fractions, first reduce them to a common denominator. When this has been done, that is the greatest of the fractions which has the greatest numerator.

Their sum has the sum of the numerators for its numerator, and the common denominator for its denominator.

Their difference has the difference of the numerators for its numerator, and the common denominator for its denominator.

EXERCISES.

114. Suppose it required to add a whole number to a fraction, for example, 6 to ⁴/₉. By (106) 6 is ⁵⁴/₉, and ⁵⁴/₉ + ⁴/₉ is ⁵⁸/⁹; that is, 6 + ⁴/⁹, or as it is usually written, (6⁴/₉), is ⁵⁸/₉. The rule in this case is: Multiply the whole number by the denominator of the fraction, and to the product add the numerator of the fraction; the sum will be the numerator of the result, and the denominator of the fraction will be its denominator. Thus, (3¼) = ¹³/₄, (22⁵/₉) = ²⁰³/₉, (74²/₅₅) = ⁴⁰⁷²/₅₅. This rule is the opposite of that in (105).

115. From the last rule it appears that

Hence, when a whole number is to be added to a fraction whose denominator is 1 followed by ciphers, the number of which is not less than the number of figures in the numerator, the rule is: Write the whole number first, and then the numerator of the fraction, with as many ciphers between them as the number of ciphers in the denominator exceeds the number of figures in the numerator. This is the numerator of the result, and the denominator of the fraction is its denominator. If the number of ciphers in the denominator be equal to the number of figures in the numerator, write no ciphers between the whole number and the numerator.

EXERCISES.

Reduce the following mixed quantities to fractions:

116. Suppose it required to multiply ⅔ by 4. This by (48) is taking ⅔ four times; that is, finding ⅔ + ⅔ + ⅔ + ⅔. This by (112) is ⁸/₃; so that to multiply a fraction by a whole number the rule is: Multiply the numerator by the whole number, and let the denominator remain.

117. If the denominator of the fraction be divisible by the whole number, the rule may be stated thus: Divide the denominator of the fraction by the whole number, and let the numerator remain. For example, multiply ⁷/₃₆ by 6. This (116) is ⁴²/₃₆, which, since the numerator and denominator are now divisible by 6, is (108) the same as ⁷/₆. It is plain that ⁷/₆ is made from ⁷/₃₆ in the manner stated in the rule.

118. Multiplication has been defined to be the taking as many of one number as there are units in another. Thus, to multiply 12 by 7 is to take as many twelves as there are units in 7, or to take 12 as many times as you must take 1 in order to make 7. Thus, what is done with 1 in order to make 7, is done with 12 to make 7 times 12. For example,

When the same thing is done with two fractions, the result is still called their product, and the process is still called multiplication. There is this difference, that whereas a whole number is made by adding 1 to itself a number of times, a fraction is made by dividing 1 into a number of equal parts, and adding one of these parts to itself a number of times. This being the meaning of the word multiplication, as applied to fractions, what is ¾ multiplied by ⅞? Whatever is done with 1 in order to make ⅞ must now be done with ¾; but to make ⅞, 1 is divided into 8 parts, and 7 of them are taken. Therefore, to make ¾ × ⅞, ¾ must be divided into 8 parts, and 7 of them must be taken. Now ¾ is, by (108), the same thing as ²⁴/₃₂. Since ²⁴/₃₂ is made by dividing 1 into 32 parts, and taking 24 of them, or, which is the same thing, taking 3 of them 8 times, if ²⁴/₃₂ be divided into 8 equal parts, each of them is ³/₃₂; and if 7 of these parts be taken, the result is ²¹/₃₂ (116): therefore ¾ multiplied by ⅞ is ²¹/₃₂; and the same reasoning may be applied to any other fractions. But ²¹/₃₂ is made from ¾ and ⅞ by multiplying the two numerators together for the numerator, and the two denominators for the denominator; which furnishes a rule for the multiplication of fractions.

119. If this product ²¹/₃₂ is to be multiplied by a third fraction, for example, by ⁵/₉, the result is, by the same rule, ¹⁰⁵/₂₈₈; and so on. The general rule for multiplying any number of fractions together is therefore:

Multiply all the numerators together for the numerator of the product, and all the denominators together for its denominator.

120. Suppose it required to multiply together ¹⁵/₁₆ and ⁸/₁₀. The product may be written thus:

which reduced to its lowest terms (109) is ¾. This result might have been obtained directly, by observing that 15 and 10 are both measured by 5, and 8 and 16 are both measured by 8, and that the fraction may be written thus:

• 3 × 5 × 8
• 2 × 8 × 2 × 5.

Divide both its numerator and denominator by 5 × 8 (108) and (87), and the result is at once ¾; therefore, before proceeding to multiply any number of fractions together, if there be any numerator and any denominator, whether belonging to the same fraction or not, which have a common measure, divide them both by that common measure, and use the quotients instead of the dividends.

A whole number may be considered as a fraction whose denominator is 1; thus, 16 is ¹⁶/₁ (106); and the same rule will apply when one or more of the quantities are whole numbers.

EXERCISES.

From 100 acres of ground, two-thirds of them are taken away; 50 acres are then added to the result, and ⁵/₇ of the whole is taken; what number of acres does this produce?—Answer, (59¹¹/₂₁).

121. In dividing one whole number by another, for example, 108 by 9, this question is asked,—Can we, by the addition of any number of nines, produce 108? and if so, how many nines will be sufficient for that purpose?

Suppose we take two fractions, for example, ⅔ and ⅘, and ask, Can we, by dividing ⅘ into some number of equal parts, and adding a number of these parts together, produce ⅔? if so, into how many parts must we divide ⅘, and how many of them must we add together? The solution of this question is still called the division of ⅔ by ⅘; and the fraction whose denominator is the number of parts into which ⅘ is divided, and whose numerator is the number of them which is taken, is called the quotient. The solution of this question is as follows: Reduce both these fractions to a common denominator (111), which does not alter their value (108); they then become ¹⁰/₁₅ and ¹²/₁₅. The question now is, to divide ¹²/₁₅ into a number of parts, and to produce ¹⁰/₁₅ by taking a number of these parts. Since ¹²/₁₅ is made by dividing 1 into 15 parts and taking 12 of them, if we divide ¹²/₁₅ into 12 equal parts, each of these parts is ¹/₁₅; if we take 10 of these parts, the result is ¹⁰/₁₅. Therefore, in order to produce ¹⁰/₁₅ or ⅔ (108), we must divide ¹²/₁₅ or ⅘ into 12 parts, and take 10 of them; that is, the quotient is ¹⁰/₁₂. If we call ⅔ the dividend, and ⅘ the divisor, as before, the quotient in this case is derived from the following rule, which the same reasoning will shew to apply to other cases:

The numerator of the quotient is the numerator of the dividend multiplied by the denominator of the divisor. The denominator of the quotient is the denominator of the dividend multiplied by the numerator of the divisor. This rule is the reverse of multiplication, as will be seen by comparing what is required in both cases. In multiplying ⅘ by ¹⁰/₁₂, I ask, if out of ⅘ be taken 10 parts out of 12, how much of a unit is taken, and the answer is ⁴⁰/⁶⁰, or ⅔. Again, in dividing ⅔ by ⅘, I ask what part of ⅘ is ⅔, the answer to which is ¹⁰/₁₂.

122. By taking the following instance, we shall see that this rule can be sometimes simplified. Divide ¹⁶/₃₃ by ²⁸/₁₅. Observe that 16 is 4 × 4, and 28 is 4 × 7; 33 is 3 × 11, and 15 is 3 × 5; therefore the two fractions are

and their quotient, according to the rule, is

• 4 × 4 × 3 × 5
• 3 × 11 × 4 × 7,

in which 4 × 3 is found both in the numerator and denominator. The fraction is therefore (108) the same as

The rule of the last article, therefore, admits of this modification: If the two numerators or the two denominators have a common measure, divide by that common measure, and use the quotients instead of the dividends.

123. In dividing a fraction by a whole number, for example, ⅔ by 15, consider 15 as the fraction ¹⁵/₁. The rule gives ²/⁴⁵ as the quotient. Therefore, to divide a fraction by a whole number, multiply the denominator by that whole number.

EXERCISES.

A can reap a field in 12 days, B in 6, and C in 4 days; in what time can they all do it together?[16]—Answer, 2 days.

In what time would a cistern be filled by cocks which would separately fill it in 12, 11, 10, and 9 hours?—Answer, (2⁴⁵⁴/₇₆₃) hours.

124. The principal results of this section may be exhibited algebraically as follows; let a, b, c, &c. stand for any whole numbers. Then

125. These results are true even when the letters themselves represent fractions. For example, take the fraction

• a/b
• c/d

whose numerator and denominator are fractional, and multiply its numerator and denominator by the fraction

which, dividing the numerator and denominator by ef (108), is

• ad
• bc

But the original fraction itself is

• ad
• bc

hence

which corresponds to the second formula[17] in (124). In a similar manner it may be shewn, that the other formulæ of the same article are true when the letters there used either represent fractions, or are removed and fractions introduced in their place. All formulæ established throughout this work are equally true when fractions are substituted for whole numbers. For example (54), (m + n)a = ma + na. Let m, n, and a be respectively the fractions

Then m + n is

and (m + n)a is

Therefore (m + n)a, or

In a similar manner the same may be proved of any other formula.

The following examples may be useful:

Thus,

The rules that have been proved to hold good for all numbers may be applied when the numbers are represented by letters.