Chapter I: The Anglo-Saxon Period, 449-1066 (12)

=Ulysses= (_ū-lis´ēz_), or =Ulixes= (_u-lix´ēz_); called =Odysseus= (_od-is´sūs_) by the Greeks.--A king of Ithaca, famed among the Grecian heroes of the Trojan war for his craft and eloquence; the son of Laertes, husband of Penelope, and father of Telemachus and Telegonus (by Circe). In order to escape from going with the other Greek heroes against Troy, he feigned madness, ploughing the sea-shore with a horse and bull yoked together and sowing salt. The imposture, however, was laid bare by Palamedes (_q.v._), who placed Telemachus, the infant son of Ulysses, in the furrow, when the latter at once turned aside the plough; but the wily Ulysses had his revenge on Palamedes. Ulysses, in his turn, sought out and obtained the indispensable assistance of Achilles (_q.v._). At the siege of Troy his cunning and valor were of the greatest service to the Greeks. In company with Diomedes he slew the horses of Rhesus, and also carried off the Palladium (_q.v._). Perhaps the crowning effort of his ingenuity was the invention of the famous wooden horse, by means of which the city of Troy (_q.v._) was ultimately taken by the Greeks. After the taking of Troy Ulysses set out for Ithaca, which, however, he did not reach for twenty years. During this time he passed through the adventures which form the subject of Homer’s glorious poem, the _Odyssey_, which takes its name from Odysseus, the Greek name for Ulysses. He thus visited Circe (_q.v._), Polyphemus (_q.v._), the Lotophagi, and other persons and places. In order to get safely past the island of the Sirens, he, with his usual sagacity, devised special means, which proved entirely successful (see “Sirens”). He lost six of his companions while sailing between Scylla (_q.v._) and Charybdis. He then suffered shipwreck, he alone escaping by means of the mast and planks. In ten days he was drifted on to the island of Ogygia, inhabited by Calypso (_q.v._), with whom he stayed for eight years. He then constructed a raft, and made his way to the island of Scheria (_q.v._), whence he obtained a ship that carried him to Ithaca. He did not, however, make himself known at once to his wife and son. In order to see how the land lay, he disguised himself as a beggar, but was kindly received by the old swineherd. Meanwhile his son Telemachus, now grown up to manhood, returned from a journey to Pylos and Sparta, undertaken with a view to gleaning what information he could as to the probable whereabouts of his father. Ulysses then made himself known to Telemachus, and the two resolved on a plan of revenge on the numerous unfortunate suitors for the hand of the virtuous and constant Penelope (_q.v._). With great difficulty she was induced (being, as yet, unaware of the safe arrival of her husband) to promise her hand to that suitor who could shoot with the bow of Ulysses. Not one of them, however, was able to draw this bow, whereupon Ulysses himself took it up and slew them all. He then made himself known to Penelope, and went to see his father Laertes, bowed down with grief and years. Now Circe, who had had a son, Telegonus, by Ulysses, sent him in search of his father. Telegonus encountered a storm which cast his ship on the coast of Ithaca, and being pressed by hunger, he began to plunder the fields. Ulysses and Telemachus hearing of this, went out against the spoliator; but Telegonus, not knowing Ulysses, ran him through the body with a spear given to him by his mother. Thus the famous hero died at the hands of his own son. Telegonus afterwards married Penelope, and became by her the father of Italus.

=Urania= (_ū-rā´ni-a_).--The muse of astronomy. See “Musæ.”

=Uranus= (_ū´ra-nus_), or =Heaven=.--Husband of Gæa (Earth), and father of Oceanus, Hyperion, Rhea, Themis, Cronos, and other children. At the instigation of Gæa he was dethroned by Cronos.

=Utgard-Loki.=--The chief of the giants, in Norse mythology.

=V=

=Varuna=, or =Vrauna=.--In Hindu mythology, the deity who presides over the waters of the ocean, corresponding with _Neptune_ of classic mythology.

=Valhalla.=--In Scandinavian mythology the palace of immortality wherein are received the souls of heroes slain in battle.

=Valkyrs.=--The battle-maidens of Scandinavian mythology. They were mounted on swift horses and held drawn swords. They rushed into battle and selected those destined to death and conducted them to Valhalla. The number of Valkyrs differs greatly according to the various mythologists and ranges from three to sixteen, the greater part of them, however, naming only nine.

=Venus= (_vē´nus_); called by the Greeks =Aphrodite= (_af-ro-dī´tē_)--_i. e._ “sea-foam.”--The goddess of love and beauty. She was supposed to have sprung from the foam of the sea: hence her Greek name. She was the wife of Vulcan, but was very unfaithful to him. She loved the gods Mars, Bacchus, Neptune and Mercury, and the mortals Adonis and Anchises. She was considered by Paris (_q.v._) the most beautiful of the goddesses and had awarded to her the celebrated Golden Apple. Anyone who wore her magic girdle immediately became beautiful and the object of love and desire. She is generally accompanied by her son Cupid. The month of April, as the commencement of spring, was considered peculiarly sacred to the goddess of love. The myrtle, rose, apple and poppy, and the sparrow, dove, swan and swallow, were all sacred to her. She was probably originally identical with Astarte, a Syrian goddess, called by the Hebrews Ashtoreth. As might have been anticipated, the representation of the Queen of Beauty on canvas and in marble has resulted in some of the finest works of the most celebrated painters and sculptors of antiquity. Among the former, Apelles’ masterpiece of Venus rising from the sea deserves special mention; and among the latter the “Cnidian Venus” (so called because it stood in her temple at Cnidus), by Praxiteles, is unquestionably the most famous. Phryne (_q.v._) sat as model for both of these noble works of art. The fame of the “Cnidian Venus” was so great that travelers from all parts of the civilized world resorted to Cnidus in order to see it. In fact, Pliny and others declared it to be the finest statue in the world. The “Venus of Milo” is, however, the noblest extant representation of Venus. It was found, in 1820, in the island of Melos, the modern Milo (hence the epithet), which is one of the group of islands named the Cyclades, in the Ægean Sea. It now forms one of the treasures of the Louvre, Paris.

=Vertumnus= (_ver-tum´nus_).--The god of the changing year--that is, of the seasons and their productions. His festival was celebrated by the whole Roman people on the 23rd of August.

=Vesta= (_ves´ta_); called by the Greeks =Hestia= (_hes´ti-a_)--_i. e._ “the hearth.”--One of the twelve great Roman deities, the goddess and guardian of the hearth and home. She was the daughter of Saturn and Rhea. In her temple in the Forum at Rome stood no statue, the goddess being represented by the eternal fire burning on her altar as her abiding symbol. This fire was kept up and attended to by a number of virgin priestesses, called Vestals, who were chaste and pure like the goddess herself. On March 1 in every year the sacred fire was renewed, and on June 15 her temple was cleaned and purified.

=Vidar.=--The Scandinavian god of wisdom, noted for his thick shoes, and not infrequently called “The god with the thick shoes.”

=Vishnu.=--In Hindu mythology one of the great deities of the Hindu triad, ranking as the _Preserver_, after Brahma, the _Creator_, and before Siva, the _Destroyer_. It is believed that he has appeared on earth nine times, his tenth _avatar_, or incarnation, having yet to come.

=Volumnia= (_vol-um´ni-a_).--Wife of Coriolanus (_q.v._).

=Vulcanus= (_vul-kā´nus_), or =Vulcan=; called =Hephæstus= (_hē-fēs´tus_) by the Greeks. The god of fire. He was the son of Jupiter and Juno, and was lame from his birth. Besides being the god of fire, he was master of the arts which need the aid of fire, especially of working in metal. He made all the palaces of the gods on Olympus, the armor of Achilles, the fatal necklace of Harmonia, the fire-breathing and brazen-hoofed bulls of Æëtes (see “Argonautæ”), etc. The Cyclops were his workmen, and his workshops were situated under Mount Ætna in Sicily. Vulcan’s wife was Venus. His favorite abode on the earth was the island of Lemnos. His great festival was celebrated on the 23rd of August.

=W=

=White Lady.=--In German folk-lore, the ancient Teutonic goddess Holda or Berchta, who was the receiver of the souls of maidens and children, and who still exists as the White Lady, not infrequently, in German legends, transforming herself, or those whom she decoys into her home, into a white mouse.

=Wild Huntsman, The.=--A spectral hunter in folk-lore, especially in German folk-lore; the subject of a ballad by Bürger.

=Woden= (_wō´den_), or =Wotan=.--The Anglo-Saxon form of the Scandinavian god Odin; Wednesday is called after him.

=Y=

=Yama.=--In the _Rigveda_, the name of the god who rules in heaven over the blessed--the Manes, Fathers, or Pitris--and is therefore called king.

=Yggdrasil.=--In Scandinavian mythology the great ash tree which binds together heaven, earth, and hell. Its branches extend over the whole earth, its top reaches heaven, and its roots hell. The three nornas, or fates, sit under the tree, spinning the events of man’s life.

=Z=

=Zem.=--The sacred well of Mecca. According to Arab tradition, this is the very well that was shown to Hagar when with Ishmael in the desert. It is supposed to be in the heart of the city of Mecca.

=Zephyrus= (_zef´i-rus_).--The west wind, or properly, the northwest.

=Zeus= (_zūs_).--See “Jupiter.”

=Zohak.=--The giant of Persian mythology who keeps the “mouth of hell.” He was the fifth of the Pischdadian dynasty, and was a lineal descendant of Shedâd, king of Ad. He murdered his predecessor, and invented both flaying men alive and killing them by crucifixion. The devil kissed him on the shoulders, and immediately two serpents grew out of his back and fed constantly upon him. He was dethroned by the famous blacksmith of Ispahan, and appointed by the devil to keep hell-gate.

=Zohara.=--An oriental queen of love, and mother of mischief. When Harût and Marût were selected by the host of heaven to be judges on earth, they judged righteous judgment till Zohara, in the shape of a lovely woman, appeared before them with her complaint. They then both fell in love with her and tried to corrupt her, but she flew from them to heaven; and the two angel-judges were forever shut out.

=Zulzul.=--According to Chinese mythology the sage whose life was saved in the form of a rat by Gedy, the youngest of the four sons of Corcud. Zulzul gave him, in gratitude, two poniards, by the help of which he could climb the highest tree or most inaccessible castle.

=EXPLANATORY CHART OF GREEK AND ROMAN MYTHOLOGY: SHOWING THE ORIGIN, RELATIONSHIP AND DESCENT OF CHIEF MYTHS=

The relationship of these mythical personages are quite unlike those
of mortals and are full of inconsistencies. To reconcile all the
contradictions of the poets and mythologists is impossible. Perhaps
this chart is as consistent with their fabulous tales as can well be
made.

CHAOS Produced EREBUS, god of _darkness_, NOX, goddess of _night_, and
TERRA, _Earth_.

TERRA or TITÆA produced CŒLUS or URANUS, _Heaven_.

CŒLUS or URANUS, i.e. _Heaven_, and TERRA or TITÆA, i.e. _Earth_. had
|
+->=TITAN= Oldest of the twelve Titans.
|
+->=THE CYCLOPS= Giants, at first three in number: Arges, Brontes,
| Steropes.
|
+->=BRIAREUS= A famous giant called by men Ægæon, and by the gods
| Braireus.
|
+->=TETHYS= Wife of _Oceanus_; for offspring, see =Oceanus=.
|
+->=THEA= Wife of _Hyperion_; the mother of rivers, and of about three
| thousand daughters, called _Ocean´i-des_.
|
+->=SATURN= or =Cronos=, god of Time, had by _Rhea_ same as _Ops_,
| same as _Cybele_.
| |
| +->=JUNO=, wife and sister of Jupiter, queen of the gods, and of
| | Heaven and Earth.
| |
| +->=JUPITER= or =Zeus=, the most powerful of all the gods; king of
| | gods and men, had
| | |
| | +->_By Them´is._
| | | |
| | | +->=Astræa=, the goddess of _justice_;
| | | |
| | | +->=Nemesis=, of _vengeance_.
| | |
| | +->_By Juno._
| | | |
| | | +->=Mars=, the god of _war_; by _Venus_, Anteros, Harmonia;
| | | | the goddess of _youth_; once cupbearer to Jupiter.
| | | |
| | | +->=Hebe=, by her husband _Hercules_, Alexiares and Anicetus.
| | | |
| | | +->=Typhon=, by the monster _Echidna_, Chimæra and Sphinx.
| | | |
| | | +->=Vulcan=, the god of _fire_ and of _blacksmiths_, and
| | | husband of Venus; by his wife _Venus_, Cupid; by
| | | _Medusa_, Cacus, by _Juno_, Cæculus.
| | |
| | +->_By Lato´na._
| | | |
| | | +->=Apollo=, the god of _poetry_, _music_, _eloquence_,
| | | | _medicine_, _the fine arts_, _augury_, and _archery_.
| | | |
| | | +->=Diana=, the goddess of _hunting_, the patroness of
| | | chastity, presided also over childbirth.
| | |
| | +->_By Ma´ia._
| | | |
| | | +->=Mercury=, the _messenger_ of the gods, the god of
| | | _eloquence_ and _commerce_, the patron of _travellers_,
| | | _thieves_, and _knaves_, and the conductor of the souls of
| | | the dead to the infernal regions. By _Penelope_, Pan. By
| | | the Greeks he was called _Hermes_.
| | |
| | +->_By Mnemos´y-ne._
| | | |
| | | +->=The Nine Muses.=
| | | |
| | | +->_Cli´o_ presided over History. }
| | | | }
| | | +->_Calli´o-pe_ presided over }
| | | | eloquence and epic poetry. }
| | | | }
| | | +->_Er´ato_ presided over lyric }
| | | | and amorous poetry. }
| | | | }
| | | +->_Thali´a_ presided over }
| | | | pastoral and comic poetry and }
| | | | festivals. }
| | | | }See
| | | +->_Melpom´e-ne_ presided over }Dictionary of
| | | | tragedy. }Mythology.
| | | | }
| | | +->_Terpsich´o-re_ presided over }
| | | | dancing. }
| | | | }
| | | +->_Euter´pe_ presided over music.}
| | | | }
| | | +->_Polyhym´nia_ presided over }
| | | | singing and rhetoric. }
| | | | }
| | | +->_Ura´nia_ presided over }
| | | astronomy. }
| | |
| | +->_By Euryn´o-me._
| | | |
| | | +->=Graces.=
| | | |
| | | +->_Agla´ia_ }Three beautiful virgins, attendants on
| | | | }Venus; presided over kindness and good
| | | +->_Thali´a_ }offices, and were supposed to give to
| | | | }beauty its charms; represented dancing
| | | +->_Euphros´y-ne_}in a circle with their hands joined.
| | |
| | +->_By Sem´e-le._
| | | |
| | | +->=Bacchus=, god of _wine_; by his wife _Ariadne_, Thoas,
| | | Œnopion, Ceranus, Tauropolis, and others.
| | |
| | +->_By Metis._
| | | |
| | | +->=Minerva=, the goddess of _wisdom_, _war_, and the
| | | _liberal and useful arts_.
| | |
| | +->_By Dione._
| | | |
| | | +->=Venus=, said to have been borne in the foam of the sea;
| | | the goddess of _love_ and _beauty_, and mistress of the
| | | graces; wife of _Vulcan_; for offspring, see =Vulcan=.
| | |
| | +->_By Ceres._
| | | |
| | | +->=Pros´erpine=, wife of Pluto, _queen_ of hell, presided
| | | over death. She was stolen away by Pluto while gathering
| | | flowers in Sicily, and became the mother of the Fates and
| | | Furies, which see under Dictionary.
| | |
| | +->_By Euro´pa._
| | | |
| | | +->=Minos=, =Rhadamanthus=, and =Æ´acus=, three inflexible
| | | judges of Hades.
| | |
| | +->_By Leda._ }
| | | | }
| | | +->=Castor= and =Pollux=. }
| | | }
| | +->_By Dan´a-e._ }
| | | | }See Dictionary
| | | +->=Per´seus.= }of Mythology.
| | | }
| | +->_By Anti´o-pe._ }
| | | | }
| | | +->=Amphi´on= and =Zethus=.}
| | |
| | +->_By Segesta._
| | | |
| | | +->=Æolus=, whose offspring were the various Winds.
| | |
| | +->_By Alcmena._
| | |
| | +->=Hercules=, whose descendants were the Heraclidæ.
| |
| +->=VESTA=, the goddess of _fire_, and patroness of _Vestal
| | Virgins_, who had the care of the sacred fire in the temple of
| | Vesta at Rome, which was kept continually burning.
| |
| +->=CERES=, the goddess of _corn_ and _harvest_. The famous
| | _Eleusinian mysteries_ were celebrated in honor of Ceres, during
| | the representation of which it was death to speak; as it was
| | also to reveal afterwards what took place.
| |
| +->=LATONIA=, celebrated for her beauty, and for being greatly
| | beloved by Jupiter and persecuted by Juno.
| |
| +->=NEPTUNE=, the god of the _sea_, the father of rivers and
| | fountains, and, next to Jupiter, the most powerful deity; had by
| | _Amphitrite_, TRITON, his father’s companion and herald.
| |
| +->=PLUTO=, the god of the _infernal regions_, of _death_ and
| _funerals_; the dog _Cer´berus_, a frightful mastiff with three
| heads, and a tail like a serpent, watches at his feet, and three
| _Har´pies_, winged monsters, hover about him.
|
+->=MNEMOSYNE= Mother of the nine _Muses_.
|
+->=THEMIS= Mother of Astræa, goddess of Justice.
|
+->=CYBELE= OPS or RHEA, wife of _Saturn_; the goddess of _all
| things_; styled _Magna Mater_ or _Great Mother_, _Bona Mater_ or
| _Good Mother_; for off-spring, see =Saturn=.
|
+->=OCEANUS= The god of water, to whom the ancients recommended
| themselves when going on a voyage, had by Tethys.
| |
| +->AMPHITRITE had by _Neptune_ TRITON, who had no offspring.
| |
| +->CYLMENE had by _Japetus_ ATLAS, also Menœtius, Prometheus,
| | Epimetheus, and others.
| |
| +->PHORCYS had by _Ceto_ {The Gorgons, viz., Medusa,
| | {Stheno, and Euryale; three
| | {sisters whose heads were covered
| | {with vipers.
| | {The Graiæ, viz., Pephredo, Enyo,
| | {and Dinon.
| |
| +->ACHELOUS had by _Calliope_. The _Sirens_ were three sea
| | nymphs, named _Parthen´ope_,
| | _Lige´ia_, and _Leuco´sia_, having
| | the form of a woman above the
| | waist, and the rest of the body
| | like a flying fish.
| |
| +->The Harpies, viz., Aello, Ocypete, and Celæus.
|
+->=HYPERION=, god of the Sun, had by _Thea_, AURORA, the goddess of
| the _morning_; represented riding in a rose-colored chariot drawn
| by white horses, usually covered with a veil, the morning star
| appearing overhead. She was called _rosy-fingered_, because she
| scattered roses; by _Tithon´us_, a mortal, she had Memnon and
| Æmathion.
|
+->=JAPETUS=, father of mankind, had by _Clymene_, ATLAS, also
Prometheus, Epimetheus, Menœtius, and others, called _Japitonides_.

}_Nox_ or _Night_, _Mors_ or _Death_,
EREBUS } {Light, or Day,}_Somnus_ or _Sleep_, and _Morpheus_ (the
and }had{Somnus, Mors, }minister of Somnus, who brought dreams to
NOX } {and Charon, }men) were infernal divinities.
{the Ferryman }_Momus_, god of laughter and satire, son of
}Somnus and Nox.

Ancient Roman Sun-god--=Janus=, the god of the _year_, presided over
the gates of heaven, and over peace and war; represented with two
faces. His temple in Rome was open in time of war and shut in time of
peace.

=IMMENSELY ENLARGED REPRESENTATIONS OF ATOMS=

=(1) of Ordinary Matter; (2) of Radium=]

BOOK OF THE SCIENCES AND INVENTION

HISTORICAL DEVELOPMENT OF THE SCIENCES

OUTLINES OF SCIENCE FOR SCHOOLS

PRACTICAL MATHEMATICS FOR DAILY USE: BUSINESS AND INDUSTRIAL
ARITHMETIC, EVERYDAY APPLICATIONS OF PERCENTAGE, WEIGHTS AND
MEASURES, MENSURATION AND ITS APPLICATIONS

COMMERCIAL AND INDUSTRIAL LAW

PHYSICS: ITS PRINCIPLES AND APPLICATIONS

CHEMISTRY: ITS THEORY AND USES

THE CHEMISTRY OF COMMON THINGS

REVISED TABLE OF CHEMICAL ELEMENTS

GREAT INVENTIONS AND SCIENTIFIC DISCOVERIES

RELATION OF THE GOVERNMENT TO SCIENCE

BOOK OF SCIENCE AND INVENTION

Science in its widest significance is sometimes defined as the correlation of all knowledge. In this sense it would include _philosophy_. In a more restricted and generally accepted sense, the term is applied to the _systematized_ divisions of knowledge.

Science and philosophy _resemble_ each other in so far as they both have to do with knowledge; but while the latter deals with the whole sum of knowledge and goes back to generalized first principles, the former takes up special branches of it. That is, a science is such in fact when a sufficient number of interrelated facts are so arranged and classified by referring them to the general truths and principles on which they are founded that they constitute a well-certified and more or less complete branch of knowledge.

From the present development of knowledge the separate entities of the universe are five--namely, ether, matter, energy, life, and mind. The first three are inseparable agents in the simplest phenomena that occur in nature. They may ultimately be reduced to two, or, conceivably, to one. It is with these that the various branches of science have to deal--to observe, to experiment, to classify, to define.

CLASSIFICATION OF THE SCIENCES.--The sciences may be grouped in two ways. _First_, from what has been said above, they may be divided into:

(a) the _physical_ sciences, which have to do with inorganic nature--that is with the laws and properties of matter, energy, and ether;

(b) the _biological_ sciences, which consider the laws of life; and

(c) the _psychical_ sciences, which deal with the phenomena of mind.

_Second._--Another, and probably more practical, division is that of (a) _pure_ or _theoretic_ sciences, and (b) _applied_ or _practical_ sciences. The latter consist of those branches which deal with facts, events, or phenomena as explained, accounted for, or produced by means of powers, causes, or laws; the former as the knowledge of these powers, causes, or laws, considered apart or as pure from all applications. To the class of pure or fundamental sciences belong mathematics, physics, chemistry, psychology, and sociology; to the applied or concrete belong geology, mineralogy, botany, zoology, meteorology, geography, ethics, politics, law, jurisprudence, logic, grammar, rhetoric, philology, and political economy; navigation, engineering, and practical mechanics; surgery, medicine, materia medica, etc.

METHODS OF SCIENCE.--The great method of scientific inquiry is experiment--the laboratory. Contrasted with _experiment_ is _observation_. But even in astronomy, emphatically an observational science, experiment plays an important part. The dynamical knowledge which Newton developed into the cosmic law of gravitation was founded on experiment. Meteorology, again, has made great strides in these days by appealing to laboratory experiments for elucidation of its phenomena. Likewise in biology, botany, and zoology experiment has led to striking discoveries; while such branches as embryology and bacteriology are as truly experimental as chemistry itself.

In the psychical group of sciences the method of experimenting still awaits development. The complexity of the problems presented, and the manner in which they affect the welfare and happiness of humanity, render social and political experimenting excessively hazardous. Such sciences as those studied by the economist, the ethnologist, the moralist, or the theologian are of necessity essentially observational.

APPLIED ARITHMETIC, WEIGHTS AND MEASUREMENTS

It would be difficult to overestimate the extent to which mathematics enters into the conditions of everyday life. In its elementary stages, as the science of number, it teaches us the relations of magnitude, and enables us to build up a system of calculation and measurement which, applied to the relations observed to exist in nature, gives results of far-reaching importance.

The properties of number are investigated in arithmetic, and methods examined by which those engaged in practical science are able to work out their results to any degree of approximation.

With the help of algebra, we arrive at a system of logarithms by which many of these results may be reached with the minimum of labor.

The measurement of lines and angles, by methods investigated in geometry and trigonometry, enables us to calculate areas, and work out various problems met with in surveying, and is of the first importance in astronomy.

_Arithmetic_, which deals with the properties of numbers, forms the basis of all mathematical calculation. (For the _primary_ treatment of numbers, see under The Child World.)

COMMON FRACTIONS

A Fraction is one or more of the equal parts into which a unit has been divided. A _Common Fraction_ is expressed by two numbers; the one written above the line is called the Numerator, the one below, the Denominator: both, called the Terms, denote the value of the fraction.

Thus, in the fraction 3/4, the denominator 4, denotes that a unit or whole thing has been divided into four equal parts; and the numerator 3, shows that three of those parts are taken or expressed in the fraction.

A _Proper Fraction_ is one whose numerator is less than its denominator; as 1/2, 3/4, 7/8, etc. Its value is always less than 1.

An _Improper Fraction_ is one whose numerator is equal to, or greater than its denominator, as 5/5, 9/7, 30/12, etc. Its value is never less than 1.

A _Mixed Number_ is a whole number and a fraction; as 3-2/5, 10-1/2, 6-2/3.

The mixed number means that there are whole things taken together with a fraction of another.

A _Complex Fraction_ is one in which the numerator or denominator, or both, are fractions.

3-1/7 1 15/17
Thus, -----, ---------, -----, are complex fractions.
2-3/8 5/6 × 3/4 8

SIMPLE FRACTIONS

A very good method of learning the combinations in small fractions is by the use of paper or cardboard disks.

Cut out a large number of them, and, in order to avoid trouble later on, it might be better to have the disks all of one size--about 4 inches in diameter.

LEARNING THE FRACTION 1/2 WITH DISKS

EXPLANATION:--Take a circular disk and cut it into two equal parts. Then proceed in this manner: What is this part called? What is other part called? How many halves in the whole circle? One-half and one-half are what? One-half taken away from one leaves what? If I take a half two times, what do I get? How many halves in a whole?

Now I will write these--

1 1
- + - =
2 2

1 ÷ 2 =

1
1 less - =
2

1
1 - - =
2

1
2 × - =
2
1
1 divided by - =
2

Give me the answers and I will write them.

Drawings showing the “placing” of disks for number combinations can then be made; as,

Make similar drawings to tell about halves.

Proceed like this--How many halves in a pie? If a pie cost 10 cents, what will half a pie cost? Who can tell other stories about halves? etc.

Learn fourths along with halves.

LEARNING THE FRACTION 1/3 AND OTHERS WITH DISKS

Cut several disks into thirds and have children practice on cutting, so that they will be able to make the three parts of each disk equal. Frequently children will find pleasure in “teaching” one another.

Then proceed like this: What do you call each of these parts? Why are they called thirds? How many thirds in a circle? I am going to take a circle and cut it any way, so as to make three parts; do I call these unequal parts thirds? Why not? Let me write one-third on a piece of paper for you. (Write, 1/3.) Draw a circle for me. Instead of cutting it, draw lines where you would cut it to make thirds. Write one-third (1/3) on each third of a circle. I write this (1/3 + 1/3). Who can tell me what the answer is? Are two-thirds and two-thirds more than one? How much more? I have two-thirds of an apple and give Mary one-third, how much have I left? Who can give other story problems about thirds? Everybody try, etc.

Learn sixths along with thirds. Use disks, dots, marks, sticks, and inches to illustrate.

Remember that no advance should be made until each little part is understood. Then have fifths compared with fourths, thirds, and halves.

Teach tenths along with fifths.

When twelfths are taught, show the relations between twelfths and sixths, fourths, thirds, and halves.

EQUAL FRACTIONS IN DIFFERENT FORMS

Have the children see how fractions may differ in form but still remain the same in value.

Begin with his knowledge of smaller fractions as

1 2 3 4 5
-, -, -, -, and -- of an apple.
2 4 6 8 10

Let them show by the use of drawings that fractions may have large or small terms but be equal in value.

Write a number of proper fractions, improper fractions, and mixed numbers, and have the children pick out those of each kind; as,

3 5 19 20 18 11
-, 27-1/2, --, --, --, --, --, 3-1/2, 16-2/3
8 11 20 20 16 5

PRINCIPLES OF FRACTIONS

_1. A fraction’s value is the quotient obtained by dividing the numerator by the denominator._

6 6
- = 3 3 is the value of -
2 2

2 2 2 2
- = - - is the value of -
3 3 3 3

_2. Multiplying the denominator of a fraction divides the fraction by that number._

1 1
- = -
2 × 4 8

3 3
- = --
7 × 3 21

2 2
- = --
3 × 9 27

_3. Dividing the denominator of a fraction multiplies the fraction by that number._

3 3
- = -
8 ÷ 4 2

10 10
-- = --
9 ÷ 3 3

3 3
-- = -
10 ÷ 5 2

_4. Multiplying the numerator of a fraction multiplies the fraction by that number._

2 × 2 4
- = -
3 3

1 × 8 8
- = -
9 9

5 × 3 15
- = --
8 8

_5. Dividing the numerator of a fraction divides the fraction by that number._

4 ÷ 2 2
- = -
7 7

12 ÷ 12 1
-- = --
16 16

3 ÷ 3 1
- = -
7 7

_6. Multiplying both numerator and denominator of a fraction by the same number does not change the value of the fraction._

1 × 3 3 1
- = - = -
3 × 3 9 3

6 × 2 12 6
- = -- = -
7 × 2 14 7

_7. Dividing both numerator and denominator of a fraction by the same number does not change the value of the fraction._

12 ÷ 3 4 12
-- = - = --
15 ÷ 3 5 15

18 ÷ 9 2 18
-- = - = --
27 ÷ 9 3 27

REDUCTION OF FRACTIONS

is the process of changing their forms without altering their values.

To reduce a fraction to its lowest terms:

RULE.--Divide both terms by their greatest common divisor.

Reduce 8/12 to its lowest terms.

WORK: 4) 8/12 (2/3 _Ans._ 2/3

Four is the G. C. D. of 8 and 12; hence 8/12 ÷ 4 = 2/3.

Reduce 35/56 to its lowest terms.

WORK: 7) 35/56 (5/8 _Ans._ 5/8

Seven is the G. C. D. of 35 and 56; hence 35/56 ÷ 7 = 5/8.

A fraction whose terms have no common divisor is in its lowest terms, as 9/16.

To reduce an improper fraction to a whole or mixed number:

RULE.--Divide the numerator by the denominator; the quotient will be the whole or mixed number.

How many units in 30/6?

WORK: 30 ÷ 6 = 5 _Ans._ 5.

There are as many units in 30 sixths as 6 is contained times in 30.

Reduce 75/4 to a mixed number.

WORK: 75 ÷ 4 = 18 + 3 _Ans._ 18-3/4.

In 75 fourths there are 18 units, and 3 fourths over, which equals 18-3/4.

To reduce a mixed number to an improper fraction:

RULE.--Multiply the whole number by the denominator of the fraction; add the numerator to the product, and write the sum over the denominator.

Reduce 18-3/4 to an improper fraction.

WORK: 18 × 4 = 72

72/4 + 3/4 = 75/4

_Ans._ 75/4.

In 18 are 72 fourths, plus the 3 fourths, equals 75 fourths.

To reduce two or more fractions to their least common denominator:

RULE.--Find the least common multiple of the given denominators for a common denominator. Then for each new numerator take such a part of this common denominator as the fraction is part of 1.

Reduce 1/2, 2/3 and 3/4 to their L. C. D.

WORK:

1 6
- = --
2 12

2 8
- = --
3 12

3 9
- = --
4 12

_Ans._ 6/12, 8/12 and 9/12.

The L. C. M. of the denominators 2, 3 and 4 is 12. Hence, 12 is the L. C. D. to which the given fractions can be reduced. Then to change 1/2 to 12ths, say, 1/2 of 12 is 6, and write it over 12; to change 2/3 to 12ths, say 2/3 of 12 is 8, and write it over 12; to change 3/4 to 12ths, say, 3/4 of 12 is 9, and write it over 12.

Fractions must be reduced to a common denominator to be added or subtracted.

ADDITION OF FRACTIONS

If two or more fractions have the same denominator, their sum is obtained by adding the numerators.

WORK: 1 4 5 1 + 4 + 5 10 3
- + - + - = --------- = -- = 1-
7 7 7 7 7 7

If the fractions have different denominators, we must first express them as equivalent fractions with the same denominator.

EXAMPLE 1: Find the value of

1 3 5 2
- + - + -- + -
9 7 21 3

The lowest common multiple is 63. The several denominators, when divided into 63, give 7, 9, 3, 21 respectively, for quotients. Therefore, we multiply the numerators and denominators of the fractions by 7, 9, 3, 21, and add the numerators to obtain the required sum. The result must be reduced to a mixed number or to lower terms, if necessary.

WORK:

1 3 5 2
- + - + -- + -
9 7 21 3

7 + 27 + 15 + 42
= ----------------
63

= 91/63 = 1-28/63 = 1-4/9 _Ans._

In adding mixed numbers, first add the whole numbers, then the fractions, finally adding the two results.

EXAMPLE 2: Add together 3-1/8 + 7/24 + 7-11/15 + 4-3/20. Given expression:

1 7 11 3
= 3 + 7 + 4 + - + -- + -- + --
8 24 15 20

15 + 35 + 88 + 18
= 14 + -----------------
120

156 36 3
= 14 + --- = 14 + 1--- = 15-- _Ans._
120 120 10

SUBTRACTION OF FRACTIONS

The principle is the same as in addition. Reduce the fractions, if they have different denominators, to a common denominator, and then take the difference of the numerators. In the case of mixed numbers, subtract the whole numbers and the fractions separately.

EXAMPLE 1: Take 4-5/21 from 6-3/7.

3 5 3 5
6- - 4-- = 6 - 4 + - - --
7 21 7 21

9 - 5
= 2 + -----
21

4 4
= 2 + -- = 2-- _Ans._
21 21

If the fractional part of the number to be subtracted be greater than the fractional part of the other number, we proceed as follows:

EXAMPLE 2: From 7-4/15 take 4-11/25.

4 11 4 11
7-- - 4-- = 7 - 4 + -- - --
15 25 15 25

20 - 33
= 3 + -------
75

75 + 20 - 33
= 2 + ------------
75

62 62
= 2 + -- = 2-- _Ans._
75 75

EXAMPLE 3: Simplify 3-2/9 + 4-5/7 - 5-13/21 + 2/35 - 1-14/15. Given expression:

2 5 13 2 14
= 3 + 4 - 5 - 1 + - + - - -- + -- - --
9 7 21 35 15

70 + 225 - 195 + 18 - 294
= 1 + -------------------------
315

313 - 489[15]
= 1 + ---------
315

628 - 489 139
= --------- = --- _Ans._
315 315

[15] Obtained by adding all the numerators with + before them, and
then all those with - before them.

MULTIPLICATION OF FRACTIONS

(i) When the multiplier is a whole number. This, as in the case of whole numbers, means that we have to find the sum of a given number of repetitions of the fraction.

EXAMPLE 1:

7 7 7 7 7 28
- × 4 means - + - + - + -, _i.e._, --; or
9 9 9 9 9 9

7 × 4
-----
9

Hence, to multiply a fraction by a whole number, simply multiply the numerator by that number.

Since the multiplier thus becomes a factor of the numerator, we cancel any common factors contained in the multiplier and the denominator; and this may be done before we perform the actual multiplication:

EXAMPLE 2: Multiply 19/46 by 69.

19 19 × 69 19 × 3 57
-- × 69 = ------- + ------ (cancelling 23), = -- = 28-1/2 _Ans._
46 46 2 2

It follows that if the multiplier be itself a factor of the denominator, we may, to multiply a fraction by a whole number, divide the denominator by that number.

(ii) When the multiplier is a fraction.

EXAMPLE: In performing the operation 7 × 9, it is plain that we do to 7 what we do to a unit to obtain 9. Similarly, 3/5 × 4/11 may be looked upon as doing to 3/5 what we do to the unit to obtain 4/11.

Now, to obtain 4/11 from the unit, we must divide the unit into 11 equal parts and take 4 of them.

Therefore, to find the value of 3/5 × 4/11 we must divide 3/5 into 11 equal parts and take 4 of them.

But 3/5 = 33/55 = 3/55 × 11, so that, the eleventh part of 3/5 is 3/55; and, if we take 4 of these parts, we get 3/55 × 4 or 12/55.

3 4 12
Thus, - × -- = --. Now 12 = 3 × 4, and 55 = 5 × 11.
5 11 55

Hence we have the following rule: To multiply two fractions together, multiply the numerators for a new numerator and the denominators for a new denominator.

As in Example 2 the work is shortened if we cancel common factors from the numerators and denominators.

EXAMPLE: Multiply 22/91 by 13/77.

2
2̸2̸ × 1̸3̸ 2
The product = -------- = -- _Ans._
9̸1̸ × 7̸7̸ 49
7 7

Here, the 22 of the numerator and the 77 of the denominator contain a common factor, 11. Therefore, we cross out the 22 and write 2 above it, and cross out the 77 and write 7 under it. Similarly, we cancel the factor 13 from 13 and 91. There is now 2 left for numerator and 7 × 7 for denominator.

To multiply more than two fractions together, we proceed in the same way.

In multiplication of fractions, mixed numbers must first be expressed as improper fractions.

EXAMPLE: Simplify 5-1/7 × 11/27 × 1-11/24.

3̸ 5
3̸6̸ 11 3̸5̸ 55 1
Given expression = -- × -- × -- = -- = 3-- _Ans._
7̸ 2̸7̸ 2̸4̸ 18 18
9 2

DIVISION OF FRACTIONS

(i) When the divisor is a whole number. Suppose we have to divide 7/9 by 4.

We know 7/9 = 28/36. This fraction means that the unit is divided into 36 equal parts, and 28 of the parts taken. If we divide the 28 parts by 4, we get 7 of them--_i.e._ 7/36. Hence 7/9 ÷ 4 = 7/36.

Therefore, to divide a fraction by a whole number, we multiply the denominator by that number.

In the same way as already explained for multiplication, we cancel any common factors contained in the divisor and the numerator. Hence, if the numerator be exactly divisible by the divisor, we may divide a fraction by a whole number by dividing the numerator by that number.

EXAMPLE 1:

3
27 2̸7̸ 3
-- ÷ 18 = ------- = -- _Ans._
31 31 × 1̸8̸ 62
2

EXAMPLE 2:

36 4
-- ÷ 9 = -- _Ans._
41 41

(ii) When the divisor is a fraction.

In the operation 24 ÷ 3, we have to find the number which, when multiplied by 3, will give 24. Similarly, to find the value of 3/7 ÷ 5/9 we have to find the fraction which, when multiplied by 5/9, will give 3/7.

3 × 9
But ----- is the fraction which gives 3/7 when multiplied by 5/9.
7 × 5

3 5 3 × 9
Therefore, - ÷ - = -----.
7 9 7 × 5

Hence, to divide by a fraction, invert the divisor and multiply.

As in multiplication, mixed numbers must first be reduced to improper fractions.

EXAMPLE 3: Divide 3-1/14 by 5-5/42.

3
1 5 43 215 4̸3̸ 4̸2̸ 3
3-- ÷ 5-- = -- ÷ --- = -- × --- = - _Ans._
14 42 14 42 1̸4̸ 2̸1̸5̸ 5
5

DECIMAL FRACTIONS

Differ in form from common fractions, in not having a written denominator; and from whole numbers, by having the decimal point (.) prefixed; which also separates the integral part from the decimal. The word decimal is derived from the Latin word _decem_, which signifies ten. The denominator of a decimal is always 10, or some power of 10, as 100, 1000, etc.

A Complex Decimal is a decimal with a common fraction at the right, as, .12-1/2.

A Mixed Decimal is a whole number with a decimal fraction to its right, as, 34.5.

The denominations of United States money are based on the decimal system--the dollar occupying the unit’s place, the dime the tenth’s place, the cent the hundredth’s place, and the mill the thousandth’s place.

_The rules given for addition, subtraction, and so on, also apply to decimals._

ADDITION IN DECIMALS

EXAMPLE: 27.295 + .0287 + 591.68 + 9.1846.

27.295
.0287
591.68
9.1846
--------
628.1883 _Ans._

Write the numbers so that the same powers of 10 come under one another, or, what is the same thing, write the numbers so that the decimal points come under one another. Then, adding the ten-thousandths first, 6, 13, carry 1, etc.

SUBTRACTION IN DECIMALS

EXAMPLE: Subtract .07295 from 21.651.

21.651
.07295
--------
21.57805 _Ans._

Write the first number under the second, so that the point comes under the point. Remember that we may consider there are 0’s above the 9 and 5, since in 21.651 there are no ten-thousandths and no hundred-thousandths.

Say, mentally 5 and 5 make 10, carry 1. 10 and 0 make 10, carry 1. 3 and 8 make 11, carry 1, etc.

MULTIPLICATION IN DECIMALS

RULE.--Multiply as in whole numbers, and point off from the right of the product as many places as there are decimal places in both multiplier and multiplicand--prefixing ciphers if necessary.

EXAMPLE 1: Multiply 87.432 by 564.

87.432
564
---------
43716.0
5245.92
349.728
---------
49311.648 _Ans._

Place the multiplier so that its unit’s digit comes under the right-hand digit of the multiplicand. Then place the first figure of each product underneath the multiplying digit. The decimal point of the answer will then be directly under the decimal point of the multiplicand.

EXAMPLE 2: Multiply 31.56 by 5.49.

31.56
5.49
--------
157.80
12.624
2.8404
---------
173.2644 _Ans._

As before, place the unit’s figure of the multiplier--that is, the 5--under the right-hand digit of 31.56, and proceed as above.

Note.--The number of decimal places in the product will always be equal to the sum of the number of decimal places in the multiplier and the multiplicand. Thus, in Example 2, there are two places of decimals (_i.e._ two figures to the right of the point) in 31.56, and two places of decimals in 5.49; and we found 2 + 2 = 4 places in the product 173.2644.

To multiply a decimal by 10, 100, etc.

Rule.--Remove the (.) as many places to the right as there are ciphers in the multiplier.

Work: 8.75 × 10 = 87.5
8.75 × 100 = 875.
8.75 × 1000 = 8750.

DIVISION OF DECIMALS

RULE.--Divide as in whole numbers, annexing ciphers to the dividend, if necessary; then point off from the right of the quotient as many places as the decimal places in the dividend exceed those in the divisor--prefixing ciphers if necessary.

(a) Division of a decimal by a whole number.

EXAMPLE 1: Divide 18.2754 by 4.

4)18.2754
--------
4.56885

We divide 4 into 18 (units) and have 4 (units) quotient and 3 units remainder. Since the 4 is the unit’s figure of the quotient, we write the decimal point immediately after it. Then, the 2 units remainder and the 2 tenths of the dividend make 22 tenths to be divided by 4, and so on. Having reached the 4 (ten-thousandths) of the dividend, we find 8 (ten-thousandths) quotient and 2 remainder. This remainder is 20 hundred-thousandths, which when divided by 4 gives 5 (hundred-thousandths) and no further remainder.

EXAMPLE 2: Divide 18.2758 by 11.

11)18.2758
--------
1.66143636

Here we find the digits 3, 6 repeated indefinitely in the quotient. Decimals of this sort will be fully considered later.

EXAMPLE 3: Divide 354.43 by 184.

184)354.43(1.92625 _Ans._
----
1704
----
483
---
1150[16]
----
460
---
920
---

Here we find the first figure of the quotient is obtained by dividing 184 into 354 units. Having now reached the decimal point in the dividend we also put the decimal point in the answer, and go on as before.

[16] At this stage there is a remainder 115 hundredths. We bring
down 0 from the dividend, and obtain 1150 thousandths, etc.

(b) Division of a decimal.

EXAMPLE 4: Divide 10.6603 by 7.85.

Thus:

785)1066.03(1.358 _Ans._
-----
2810
----
4553
----
6280
----

Here 7.85 is 785 hundredths, and 10.6603 is 1066.03 hundredths; so that the required quotient is obtained by dividing 1066.03 by 785.

Therefore, to divide by a decimal, move the point as many places to the right as will make the divisor a whole number; move the point in the dividend the same number of places to the right. Then proceed as in Example 3.

EXAMPLE 5: Divide 176.4 by .00012.

12)17640000
---------
_Ans._ 1470000

Here, to make the divisor a whole number, we have to move the point 5 places. Therefore we also move the point 5 places to the right in the dividend, first writing enough 0’s after the 176.4 to enable us to do so.

To divide a decimal by 10, 100, etc.

RULE.--Remove the (.) as many places to the left as there are ciphers in the divisor.

WORK:

62.5 ÷ 10 = 6.25
62.5 ÷ 100 = .625
62.5 ÷ 1000 = .0625

Expression of decimal fractions as common fractions.

EXAMPLE: Express 5.375 as a common fraction.

.375 = 375 thousandths.

375 3
Therefore 5.375 = 5---- = 5- _Ans._
1000 8

RULE.--_Take the digits of the decimal for numerator; for the denominator put down 1 followed by as many ciphers as there are digits in the decimal. Reduce this fraction to its lowest terms._

Expression of common fractions as decimals.

We have seen that a common fraction represents the quotient of the numerator divided by the denominator. Therefore, to convert a common fraction to a decimal fraction, we divide the numerator by the denominator.

EXAMPLE: Express 3/32 as a decimal.

4) 3.0
---------
8) .75
---------
.09375 _Ans._

It will be found in many cases that there is always a remainder, so that the quotient can be continued indefinitely.

CIRCULATING DECIMALS

The learner has already discovered that some common fractions cannot be changed to exact decimal fractions, as--

1/3 =.33333 on to infinity.
2/3 =.66666 on to infinity.
7/33 =.212121, etc.

These decimals are known as _Circulates_, _Recurring_ or _Circulating_ decimals.

The part which recurs is called the _Repetend_.

This is marked by putting a dot over the first and last figures of it. For instance, if we write the 21 in the last case above, this way: 2̊1̊, it indicates that, if written out, the result would be 21212121, etc., on to infinity.

Where a circulating decimal occurs in work, it is best to reduce it to a common fraction. If need be, it may be expressed in the result as a circulate to any number of decimal places.

To change a pure circulate to a common fraction.

RULE.--Omit the (.) and write the figures of the repetend for the numerator, and as many 9’s for the denominator as there are places in the repetend.

EXAMPLES: Change the pure circulates .3̊, .2̊7̊, .1̊42857̊, to common fractions.

(3 1)
.3̊, (- = -) _Ans._ 1/3.
(9 3)

(27 3)
.2̊7̊, (-- = --) _Ans._ 3/11
(99 11)

(142857 1)
.1̊42857̊, (------ = -) _Ans._ 1/7
(999999 7)

To change a mixed circulate to a common fraction.

RULE.--From the whole decimal subtract the finite part, and make the remainder the numerator. For the denominator, write as many 9’s as there are figures in the repetend, and annex as many 0’s as there are finite places.

EXAMPLE: Change the mixed circulates .16̊ and .416̊ to common fractions.

15 1
16 - 1 = 15, --- = -. _Ans._ 1/6.
90 6

375 5
416 - 41 = 375, --- = -- _Ans._ 5/12.
900 12

To add, subtract, multiply and divide circulates, reduce them to common fractions, then apply the respective rules.

SHORT METHODS IN MERCHANDISING

When one of the numbers is an _aliquot part_ of 100, the process of multiplication and division can often be very much shortened, as shown below.

Find cost of 27 yards of goods at 16-2/3c ($1/6) per yard. At $1 per yard, 27 yards cost $27; at $1/6, (27 ÷ 6), $4-1/2. _Ans._ $4-1/2.

Find cost of a bale of cotton, 528 pounds at 8-1/3c ($1/12) per pound. At $1 per pound, 528 pounds cost $528; at $1/12 (528 ÷ 12) $44. _Ans._ $44.

Find cost of 1845 pounds of iron, at 3-1/3c ($1/30) per pound. Take 1/30 of 1845, since 3-1/3c is 1/30 of $1. (1845 ÷ 30 = 61-1/2). _Ans._ $61-1/2.

Find cost of 16 pounds of butter at 37-1/2c ($3/8) per pound. Here we take 3/8 of 16. Say 1/8 of 16 is 2, and 3/8 is (2 × 3) 6. Or say 3 times 16 is 48, and 1/8 of 48 is 6. _Ans._ $6.

Find cost of 17-1/2 bushels of apples at 75c ($3/4) per bushel. The shortest way to find 3/4 of $17.50 is to diminish it by 1/4 of itself.

4)17.50 at $1
------
4.37-1/2 at $1/4
13.12-1/2 at $3/4

_Ans._ $13.12-1/2.

At 6-1/4c per pound how much sugar will $5 buy? As 6-1/4c is 1/16 of $1, evidently each dollar will buy 16 pounds. _Ans._ 80 pounds.

In multiplying by a fraction, write the quantity in a line with the numerator and cancel common factors.

Find cost of 72 yards of carpet, at 87-1/2c ($7/8) a yard. Cancel 8, also 72 and write 9 instead. _Ans._ $63.

7 × 7̸2̸ 9 = 63
-
8̸

Of 28 pounds of coffee, at 18-3/4c ($3/16) per pound. Cancel 28 and 16, write 7 and 4. _Ans._ $5-1/4.

3 × 2̸8̸ 7 21
---- = -- or 5-1/4
1̸6̸ 4 4

At 66-2/3c ($2/3) per bushel, how many bushel of wheat will $34 buy? _Ans._ 51 bushel.

3 × 3̸4̸ 17 = 51
-
2̸

In division, invert terms of fraction.

How much syrup, at 41-2/3c ($5/12) per gallon can be bought for $15? _Ans._ 36 gallons.

12 × 1̸5̸ 3 = 36
--
5̸

TABLE OF ALIQUOT PARTS OF 100

3-1/3 is 1/30
6-1/4 is 1/16
8-1/3 is 1/12
12-1/2 is 1/8
16-2/3 is 1/6
18-3/4 is 3/16
20 is 1/5
25 is 1/4
31-1/4 is 5/16
33-1/3 is 1/3
37-1/2 is 3/8
40 is 2/5
41-2/3 is 5/12
43-3/4 is 7/16
50 is 1/2
56-1/4 is 9/16
58-1/3 is 7/12
60 is 3/5
62-1/2 is 5/8
66-2/3 is 2/3
68-3/4 is 11/16
75 is 3/4
80 is 4/5
81-1/4 is 13/16
83-1/3 is 5/6
87-1/2 is 7/8
91-2/3 is 11/12
93-3/4 is 15/16

This table embodies all the aliquot parts of 100 and their equivalent fractions which are generally used in practical calculations.

PROBLEMS IN GRAIN, STOCK, COTTON, COAL, HAY, LUMBER, ETC.

To find the value of articles sold by the unit, hundred or thousand.

RULE.--Multiply the quantity by the price, or vice versa, and point off the proper number of decimal places in the result.

Find the cost of a bale (518 pounds) of cotton at 7-3/8c per pound.

518 × .07 = 36.26
„ .00-3/8 = 1.94-1/4
---------
_Ans._ $38.20-1/4

At 7c (.07) per pound, 518 pounds cost $36.26; at 3/8c, $1.94-1/4. For 3/8 of 518, multiply by 3, and divide product by 8.

Find cost of a lot of hogs, weighing 8740 pounds, at $4.35 per hundredweight.

87.40
4.35
--------
380.1900