The Enormous State University History Department offers three coursesAncient, Medieval, and Modern Historyand the chairperson is trying to decide how many sections of each to offer this semester. The department is allowed to offer45sectionstotal, there are4,700students who would like to take a course, and there are58professors to teach them. Sections of Ancient History have 100 students each, sections of Medieval History hold 50 students each, and sections of Modern History have 200 students each. Modern History sections are taught by a team of two professors, while Ancient and Medieval History need only one professor per section. How many sections of each course should the chair schedule in order to offer all the sections that they are allowed, accommodate all of the students, and give one teaching assignment to each professor? HINT [See Example 2.] (Assume each section has the maximum number of students allowed.)

Ancient History ... sections

Medieval History... sections

Modern History... sections

Suppose in one year, total revenues from digital sales of pop/rock, tropical (salsa/merengue/cumbia/bachata), and urban (reggaeton) Latin music in a certain country amounted to $24million. Pop/rock music brought in twice as much as the other two categories combined and $9million more than tropical music. How much revenue was earned from digital sales in each of the three categories?

pop/rock music $... million

tropical music $... million

urban Latin music $... million

A bagel store orders cream cheese from three suppliers, Cheesy Cream Corp. (CCC), Super Smooth & Sons (SSS), and Bagel's Best Friend Co. (BBF). One month, the total order of cheese came to 100 tons. (The store does booming trade.) The costs were $80, $50 and $65 per ton from the three suppliers respectively, with the total cost amounting to $6,575. Given that the store ordered the same amount from CCC and BBF, how many tons of cream cheese were ordered from each supplier?

Cheesy Cream Corp.... tons

Super Smooth & Sons... tons

Bagel's Best Friend Co.... tons

Halmar the Great has boasted to his hordes of followers that many a notorious villain has fallen to his awesome sword: His total of700victims consists of evil sorcerers, trolls, and orcs. These he has slain with a total of800mighty thrusts of his sword; evil sorcerers and trolls each requiring two thrusts (to the chest) and orcs each requiring one thrust (to the neck). When asked about the number of trolls he has slain, he replies, "I, the mighty Halmar, despise trollsninetimes as much as I despise evil sorcerers. Accordingly,ninetimes as many trolls as evil sorcerers have fallen to my sword!" How many of each type of villain has he slain?

sorcerers

trolls

orcs

The Fancy French Perfume Company recently had its secret formula divulged. It turned out that it was using, as the three ingredients, rose oil, oil of fermented prunes, and alcohol. Moreover, each14-ounceecono-size bottle contained4more ounces of alcohol than oil of fermented prunes, while the amount of alcohol was equal to the combined volume of the other two ingredients. How much of each ingredient did the company use in an econo-size bottle?

rose oil... oz

oil of fermented prunes... oz

alcohol ...oz