By Susan Montgomery
The final ten years have visible a few major advances in Hopf algebras. the simplest identified is the advent of quantum teams, that are Hopf algebras that arose in mathematical physics and also have connections to many components of arithmetic. additionally, numerous conjectures of Kaplansky were solved, the main awesome of that is one of those Lagrange's theorem for Hopf algebras. paintings on activities of Hopf algebras has unified prior effects on workforce activities, activities of Lie algebras, and graded algebras. This ebook brings jointly lots of those contemporary advancements from the point of view of the algebraic constitution of Hopf algebras and their activities and coactions. Quantum teams are handled as an immense instance, instead of as an lead to themselves. the 2 introductory chapters overview definitions and simple evidence; another way, lots of the fabric has no longer formerly seemed in publication shape. offering an obtainable creation to Hopf algebras, this e-book could make a great graduate textbook for a direction in Hopf algebras or an advent to quantum teams.
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15. 5% 16. 180% 1 17. 1% or 11 % 9 miL28711_ch00_001-048 9/20/06 28 06:52 AM BA Page 28 CONFIRMING PAGES Chapter R Reference Example 7 Applying Percentages Shauna received a raise, so now her new salary is 105% of her old salary. Find Shauna’s new salary if her old salary was $36,000 per year. Solution: The new salary is 105% of $36,000. 05 ϫ 36,000 ϭ 37,800 The new salary is $37,800 per year. Skill Practice 18. If 60% of the 35 students in an Algebra class are women, find the number of women. Example 8 Applying Percentages A couple leaves a 15% tip for a $32 dinner.
25% 4 5 5 ϭ ϫ 100% 3 3 7 8 The value 254 can be written in decimal form by dividing 25 by 4. Multiply by 100%. ϭ 100 5 ϫ % 3 1 Write the whole number as a fraction. ϭ 500 % 3 Multiply fractions. 6% 3 Skill Practice 15. Simplify. The value 500 3 can be written in decimal form by dividing 500 by 3. Convert the fraction to a percent. 16. 9 5 17. 1 9 6. Applications of Percents Many applications involving percents involve finding a percent of some base number. For example, suppose a textbook is discounted 25%.
Skill Practice Answers 3 10 12. 56% 9. 241 200 13. 436% 10. 1 4 14. 2% 11. 2 5 b. 1 16 c. 3 CONFIRMING PAGES 27 Decimals and Percents Solution: a. 2 2 ϭ ϫ 100% 5 5 ϭ 100 2 ϫ % 5 1 Multiply by 100%. Write the whole number as a fraction. 20 2 100 ϭ ϫ % 5 1 Multiply fractions. 1 ϭ b. 40 % or 40% 1 1 1 ϭ ϫ 100% 16 16 ϭ 1 100 ϫ % 16 1 Simplify. Multiply by 100%. Write the whole number as a fraction. 25 1 100 ϭ ϫ % 16 1 Multiply the fractions. 4 c. 25% 4 5 5 ϭ ϫ 100% 3 3 7 8 The value 254 can be written in decimal form by dividing 25 by 4.