Introduction:
Mathematics, often likened to a language, requires a blend of practice and comprehension. The realms of complex fractions and unit rates can certainly seem intricate and perplexing. Yet, there’s no need for trepidation. This blog post serves as your compass, guiding you through Lesson 2 on Complex Fractions and Unit Rates with comprehensive solutions in the answer key. Our mission is to furnish students with the essential tools for triumph in mathematical exams and assignments. Let’s embark on this mathematical voyage!
Problem 1: Deciphering Unit Rates
Solution: Consider the query: “What is the unit rate of 8 ounces of water in a 64-ounce jug?” To resolve this, establish the ratio of 8 ounces to 64 ounces – 8:64 or 8/64. Simplify the ratio to 1/8. Ergo, the unit rate for 8 ounces of water in a 64-ounce jug is 1/8.
Problem 2: Simplifying Complex Fractions
Solution: Unravel the complex fraction (1/2) / [(3/4) / (2/3)] by inverting and multiplying the divisor:
(1/2) * (2/3) / (3/4)
Further simplification yields (1/3), the simplified form of the given complex fraction.
Problem 3: Conversion Chronicles
Solution: Convert 9.6 miles/hour to feet/second via conversion factors – a bridge between units. A mile equals 5280 feet, and an hour equals 3600 seconds. Hence, the conversion factor is (5280/3600). The calculation unfolds as:
9.6 (miles/hour) * (5280/3600) * (1/3600) (hours/second) = 14.08 feet/second
Thus, the answer is 14.08 feet/second.
Problem 4: Proportional Puzzles
Solution: To determine the required flour quantity for baking 36 cookies, harness the power of proportions. The flour-to-cookies ratio remains consistent. Set up the proportion:
6 cups of flour / 24 cookies = x cups of flour / 36 cookies
Cross-multiplying and solving produce:
x = 9 cups of flour
Hence, 9 cups of flour are needed for 36 cookies.
Conclusion:
Conquering complex fractions and unit rates may appear challenging, yet it is far from insurmountable. Armed with this detailed answer key to Lesson 2, you possess the tools necessary for exam and assignment success. Remember, mastery arises from persistent practice. Approach problems deliberately, absorbing the solution methods provided herein. As you navigate this mathematical odyssey, recall that diligence and perseverance pave the way to becoming a math virtuoso. Best of luck in your academic endeavors!
