Can you factor this? 6x2 + 11x + 3
Sure, you can do it algebraically, but you can also channel your inner ancient Greek and solve it using geometry. See drawing at bottom.
The answer is (3x + 1)(2x+3).
Ratios are comparisons between two like units. For instance, in this class, the ratio of males to females is 11:14 (that's in colon notation. There are no units because they cancel out.)
Rates are fractions that compare two unlike units. Ex: 30 miles/hour, 3 dollars/pound, etc.
If they can be reduced, reduce them. Don't leave any radicals in the denominator. Write them as fractions!
Proportions are equations where a rate equals a rate, or a ratio equals a ratio. You can cross-multiply to get your answer with proportions. Ex: 30 miles/hour = x miles/5 hours. (x = 150)
Practice problems:
If 8,218 schools have 270,272 girls, what is the rate of girls per school? Round answers to the nearest tenth.
Answer: 270,272 girls/ 8218 schools = 32.9 girls/school.
3,122 people smoke cigarettes. 1,615 of those people quit smoking. What is the ratio of those that smoke to those that don't?
Answer: 3,122 - 1,615 = 1,507 smokers left. The ratio of those that smoke to those that don't is 1,507/1,615 or 0.9 or (best answer) 9:10.
A triangle's side proportions are 4:6:9, and its perimeter is 190 inches. What are the lengths of its sides?
Answer: 4 + 6 + 9 obviously doesn't equal 190 inches, so look at it like 4x + 6x + 9x = 190. x=10, so the side lengths are 40, 60, and 90.
A triangle's side proportions are 3:2:8 and its perimeter is 72. What are its side lengths?
Answer: 3x + 2x + 8x = 72. 13x = 72, and x = 5.54. So 3(5.54), 2(5.54), and 8(5.54) are the side lengths.
The replica of a statue is 10 inches tall. The real statue is 10 feet tall. What is the ratio of the replica's height to the statue's?
Answer: Convert 10 feet to inches. So 10 in/10 feet = 10 in/120 in. Simplifies to 1/12.
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