University:
Georgia Southern University
Course:
MATH 2242 | Calculus II
Academic year:

2024
Views:

375

Pages:

4
Author:

Jarrett Brock

Question 1 40 is what percent of 50? Answer 1 To calculate the percentage: 40 50 = 𝑥 100 . 50 40 = 100 x . Cross-multiply: 40 × 100 = 50 × 𝑥 ⟹ 𝑥 = 4000 50 = 80 % . 40×100=50×x⟹x= 50 4000 =80%. Answer 2 Divide directly: 40 50 = 0.8. 50 40 =0.8. Convert to a percentage: 0.8 × 100 = 80 % . 0.8×100=80%. Final Answer: 40 is 80% of 50. Question 2 Solve the proportion 𝑥 2 = 3 𝑥 + 1 2 x = x+1 3 . Answer 1 Using the property of proportions (cross-multiplication): 𝑥 ( 𝑥 + 1 ) = 2 × 3. x(x+1)=2×3. Simplify: 𝑥 2 + 𝑥 = 6. x 2 +x=6. Move all terms to one side: 𝑥 2 + 𝑥 − 6 = 0. x 2 +x−6=0. Factorize: ( 𝑥 + 3 ) ( 𝑥 − 2 ) = 0. (x+3)(x−2)=0. Solve: 𝑥 = − 3 or 𝑥 = 2. x=−3orx=2. Final Answer: The solution is 𝑥 = − 3 x=−3 or 𝑥 = 2 x=2. Answer 2 Using the same equation 𝑥 2 = 3 𝑥 + 1 2 x = x+1 3 , expand the denominators to find a common base: Multiply both sides by 2 ( 𝑥 + 1 ) 2(x+1): 𝑥 ( 𝑥 + 1 ) = 6. x(x+1)=6. Follow the same steps as Answer 1 to simplify and solve: 𝑥 2 + 𝑥 − 6 = 0 ⟹ 𝑥 = − 3 or 𝑥 = 2. x 2 +x−6=0⟹x=−3orx=2. Question 3 The difference between ratios and rates. Answer 1 Ratio: A ratio compares two numbers, showing how many times one value contains or is contained in the other. Example: Ram had 10 cookies, and 3 of them were expired. The ratio of expired cookies to the total is: 3 10 . 10 3 . Rate: A rate compares two quantities of different units. Example: A person blinks 360 times in a minute. The rate of blinking is: 360 blinks 60 seconds = 6 blinks per second . 60seconds 360blinks =6blinks per second. Answer 2 A ratio compares parts of the same category. A rate compares different units like miles per hour or cost per item. Question 4 The sum of three consecutive odd integers equals 24 less than four times the third integer. Answer 1 Let the three consecutive odd integers be 𝑥 , 𝑥 + 2 , 𝑥 + 4 x,x+2,x+4. Step 1: Write the equation The sum of the first two equals 24 less than four times the third: 𝑥 + ( 𝑥 + 2 ) = 4 ( 𝑥 + 4 ) − 24. x+(x+2)=4(x+4)−24. Step 2: Simplify Combine like terms: 2 𝑥 + 2 = 4 𝑥 + 16 − 24. 2x+2=4x+16−24. Simplify further: 2 𝑥 + 2 = 4 𝑥 − 8. 2x+2=4x−8. Step 3: Solve for 𝑥 x Move terms: 2 𝑥 = 4 𝑥 − 10 ⟹ − 2 𝑥 = − 10. 2x=4x−10⟹−2x=−10. Divide: 𝑥 = 5. x=5. Thus, the integers are 5 , 7 , and 9 5,7,and9. Final Answer: The integers are 5, 7, and 9. Question 5 A real-life example of ratios. Answer A ratio represents the relationship between two quantities. Example: Ram has 10 cookies, and 3 out of 10 cookies were expired. The ratio of expired cookies to the total is: 3 10 . 10 3 .