Ratios, rates, proportions #2 - Questions and Answers

Question 1 What percent of 600 is 192? Answer 1 192 600 × 100 % = 32 % 600 192 ​ ×100%=32% Answer 2 192 is 32% of 600. Answer 3 Let 600 be 100% and 192 → ? Multiply 100% and 192, then divide by 600: 100 % × 192 600 = 19200 % 600 = 32 % . 600 100%×192 ​ = 600 19200% ​ =32%. Question 2 A small drugstore orders copies of a certain magazine for its magazine rack each week. Let 𝑋 X=demand for the magazine, with pmf 𝑥 x 1 2 3 4 5 6 𝑝 ( 𝑥 ) p(x) 1 15 15 1 ​ 2 15 15 2 ​ 3 15 15 3 ​ 4 15 15 4 ​ 3 15 15 3 ​ 2 15 15 2 ​ Suppose the store owner actually pays $1.00 for each copy of the magazine and the price to customers is $2.00. If magazines left at the end of the week have no salvage value, is it better to order three or four copies of the magazine? Answer 1 Given 𝑋 X: 1, 2, 3, 4, 5, 6 𝑃 ( 𝑥 ) P(x): 1 15 , 2 15 , 3 15 , 4 15 , 3 15 , 2 15 15 1 ​ , 15 2 ​ , 15 3 ​ , 15 4 ​ , 15 3 ​ , 15 2 ​ When three copies are ordered, the net revenue is ℎ ( 𝑥 ) = Total Revenue − Cost = 2 𝑥 − 3 h(x)=Total Revenue−Cost=2x−3 ℎ ( 𝑥 ) : − 1 , 1 , 3 , 3 , 3 , 3 h(x):−1,1,3,3,3,3 So 𝐸 ( ℎ ( 𝑥 ) ) E(h(x)): 𝐸 ( ℎ ( 𝑥 ) ) = ∑ 𝑥 ℎ ( 𝑥 ) 𝑝 ( 𝑥 ) = ( − 1 × 1 15 ) + ( 1 × 2 15 ) + ( 3 × 3 15 ) + ( 3 × 4 15 ) + ( 3 × 3 15 ) + ( 3 × 2 15 ) = 2.667. E(h(x))= x ∑ ​ h(x)p(x)=(−1× 15 1 ​ )+(1× 15 2 ​ )+(3× 15 3 ​ )+(3× 15 4 ​ )+(3× 15 3 ​ )+(3× 15 2 ​ )=2.667. When four copies are ordered, the net revenue is ℎ ( 𝑥 ) = Total Revenue − Cost = 2 𝑥 − 4 h(x)=Total Revenue−Cost=2x−4 ℎ ( 𝑥 ) : − 2 , 0 , 2 , 4 , 4 , 4 h(x):−2,0,2,4,4,4 So 𝐸 ( ℎ ( 𝑥 ) ) E(h(x)): 𝐸 ( ℎ ( 𝑥 ) ) = ∑ 𝑥 ℎ ( 𝑥 ) 𝑝 ( 𝑥 ) = ( − 2 × 1 15 ) + ( 0 × 2 15 ) + ( 2 × 3 15 ) + ( 4 × 4 15 ) + ( 4 × 3 15 ) + ( 4 × 2 15 ) = 2.93. E(h(x))= x ∑ ​ h(x)p(x)=(−2× 15 1 ​ )+(0× 15 2 ​ )+(2× 15 3 ​ )+(4× 15 4 ​ )+(4× 15 3 ​ )+(4× 15 2 ​ )=2.93. Conclusion: Since 𝐸 ( ℎ ( 𝑥 ) ) = 2.93 E(h(x))=2.93 for four copies is higher than 𝐸 ( ℎ ( 𝑥 ) ) = 2.667 E(h(x))=2.667 for three copies, it is better to order four copies. Answer 2 3 magazines: Total cost: 3 × 2 = 6 3×2=6. Net revenue for different demands is calculated, yielding an expected value: 𝐸 ( 𝑅 ) = 4.93. E(R)=4.93. 4 magazines: Total cost: 4 × 2 = 8 4×2=8. Net revenue for different demands gives: 𝐸 ( 𝑅 ) = 5.33. E(R)=5.33. Conclusion: Ordering 4 magazines is better because the expected revenue is higher. Question 3 Whether the ratios 15 36 36 15 ​ and 35 42 42 35 ​ are equivalent ratios. Answer Concept: Equivalent fractions are reduced to the same fraction in their simplest forms. Simplify 15 36 36 15 ​ : The HCF of 15 and 36 is 3: 15 36 = 5 12 . 36 15 ​ = 12 5 ​ . Simplify 35 42 42 35 ​ : The HCF of 35 and 42 is 7: 35 42 = 5 6 . 42 35 ​ = 6 5 ​ . Since 5 12 ≠ 5 6 12 5 ​  = 6 5 ​ , the ratios are not equivalent. Question 4 A piston-cylinder device initially contains 0.07 𝑚 3 0.07m 3 of nitrogen gas at 130 kPa and 18 0 ∘ 180 ∘ . The nitrogen is expanded to a pressure of 80 kPa isentropically. Determine the final temperature and boundary work. Answer 1 Final volume 𝑉 2 V 2 ​ : 𝑉 2 = 𝑉 1 ( 𝑃 1 𝑃 2 ) 1 𝛾 . V 2 ​ =V 1 ​ ( P 2 ​ P 1 ​ ​ ) γ 1 ​ . Work 𝑊 𝑏 W b ​ : 𝑊 𝑏 = 𝑃 2 𝑉 2 − 𝑃 1 𝑉 1 1 − 𝛾 . W b ​ = 1−γ P 2 ​ V 2 ​ −P 1 ​ V 1 ​ ​ . Question 5 Given: Khannouchi ran 26.2 miles in 2 hours, 5 minutes, and 38 seconds. a) Find speed in miles per hour. b) Find speed in miles per minute. c) Find speed in feet per minute. Answer Total time: 2 + 5 60 + 38 3600 = 2.094 2+ 60 5 ​ + 3600 38 ​ =2.094 hours. a) Speed (mph): 26.2 2.094 = 12.51   mph . 2.094 26.2 ​ =12.51mph. b) Speed (miles per minute): 12.51 60 = 0.21   miles/min . 60 12.51 ​ =0.21miles/min. c) Speed (feet per minute): 0.21   miles/min × 5280 = 1108.8   ft/min . 0.21miles/min×5280=1108.8ft/min.