Questions and Answers #27 Ratios, Rates, Proportions

Questions and Answers Sheet 27 Ratios, Rates, Proportions Question #1 Which ratio forms a proportion with 16 20 ? Answer: You may simplify the ratio-fraction: 4 4 4 = 4 5 5 = Answer: Any ratio that has a proportion of 4×𝑘 5×𝑘 , where k is a whole number, the simplest being Question #2 How do you determine whether the ratios are equivalent: 12 5 and 36 ? 15 Answer: The ratio 12 5 is in simplest form. That is no other factor except 1 will divide into them. 36 If we now simplify the ratio 15 36 and 15 both have a common factor of 3 which we can divide them by. 36 36÷3 12 ⇒ = = 15 15÷3 5 Usually written using cancelling as follows. 12 36 36 12 = = 5 15 15 5 36 Since 15 reduces to the same value 12 5 then the ratios are equivalent Question #3 How do you determine whether the ratios are equivalent: Answer: 3 6 : ∷ 7: 8 4 7 Solution: 3 6 : 4 7 3×7 6×4 ⇔ or : 4×7 7×4 21 24 : 28 28 or 21:24 or (3 × 7): (3 × 8) or 7:8 . Question #4 How do you divide $95 in a ratio of 2:3 ? 3 4 and 6 7 ? 4 5 Answer: sum the 2 parts of the ratio. That is: 2+3=5 To find 1 part, divide $95 by 5 95 ⇒ 95 ÷ 5 = = 19 5 Hence 2 parts = 2 × $19 = $38 and 3 parts = 3 × $19 = $57 Thus $95 in the ratio 2:3 is $38 to $57 Question #5 What must n be if the ratio 6:7 is equivalent to 36:n? Answer: Answer: n=42 Explanation: 36 You first do which equals 6. This is the constant that you are going to multiply by 7. Since you have to 6 multiply by both sides on a ratio, you do 6 × 6 and 6 × 7. This equals 42. The final ratios are: 6: 7 and 36: 42 Question #6 What ratio 15 45 expressed in simplest form? Answer: You may see that 45 = 3 × 15 Explanation: In that case 15 45 may be expressed as: 1 15 1 = 3  15 3 If not, you can factorize both numbers into 's: = 1 3  5 1 = 3 3  5 3 Question #7 How do you write the ratios 1 1 : 2 5 in simplest form? Answer: One of the rules of ratios is that there should be no fractions, only whole numbers. 1 1 We have : 2 5 One method is to change them to a common denominator. However, let's multiply by that LCM instead, because this will allow us to cancel the denominators altogether immediately 10 1 1 10 × : × 1 2 5 5 1 10 1 1 10  :  1 1 2 5 Answer: 5:2 2 Question #8 You are running 1 10 mile sprints in gym class this week. How many miles will you run if you complete 3 sprints? Answer: How much are you running: 1 10 miles per sprint How many sprints are you completing: 3 sprints How many miles do you run in total: number of sprints × number of miles per sprint = the number of miles you'll run Therefore: 1 3sprints ⋅ 10 miles 1sprint = 3 10 Therefore, you'll run miles 3 10 miles if you complete 3 sprints. Question #9 The ratio of two people's weight remains the same, whether measured in pounds or kilograms? (1𝑘𝑔 = 2.2𝑙𝑏) Answer: If I am twice as heavy as you the ratio of our weights is 2:1 whether you weigh me in pounds, kilograms or anything else. If I am 100 kg and you are 50 kg, the ratio is 100:50=2:1 If I am 220 lb and you are 110 lb, the ratio is 220:110=2:1 (100kg=220lb, 50kg=110lb) Question #10 If 75% of x is 1500 then x is Answer: Step 1 Given, 75% of x is 1500 then We have to find x We know that, 75% of x is 1500 means Step 2 75% × 𝑥 = 1500 75 ( ) 𝑥 = 1500 100 75𝑥 = 1500 × 100 x=150000/75 x=2000 Question #11 y=|x+3|-1 Answer: Step 1 Solution:-The given equation is y=| x+3|-1 Step 2 y=|x+3|1 𝑦 = |(𝑥 + 3)| − 1 ⇒ −𝑥 − 3 − 1 ⇒ −𝑥 − 4; 𝑖𝑓 𝑥 + 3 < 0 ⇒ 𝑥 < −3 (𝑥 + 3) − 1 = 𝑥 + 3 − 1 = 𝑥 + 2; 𝑖𝑓 𝑥 + 3 ≥ 0 ⇒ 𝑥 ≥ −3| a) vertex x=(-3,-1) b) Axis of symmetry x=-3 c) zeros x=-2 and x=-4 d) y=| x+3| -1 lies on 1𝑠𝑡 , 2𝑛𝑑 and 3𝑟𝑑 quadrants and vertex lies on 3𝑟𝑑 quadrant e) range = (−∞, ∞) f) y-intercepts =2 f) slope for 𝑥 ≥ 3 y=x+2 compare with y=mx+c slope m=1 for x < 3 y=-x-4 compare with y=m'x slope m'=-1 g) Domain 𝑥 = [−1, ∞) Question #12 To calculate: The unit rate. Given information: The following fraction: 26 𝑝𝑜𝑖𝑛𝑡𝑠 3 𝑞𝑢𝑎𝑟𝑡𝑒𝑟𝑠 Answer: Calculation: To calculate unit rate, the denominator should be 1. 26 𝑝𝑜𝑖𝑛𝑡𝑠 3 𝑞𝑢𝑎𝑟𝑡𝑒𝑟𝑠 26 𝑝𝑜𝑖𝑛𝑡𝑠÷3 ⇒ ⇒ 3 𝑞𝑢𝑎𝑟𝑡𝑒𝑟𝑠÷3 8.7 𝑝𝑜𝑖𝑛𝑡𝑠 1 𝑞𝑢𝑎𝑟𝑡𝑒𝑟 [Divide numerator and denominator by 3] [Simplify] Therefore, the unit rate is 8.7 points per quarter.