Question and Answers #20 Fractions

Questions and Answers Sheet 20 Fractions Question #1 Using the quadratic formula, find the roots of: 4𝑥 2 + 7𝑥 − 15 = 0 Answer can be left as fractions. 𝑥 = −𝑏±√𝑏 2 −4𝑎𝑐 2𝑎 Answer: Step 1 The quadratic formula is used when the roots of the quadratic equation cannot solved by the factorisation method. Step 2 Here on comparsion of the quadratic equation 4𝑥 2 + 7𝑥 − 15 = 0 with the standard quadratic equation 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 , it is observed that a is 4, b is 7 and c is -15. Therefore substitute these values in the equation 𝑥 = −𝑏±√𝑏 2 −4𝑎𝑐 2𝑎 to find the value of x. 𝑥= −𝑏 ± √𝑏 2 − 4𝑎𝑐 2𝑎 𝑥= −7 ± √72 − 4(4)(−15) 2(4) = −7 ± √49 + 240 8 = −7 ± √289 8 𝑥= −7 ± 17 8 𝑥= −7 + 17 8 = 5 4 𝑥= −7 − 17 8 = −3 Question #2 Solve the equation below with factoring. Separate multiple solutions with a comma any reduce and fractions. 11𝑥 − 31 = 𝑥 2 − 3 𝑥 =? Answer: Step 1 Given equation is, 11𝑥 − 31 = 𝑥 2 − 3 Find value of x by factoring Step 2 We have, 11𝑥 − 31 = 𝑥 2 − 3 Rewrite as, ⇒ 𝑥 2 − 3 = 11𝑥 − 31 ⇒ 𝑥 2 − 11𝑥 − 3 + 31 = 0 ⇒ 𝑥 2 − 11𝑥 + 28 = 0 (Equation in standard form) ⇒ 𝑥 2 − 7𝑥 − 4𝑥 + 28 = 0 ⇒ 𝑥(𝑥 − 7) − 4(𝑥 − 7) = 0 ⇒ (𝑥 − 4)(𝑥 − 7) = 0 (Factor completely) ⇒ 𝑥 − 4 = 0 or 𝑥 − 7 = 0 (Apply the zero product rule) ⇒ 𝑥 = 4 or 𝑥 = 7 Hence, 𝑥 = 4,7 Question #3 If n divides 𝑚 + 1 ; the fraction 𝑛 can always be written as a sum of two unit fractions. 𝑚 - True - False Answer: Answer: False Explanation: n divides 𝑚 + 1 implies 𝑚+1 𝑛 = 𝑚 𝑛 + 1 𝑛 the fraction Hence 𝑛 𝑚 𝑚 𝑛 written as sum of two unit fractions is not a fraction of 𝑚+1 𝑛 . Question #4 Solve by completing the square. 3𝑓 2 + 90𝑓 = 57 Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth. 𝑓 =? or 𝑓 =? Answer: Step 1 Given equation 3𝑓 2 + 90𝑓 − 57 = 0 We know that roots of equation 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 is By comparing with the given equation we get: 𝑎 = 3, 𝑏 = 90 and 𝑐 = −57 Step 2 So the roots are: ⇒ −90 ± √902 − 4 ⋅ 3 ⋅ −57 2⋅3 ⇒ −90 ± √8100 + 684 6 ⇒ −90 ± 12√61 6 ⇒ −15 ± 2√61 ⇒ −15 + 2√61 or −15 − 2√61 ≈ 0.6204 or −30.6204 Therefore the roots are 0.620 or −30.620. Question #5 Solve using matrices. 𝑥 + 2𝑦 = 13 4𝑥 − 𝑦 = 16 The solution is ?. Answer: Step 1 The given equations are: 𝑥 + 2𝑦 = 13 ......(1) 4𝑥 − 𝑦 = 16 ........(2) Step 2 By multiplying the equation (2) by two we get: 8𝑥 − 2𝑦 = 32 .....(3) Step 3 Let's add equation (1) and (3): 𝑥 + 2𝑦 = 13 ......(1) 8𝑥 − 2𝑦 = 32 ........(3) 9𝑥 = 45 −𝑏±√𝑏 2 −4𝑎𝑐 2𝑎 Step 4 Divide both side of the resultant equation by 9: 9𝑥 45 = 9 9 𝑥=5 Final answer The value of x is = 5 Question #6 How do you write 0.4 as a fraction? Answer: 0.4 = 4 10 or you can simplify it to 2 5 Question #7 How to change 1.25 into a fraction? Answer: To convert decimals to fractions just turn it into a percent first. 1.25 in percent is 125%. 125% is Question #8 What is 3.5 as a fraction? Answer: 3.5 = 3 1 2 As 3.5 is 3 5 10 and we simplify to 3 1 2 Question #9 How do you write 2.3 as a fraction? Answer: 2.3 = 2 + 0.3 2+ 3 3 23 =2 = 10 10 10 Question #10 Convert 0.777... to a fraction. Answer: Let 𝑥 = 0.777.. 10𝑥 = 7.777. . 125 100 . 10𝑥 − 𝑥 = 7.77. . −0.7 … 9𝑥 = 7 𝑥= 7 9 Question #11 3 Convert to percentage. 20 Answer: Convert it to a decimal by dividing it. Then, multiply the decimal by 100%. 3 = 0.15 20 0.15 × 100% = 15% Question #12 1 Write 2 as improper fraction. 2 Answer: 2 1 2 is 5 as improper fraction. 2 Question #13 What is the simplest form of 64% as a fraction? Answer: 64 100 Bu, there is a common factor 4, so you can divide both numbers by 4 to have the simpliest form: 64 ÷ 4 16 = 100 ÷ 4 25 Question #14 Convert 3 25 to a decimal. Answer: We have 3 25 3 ÷ 25 = 0.12 Question #15 3 −2 How do you simplify ( ) 4 Answer: 𝑎 −𝑥 ( ) 𝑏 𝑏 𝑥 =( ) 𝑎 ? So, we have: 3 −2 4 2 ( ) =( ) 4 3 42 16 = 32 9 Question #16 Write 0.25 as a fraction. Answer: 0.25 is 25 1 and 100 as a fraction. 4 Question #17 How do you convert 0.625 into a percent and fraction? Answer: 0.625 = 625 1000 = 5 8 - fraction. 0.625 × 100% = 62.5% - percentage. Question #18 Turn 0.75 into a fraction. Answer: 0.75 = 75 100 75 15 3 = = 100 20 4 Question #19 Change 2.4 into a fraction. Answer: 2.4 is two and four tenths. So in fraction form is 2 4 10 Question #20 Convert 25% into a fraction in its simplest form. Answer: 25% = 25 100 = 5 20 = 1 4