Question and Answers #25 Fractions

Questions and Answers Sheet 25 Fractions Question #1 𝜋 Consider a sector with angle 3 radians, and radius 3.5 m. What is the arc length and area? Answer: The sector length when the angle is measured in radians is 𝑙 = 𝑟𝜃 So in this case: 𝑙 = (3.5) × 𝜋 3 = 3.66519 ⋯ And the Area is given by: 1 𝐴 = 𝑟 2 𝜃. 2 So in this case: 1 𝜋 𝐴 = (3.5)2 2 3 = 49 24 = 6.414085 ⋯ Question #2 What is the value of (n!) by formula? Answer: n factorial is given by 𝑛! = 𝑛 × (𝑛 − 1) × (𝑛 − 2) × ⋯ × 3 × 2 × 1 it can also be written as 𝑛! = 𝑛 × (𝑛 − 1)! 1! = 1 2! = 2 × 1 = 2 3! = 3 × 2 × 1 = 6 4! = 4 × 3 × 2 × 1 = 24 5! = 5 × 4 × 3 × 2 × 1 = 120 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720 and so on. Question #3 How do you simplify 1 18 + 4 ? 18 Answer: As the denominator in both fractions is same 1 18 + 4 18 = 1+4 18 = 5 18 Question #4 How do you simplify 5 12 2 + ? 3 Answer: To simplify 5 12 2 + , we should first modify the fractions so that they have common denominator. 3 As 3 is factor 12, the GCD of denominators 12 and 3 is 12 and = 5 8 13 12 + 1 + = = 12 12 12 12 = 12 1 1 + =1 12 12 12 5 12 2 5 3 12 + = + 2×4 3×4 Question #5 It costs John $13.50 to take Mary out for the evening. Sixty percent of this total was for theater tickets. What was the cost of each ticket? Answer: We can write 60% as 0.6 to be used as 0.6 × (13.50) = 8.1 dollars, corresponding to the price od two tickets: So it will be: 8.1 2 = 4.05 dollars for one ticket. Question #6 1 2 4 3 How do you simplify 1 + + ? Answer: 1 2 4 3 As GCD of the denominators 1, 3, and 4 is 12. Hence 1 + + = 1 1 2 + + 1 4 3 = 1 × 12 1 × 3 2 × 4 + + 1 × 12 4 × 3 3 × 4 = 12 3 8 + + 12 12 12 = 12 + 3 + 8 23 = 12 12 We can also write it as 1 + 3 12 + 8 12 =1+ 3+8 12 =1 11 12 Question #7 How do you simplify 2 3 1 ÷ ? 2 Answer: When we divide a fraction say inverse of 𝑐 𝑑 i.e. 𝑑 𝑎 𝑏 by another fraction say 𝑐 𝑑 this is equivalent to multiplying 𝑎 𝑏 by multiplicative 𝑐 2 1 2 2 2×2 3 2 3 1 3×1 Hence ÷ = × = = 4 3 Question #8 5 5 8 8 How do you evaluate 3 + 3 ? Answer: 5 5 3 5 3×8 5 3 = 3+ = + = + 8 8 1 8 1×8 8 = 24 5 29 + = 8 8 8 5 5 8 8 Hence 3 + 3 = 29 29 + 8 8 = 58 8 = (8 × 7) + 2 8 = 8×7 2 + 8 8 = 8  7 2 1 + 8  1 2  4 =7 1 4 Question #9 Two fifths of the instruments in the marching band are brass, one third are percussion, and the rest are woodwinds. What fraction of the band is woodwinds? Answer: We need a common denominator to equate the fifteenths. 2 6 5 15 And thus ≡ 1 5 3 15 And ≡ , 2 5 , and the 1 3 fractions. Clearly, we could express the fractions as The balance are woodwind instruments, i.e. 1 − 6 15 = 5 15 = 4 15 Question #10 How do you write 0.285 as a fraction in simplest terms? Answer: 0.285 or 0.285 is a fraction in decimal notation It is equivalent to 2 10 8 + 100 + 5 1000 or 285 1000 As the only common factor (other than 1) between 285 and 1000 is 5 to write 0.285 as a fraction in simplest term, we need to divide numerator and denominator in Hence 0.285 = 285 1000 5×57 = 5×200 = 285 1000 by 5 57 200 Question #11 85% of what number is 45? Answer: 85% is equivalent to Let 85% = 85 100 85 100 of x be 45 Hence 9× 85 100 100 17 = × 𝑥 = 45 or 85 100 ×𝑥× 100 85 = 45 × 100 85 or 𝑥 = 45 × 100 85 or 𝑥 = 5 × 9 × 100 5×17 or x = 5  9  100 or 𝑥 = 5  17 900 17 Question #12 Convert 0.9 in percentage form? Answer: 0.9 is already in decimal form and as it has 9 in tenth's place, hence 0.9 = 9 10 For converting a fraction or decimal number into percentage, one should multiply it by 100 and hence 0.9 = 9 10 9 10 = × 100% = 90% Question #13 7 How do I convert the fraction into a decimal? 8 Answer: Step 1 As the only ' factor of denominator is 2, in fact there are three twos ( 2 × 2 × 2 ) we can can convert the fraction by multiplying numerator and denominator by 5 × 5 × 5 7 8 7 7×5×5×5 8 8×5×5×5 Hence, = 875 = 1000 = 0.875 (This is always possible if only ' factors of denominator are 2's and 5's - just multiply by as many 5's and 2's) Question #14 What is 17% of 3,789? Answer: 17% of 3789 means 17 100 i.e. = of 3789 17 100 × 3789 64413 100 = 644.13 Question #15 Ten students share 8 pieces of poster board equally. What fraction of a piece of board does each student get? Answer: Step 1 8 pieces of posterboard is divided between 10 students. "eight divided by 10" 8 ÷ 10 8 ÷ 10 is the same as 8 10 Reduce the fraction to lowest terms Answer: 8 10 = 4 5 Question #16 How do you write 7 3 5 as an improper fraction? Answer: 7 is equal to 35 5 . Then 35 5 3 38 5 5 + = Or, you could multiply the denominator (5) by the whole number (7) and add the numerator (3), leaving you with 38. This is your numerator. You keep the denominator (5). Question #17 What is 5 10 of one dollar? Answer: 5 10 of a dollar will be equal to 50 cents because if we'll bring 5 10 to it's lowest terms it will be equal to: 1 2 which is half of anything, so, half of a dollar will be equal to 50 cents. Verifying it: 1 dollar = 100 cents Therefore: 50 cents × 2 = 100 cents 100 cents = 100 cents Hence, verified. Question #18 What fraction is between 1 3 1 and ? 2 Answer: For any two numbers a abd b, the average 1 1 3 2 So to find a fraction between and 1 2 (𝑎 + 𝑏) lies between them. we can form the average: 1 1 1 ( + ) 2 3 2 1 2 3 = ( + ) 2 6 6 1 5 5 = ( )= 2 6 12 Question #19 1 Paul hit 33 tennis balls during practice. He hit of them out of the tennis court. How many balls did he hit out of the 3 court? Answer: Let x be the balls Paul hit out of the court. Then we have: 𝑥 = 33 × 𝑥= 1 3 33 3 𝑥 = 11 Question #20 A restaurant uses 2 3 4 pounds of flour to make a batch of dinner rolls. About how many pounds of flour are needed if 3 batches of dinner rolls are to be made? Answer: Step 1 First method 3×2=6 3× 3 9 = 4 4 9 1 =2 4 2 Answer: 1 1 4 4 6+2 = 8 Step 2 Second method 3 2 3 8 3 11 2 = + = + = 4 1 4 4 4 4 3× 11 33 = 4 4 Answer: 33 1 =8 4 4