Question and Answers #22 Fractions

Questions and Answers Sheet 22 Fractions Question #1 How do you change 2.9 into a fraction? Answer: 2.9 is two and nine tenths. 2.9 = 2 9 10 as a mixed fraction. As an improper fraction it would be: 2 9 10 × 2 + 9 29 = = 10 10 10 Question #2 How do you write 3 1 4 as an improper fraction? Answer: Explanation: To convert a mixed fraction to an improper one, we must transfer the whole number into the numerator of the fraction. To do this, we convert the whole number 3 into a fraction with common denominator of 4 : 3⇒ 12 4 1 1 4 4 Since 3 = 3 + our final step is to add the fractions together, giving us the desired improper fraction: 12 1 13 + = 4 4 4 Question #3 How do I find the average rate of change of the function from 𝑥1 to 𝑥2? The function is 𝑓(𝑥) = −2𝑥 + 15 and 𝑥1 = 0 , 𝑥2 = 3 Answer: Explanation: Average rate of change is Algebra I slope, so 𝑚= 𝑓(3) − 𝑓(0) 9 − 15 = = −2 3−0 3 In this case 𝑓(𝑥) is linear, so the average rate of change will be equal to the slope of the line on all intervals. Question #4 How do you simplify 7 12 + 4 5 ? Answer: First of all you have to evaluate the common denominator between 12 and 5 searching the minimum comune multiple. Those numbers do not have any common factor so the MCM is 12 ⋅ 5 60 Then you have for the first fraction 60 5 12 = 5 to multiply for the numerator. For the second fraction you have = 12 to multiply for the numerator. 7 4 5 12 ⋅ 4 + = (7 ⋅ ) + 12 5 60 60 = 35 48 + 60 60 now you can sum the numerators 35 48 35 + 48 + = 60 60 60 = 83 60 Question #5 How do you simplify 48 ÷ −3 ? Answer: Explanation: 48 ÷ (−3) = −16 Two negative numbers multiply to a positive number, so if you ignore the signs and divide 48 by 3, you get 16. Because the 3 is negative, the 16 should also be negative: (−3) ÷ (−16) = +48 Question #6 1 Question: If ⍁{1}{2} of 5 is 3, then what is of 10? 3 Answer: In normal rational arithmetic Then 1 3 of 10 is 10 3 1 2 or 5 is 5 2 = 2.5 . The result could be taken as 3 if we choose to round up. = 3. 3 . If we round to the nearest value then we get 3. If we always round non- integers up, then we get 4. If we simply take the antecedant '' 1 3 1 2 of 5 is 3′′′′ to be false, then we can deduce any consequent, e.g. that of 10 is a herring or anything else we choose. Question #7 How do you convert 0.16 (6 repeating) to a fraction? Answer: First multiply by 10(10 − 1) = (100 − 10) to get an integer. The first multiplier of 10 is to shift the decimal representation one place to the left, so the repeating section begins just after the decimal point. Then the (10-1) multiplier is used to shift the digits one more place to the left (the length of the repeating pattern) and subtract the original to cancel the repeating tail. (100 − 10) ⋅ 0.16 = 16.6 − 1.6 = 15 Then divide both ends by (100 − 10) and simplify 0.16 = 15 15 1 ⋅ 15 1 = = = 100 − 10 90 6 ⋅ 15 6 Question #8 1 What fraction is equivalent to ? 3 Answer: There are infinite number of fractions that are equal to You can multiply the numerator and denominator of equal to 1 3 1 3 1 3 by any number and the resulting fraction would be so long as you are multiplying both by the same number. For example: 1 1⋅2 2 = = 3 3⋅2 6 1 1 ⋅ 4000 4000 = = 3 3 ⋅ 4000 12000 and so on and so forth. Question #9 What is the fraction of 45%? Answer: 𝑥% is equal to 45% = 𝑥 100 , so just substitute 45 for x to get: 45 100 You can divide each side by 5 to reduce the fraction to its simplest form, 9 20 Question #10 1 How do you change − into a decimal? 2 Answer: A decimal is another way to write a fraction whose denominator is a power of 10, so if you want to change a fraction to decimal first thing is to see if you can reduce or expand the fraction to get such denominator: In this case to get a deniminator of 10 you only have to multiply numerator and denominator by 5 1 1⋅5 2 2⋅5 − =− =− 5 10 = −0.5 However this method does not always work. A method that works for every fraction is to calculete the long division of numerator and denominator to see if the decimal is finite. For example: 2: 3 = 0.666 … Question #11 How do you simplify the fraction 18 24 ? Answer: Explanation: When simplifying a fraction you always would like to find a common number that goes into both the denominator and numerator, so both values can be simplified by that common number. This number has to be the highest value that can fit into both numbers. As 18 is the lowest number, that common number must be ≤ 18 . By trial and improvement, we can conclude that 6 is the highest common multiple that can fit into both numbers. Therefore we divide both 18 and 24 by 6 to simplify the fraction. 18 3 → 24 4 As.. 18 =3 6 24 =4 6 18 3 → 24 4 Question #12 How do you write 15% as a fraction? Answer: 15% = 15 100 15 5×3 = 100 5 × 20 = 3 20 Question #13 How do you convert 5 12 into a decimal and percent? Answer: Explanation: 5 12 So simply means 5 divided by 12. 5 12 = 0.4167 If you are not given a calculator to find the answer, then use long division. 5 ( ) ⋅ 100% = 41.67% 12 Question #14 What is 5 1 8 as a decimal? Answer: Explanation: First convert 5 1 8 into an improper fraction. Multiply the denominator 8 times 5, then add the numerator 1. (8 × 5) + 1 = 41 To convert this to decimal form, use a calculator (or use long division), and divide 41 by 8. 41 = 5.125 8 Question #15 Is the fraction 3 8 in lowest terms? Answer: Yes, the fraction 3 8 is in lowest form as there is no whole number by which we can divide the numerator and denominator. Question #16 How do you convert 3 20 into a decimal and percent? Answer: Divide the numerator by the denominator to change to a decimal. Then multiply the decimal by 100% to change to a percent. 3 = 0.15 20 0.15 × 100% = 15% Percent is parts per hundred so the numerator of a number with 100 as the denominator is the percent . 3 20 5 15 5 100 × = so 3 = 15% 20 Question #17 1 How do you write in percent? 6 Answer: 1 A percent is ''per 100'' or ''out of 100''. Therefore to write as a percent we need to solve: 6 1 𝑥 = 6 100 100 ⋅ 1 𝑥 ⋅ 100 = 6 100 100 =𝑥⋅1 6 𝑥 = 16 ⋅ 6 1 Therefore is 16.6% 6 Question #18 How do you convert 0.55 to a fraction in its simplest form? Answer: Explanation: 0.05 = 1 20 This is because 20 × 0.05 = 1.00 Now: 0.05 × 11 = 0.55 , which means that 0.55 = 11 20 Question #19 How do you convert 0.8 (8 repeating) to a fraction? Answer: We have a repeating decimal, 0.\overline{8} Let us put this equal to x Then, 𝑥 = 0. 8 Multiply both sides by 10: 10𝑥 = 8. 8 We can write this 8. 8 as a sum of a whole number and a decimal number: 10𝑥 = 8 + 0. 8 Now, we know that 𝑥 = 0. 8 10𝑥 = 8 + 𝑥 10𝑥 − 𝑥 = 8 + 𝑥 − 𝑥 9𝑥 = 8 9𝑥 8 = 9 9 𝑥= 8 9 Therefore, 0. 8 0. 8 = 8 9 Question #20 How do you write 0.09 as a fraction? Answer: Every place value after the decimal point can be expressed as a fraction with 10 to some power in the denominator. The first place after the decimal point is called the ''tenths place'' If I had 0.9, I would have 9 tenths, or as 9 a fraction 10 The second place after the decimal is called the ''hundredths place''. The question above asks for 0.09 . This would be 9 hundredths, or as a fraction 9 100