Arithmetic with Fractions, Schemes and Mind Maps of Calculus

Download Arithmetic with Fractions and more Schemes and Mind Maps Calculus in PDF only on Docsity! Arithmetic with Fractions Why do we need to be able to do this? Why can’t I just use my calculator? It’s often quicker to do arithmetic with fractions in your head than it is to find your calculator. Being able at least estimate the answer in your head lets you check to see whether you did your calculator work correctly. Many fractions don’t have nice looking decimal forms, so the fraction is a simpler answer. A fraction also shows a proportion directly, and those are harder to see with decimal numbers. Understanding how arithmetic works with fractions will help you when you’re working with complicated algebraic expressions later. Your teachers will expect you to be able to do fairly simple arithmetic with fractions quickly and automatically, and with no calculator help. What should you be able to do? Express a fraction in its simplest form: Commonly known as “reducing” a fraction, this means writing the fraction so there are no common factors between its numerator and its denominator. Example: Write the fraction 35/105 in its simplest form. In order to see what factors might be common to the numerator and denominator, we need to factor each one. Finding the prime factorization is a good place to start: 753 75 105 35 ⋅⋅ ⋅ = Note that there is a 5 in both the numerator and denominator, and there is a 7 in both the numerator and denominator. We can rewrite the fraction as: ⋅=⋅⋅= ⋅⋅ ⋅ = 3 1 7 7 5 5 3 1 753 75 105 35 1∙1 3 1 = The 5/5 and the 7/7 are both just ways to write 1. So 35/105 is really 1/3, multiplied by 1 two times. Its simplest form is 1/3. Express fractions with common denominators: For much of the arithmetic you will do on fractions, you need to find a common denominator. That means finding some number that is a multiple of all of the denominators you start with. Example: Find a common denominator for 7/12 and 5/8, and express each fraction over that common denominator. We need a multiple of both 12 and 8. One way to quickly find a common denominator is to simply multiply the denominators – in this case, a common denominator is 12 x 8 = 96. To express a fraction over a common denominator, multiply it by 1 (so it doesn’t change its value). The way you write the 1 is missing factor/ same missing factor. 96 60 12 12 8 5 ; 96 56 8 8 12 7 =⋅ =⋅ Find the prime factorization of two- and three-digit numbers. To find common denominators, you will want to be able to quickly factor two-and three-digit numbers. Here are some strategies 1. Have the multiplication table memorized – then you can quickly recognize 56 as 7 x 8. 2. If you need to factor a number you don’t know much about, you can divide it in turn by smaller numbers to look for factors. That sounds like a lot of work, but it doesn’t need to be. You only have to try prime numbers, less than the number’s square root. (Do you have to know the square root of the number? No, but you should be able to know what it’s between.) And if you find a factor, you can look at the smaller, probably more familiar numbers. I find it most useful to start with the smallest primes and work my way up. And, yes, I do use a calculator sometimes at this point. Example: Find the prime factorization of 132. I can see that 2 is a factor, since 132 ends in 2. I’ll divide that out to get 132 = 2 × 66. Now I can see that 66 = 6 × 11 (multiplication tables). So 132 = 2 × 6 × 11. These aren’t all prime numbers, so let me factor the 6 a bit further. 132 = 2 × 2 × 3 × 11, or 22 × 3 × 11.