QUICK MATH REVIEW & TIPS 2 Easy Path To Fractions and Decimals

Word of Advise For a good and lasting foundation in Math, know your multiplication tables by all means. Knowing multiplication translates to being able to figure out division problems in the shortest amount of time. Working with fractions, algebra, ratios and percentages also become easy to handle. Start solving Math problems using the facts, the rules & information you already know to guide you.

Fractions: The very basics A fraction is normally written in the form a b a is called the numerator b is called the denominator Any whole number can be written as a fraction by putting the number 1 as the denominator. This sometimes helps to avoid confusion when solving math problems. So 2 is the same as 2 1 and 10 is the same as 10

In the fraction: 3 7 3 is called the Numerator 7 is the Denominator

ADD, SUBTRACT AND COMPARE FRACTIONS To ADD, SUBTRACT or COMPARE fractions take steps to make sure that their denominators are the same (work out a common denominator). Then ADD, SUBTRACT or COMPARE the numerators. In establishing a common denominator, if you multiply the denominator of any of the fractions by a number you must do the same to its numerator 13 – 2 16 4 Look at the denominators and determine if one is a multiple of the other denominators. In our case 16 is a multiple of 4 (i.e. 4x4=16) 13 – 2 × 4 16 4 × 4 13 – 8 = 5 16 16 16

You can always establish a common denominator even if one of the denominators is not a multiple of the other. For example in the previous example, 16 is a multiple of 4 and that makes it easy. 5 – 3 6 5 Step 1: Look for the Least Common Multiple of the denominators 6 and 5. That will be 30 Why is it the Least Common Multiple 30? Because if we are to write down the multiples of 6 and 5 separately, we will find that 30 is the first and smallest multiple that is common to both. Step 2: Re-write each fraction using 30 as the common denominator and don’t forget to also alter the numerators to make sure that whatever you do to change the original denominator to the common denominator is also done to the numerator of the fraction 5x5 – 3x6 6x5 5x6 25 – 18 = 7 30 30 30

Arrange the following fractions in order starting with the smallest ? 5 , 2, 7, 3 16 4 8 4 Step 1: Find the common denominator. That will be 16 Write each fraction using the common denominator 5 , 2x4, 7x2, 3x4 16 4x4 8x2 4x4 5 , 8 , 14, 12 16 16 16 16 Now write the original fractions in the required order by comparing with the step above 5 , 2, 3, 7 16 4 4 8

You can always establish a common denominator for fraction problems even if none of the denominators is a multiple of the others. Simply look at the denominators closely and choose the least number that each of the denominators can divide without leaving a remainder. Arrange the following from largest to smallest: 1, 3, 2, 5 2 4 3 8

MULTIPLYING FRACTIONS In multiplying fractions it helps to simply first to reduce the values of the numbers before multiplying. If you know your multiplication table, simplifying fractions become very easy task. 8 × 45 15 56 Look closely at the numerator , denominator pairs vertically and diagonally ONLY and find out which pairs have factors common to them. In other words which vertical and diagonal pairs can be divided by the same number without leaving a remainder. You can see that the diagonal pair 15 & 45 have some common factors. For example each number is divisible by 5 so we can start off by dividing each (i.e. 15 and 45 by 5) 8 × 45 = 8 × 9 15 56 3 56 The new diagonal pair 3 & 9 each can be divided by 3 8 × 9 = 8 × 3 3 56 1 56 We see a connection between the diagonal pair 8 & 56. One is a multiple of the other or we can say that 8 can divide 56 completely. 8 × 3 = 1 × 3 1 56 1 7 The final step is to multiply the numerators of simplified fractions to get the new numerator. The multiply the denominators to get the new denominator for the resulting fraction.

The steps we followed before can be combined together as long as we pay attention to the number pairs and perform the same actions to each number pair. Simplify 24 x 18 35 42 Remember to divide vertically and diagonally only. Do you see any relationship between the vertical pair 24 & 35. The answer is NO. Do you see any relationship between the vertical pairs 18 & 42. The answer is YES. Both 18 & 42 can be divided by 2 or 3 or 6 In order to avoid too many steps, lets choose 6 to divide both 18 and 42 24 x 18 = 24 x 3 35 42 35 7 Since there is nothing else to simplify we will go ahead and multiply the numerators together. Then the denominators together. 24 x 3 = 72 35 x 7 245

DIVIDING FRACTIONS To divide two fractions: 6 ÷ 3 17 34 6 ÷ 3 17 34 Change the division operator (÷) into a multiplication operator (x). Next change the dividing fraction (the 2nd fraction) into its reciprocal. 6 x 34 17 3 You now have two fractions multiplying each other. Just simplify and multiply the two fractions. 6 x 342 171 3 62 x 2 = 2 x 2 = 4 1 31 1 1

Example: 8 ÷ 4 11 33 By applying the rule we just learned we can write down the following: 8 x 33 11 4 We notice that the resulting fractions can easily be simplified and multiplied.

PROPER, IMPROPER & MIXED FRACTIONS In a proper fraction the numerator is smaller than the denominator. 1, 3, 11 are examples of proper fractions. 5 4 18 An improper fraction is a fraction in which the numerator is larger than the denominator. 5, 11, 9 are examples of improper fractions 4 8 7 A mixed fraction is made up of a whole number and a proper fraction. Examples of mixed fractions are 1¼ , 2½, 5¾

You can change a Mixed fraction to an Improper fraction and vice versa. To change a mixed fraction to an improper fraction, multiply the whole number part by the denominator of the fraction. Then add the numerator to the result. The result will be your new numerator. The improper fraction is new numerator denominator Example: 11 = (1 x 4) +1 = 5 4 4 4 You do not need to write the above details in your conversion. With enough practice you will be able to look at any mixed fraction and calculate the numerator of the resulting improper fraction in you head.

To change an improper fraction to a mixed fraction, divide the numerator by the denominator. This should result in a whole number and a remainder. The mixed fraction will look like this: whole remainder denominator Example: 7 = 21 3 3 That is 7 ÷ 3 = 2 remainder 1

ADDING AND SUBTRACTING IMPROPER FRACTIONS When adding and subtracting improper fractions follow the same method we use for proper fractions.

ADDING AND SUBTRACTING MIXED FRACTIONS When adding mixed fractions you can simply add the whole number parts separately. Then add the fraction parts and combine the results. 3¼ + 1½ =(3+1) +¼ + ½ When subtracting mixed fractions you can avoid mistakes by first changing the mixed fractions into improper fractions. Then follow the method for subtracting proper and improper fractions. 3¼ - 1½ = 13 - 3 4 2

Tips on working with Decimals Learn to read decimal numbers correctly. This is particularly important when comparing decimal numbers. 0.5 is read as zero point five (or five tenths) 0.12 is read as zero point one two (or twelve hundredths) 17.34 is read as seventeen point three four (or seventeen and thirty-four hundredths) So 0.5 is larger than 0.12 and 1.3 is larger than 1.284 for example

23 can be written as 23.0 2 is the same as 2.0 Any whole number can be written as a decimal number by placing a decimal point (.) after the last digit followed by 0 23 can be written as 23.0 2 is the same as 2.0 123 is the same as 123.0

To divide any number by a decimal number: Step 1: Change the divisor to a non-decimal or whole number by moving the decimal point to the right, one digit at a time, until there are no more decimal places. Step 2: Move the decimal point in the number you are dividing to the right, the same number of times as in step 1. Step 3: Divide the two numbers. The answer you get is the final answer. Remember that any whole number can be written with a decimal point by placing the decimal point after the rightmost or last digit followed by 0. That is to say 4 is the same as 4.0

Example 1: 10 ÷ 0.2 Step 1: The divisor 0.2 can be made whole by moving the decimal point one place to the right. Step 2: Remember that 10 is the same as 10.0 and move the decimal point the same number of times to the right so that 10.0 becomes 100 Step 3: 100 ÷ 2 =50 It is usually more readable to rewrite such a problem as a fraction so you can easily follow the movement of the decimal point. So you would write 10 ÷ 0.2 as 10 0.2

Example 2: 3.6 ÷ 0.2 This is the same as 3.6 0.2 The divisor 0.2 can be made whole by moving the decimal point one place to the right. In 3.6 we will move the decimal point the same amount to the right so 3.6 becomes 36 36 ÷ 2 =18 =>This is the answer.

Example 3: 1.25 2.5 The divisor or denominator 2.5 can be made whole by moving the decimal point one place to the right so 2.5 becomes 25 Since we moved only one place to the right in the divisor we do the same to the numerator or product. 1.25 then becomes 12.5 12.5 = 0.5 25

Now Try These: 12 ÷ 0.03 7.5 ÷ .15 0.75 ÷ 0.5 1.25 ÷ .25

To multiply two or more numbers that contain decimals: Step 1: Count and record the total number of decimal places to the right of the decimal point in each of the given numbers. Step 2: Write down and multiply the numbers as whole numbers without the decimal point. Step 3: Put back the decimal point in your final result by jumping over each digit starting from the rightmost digit until you have counted the same number of decimal places you recorded in Step 1. Example: 0.2 x 1.5 0.2 has one digit to the right of the decimal point = 1 decimal place 1.5 has one digit to the right of the decimal point = 1 decimal place Step 1: We have a total of 2 decimal places Step 2: 2 x 15 = 30 Step 3: Final answer will be 0.30 which is the same as 0.3 ( the rightmost 0 can always be dropped without changing the results)