Arithmetic Arithmetic Decimals www.essentiallyeducation.co.uk Contents Addition & Subtraction of Decimals... 2 Multiplication of Decimals... 2 Multiplication by Powers of (10,100,1000 etc )... 3 Multiplication of Decimals by Whole Numbers or Decimals... 3 Division of Decimals Division by Powers of Ten Division of Whole Numbers & Decimals... 5 Conversion of Fractions to Decimals... 7 Conversion of Decimals to Fractions... 11 Shortening of Decimal Numbers Decimal Place... 12 Significant Figures... 13 Self Assessment Questions... 15 Self Assessment Answers... 16
Decimals The four basic rules of arithmetic, which have been applied to whole numbers and fractions, can equally be applied to decimals. The principle is the same as for whole numbers, but there are a number of differences mostly concerned with the position of the decimal point. Addition and Subtraction of Decimals This is performed exactly the same way as for whole numbers, but you must remember to keep the decimal point aligned, rather than the numbers. i.e. Addition Remember to keep the decimal points aligned! (including the answer) i.e. Subtraction Do Not Forget! to keep the decimal points aligned! (including the answer) Multiplication of Decimals There are two cases for the multiplication of decimals. Multiplication by powers of ten. i.e. 10, 100, 1000 etc Multiplication by whole numbers or decimals. 2
Multiplication of Powers of Ten (10,100,1000 etc ) To multiply by 10, 100, 1000 etc, you just move the decimal point (d.p) to the right by however many powers of ten you have. So to multiply by: 10 You move the decimal point 1 place to the right. 100 You move the decimal point 2 places to the right. 1000 You move the decimal point 3 places to the right. 10000 You move the decimal point 4 places to the right etc Here s how it works: What is 9.685 x 100? Decimal point has moved 2 Places to the right. So 9.685 x 100 = 968.5 Another Example: What is 28.375 x 1000? Decimal point has moved 3 Places to the right. So 28.375 x 1000 = 28375 Multiplication of Decimals by Whole Numbers or Decimals The process of multiplication of decimals is the same as for the multiplication of ordering numbers, but there is an extra step to work out the position of the decimal point. 3
For Example: What is 4.2 x 6.4? We proceed by working out the sum in the normal way as if there were no decimal points. Our answer so far is 2688, but we are not finished yet. Now for our extra step, to find out the position of the decimal point. Count the number of numbers to the right of the decimal point in the original sum. In this case there are 2 numbers: The decimal point is now moved back this number of places (i.e 2) from the right-hand side of the above answer. So in our example: Therefore the answer to 4.2 x 6.4 = 26.88 Here is another example: What is 14.0065 x 5.5? Write the sum out in the normal way as if there were no decimal points. 4
We now need to find out where the decimal point will go. The original sum is: = 5 numbers to the right of the decimal point. We need to count the number of numbers to the right of the decimal point. In this case there are 5 numbers. The decimal point is now moved back this number of places (i.e. 5) from the right-hand side of the above answer. Therefore, the answer to 14.0065 x 5.5 = 77.03575 Division of Decimals (a) Division of Powers of 10 Just as in multiplication of decimals we can divide decimals by powers of ten i.e. 10, 100, 1000 etc by moving the decimal point. This is simply the reverse of the multiplication case, so for division you move the decimal point to the left. i.e. 62.53 10 = 6.253 62.53 100 = 0.6253 62.53 1000 = 0.06253 5
Another example:3948.75 100 = 39.4875 3948.75 1000 = 3.94875 3948.75 10000 = 0.394875 Note: Remember that the decimal point moves to the Left. (b) Division by Whole Numbers and Decimals Division of decimals by whole numbers is carried out as for the division of ordinary numbers, and the position of the decimal point remains in the same place. Here is an example: 24.68 4 Note: There is no need to move the decimal point when dividing. Another example: 52.35 5 Unlike division of whole numbers, division of decimals does not give a remainder. Instead the remainder is carried into the next column and division is continued with the help of extra zeros until it goes exactly. This can be seen in the following example: 56.38 8 In order to divide any number by a decimal number the decimal must first be converted to a whole number, this means that there must be no numbers to the right of the decimal point. As an example, we cannot divide by the following numbers: 16.8, 52.6, 8.7, 0.93. The only way to carry out division by decimals is to multiply the decimal by powers of ten (i.e. 10, 100, 1000 etc...), in order to move the decimal point and produce a whole number. 6
i.e. 8 0.4 Step 1: multiply 0.4 by 10 to give 4 Step 2: multiply 8 by 10 to give 80. is re-written as i.e. 8 0.4 = 20 Here is another example: 7.2 0.6 Step 1: make the 0.6 into a whole number. We do this by multiplying by 10 which gives 6 (0.6 x 10 = 6). Step 2: multiply the 7.2 by 10 to give 72 (7.2 x 10 = 72) So 7.2 0.6 is re-written as 72 6 and the sum becomes: Therefore, the answer to 7.2 0.6 = 12 Conversion of Fractions to Decimals Sometimes it may be more convenient to express fractions in terms of decimals, this is particularly so n scientific and engineering applications. To convert a fraction to a decimal you divide the denominator into the numerator: i.e. 7
First re-write the quarter as: Now write the 1 as 1.0000 (adding these zeros helps the working), so the sum becomes: The division is then carried out the same way as mentioned previously, leaving the decimal point in the same position. So in this case, 4 into 1 will not go so you put a zero above the 1 do not forget the decimal point here. Now we divide the 4 into 10, which gives 2 and a remainder of 2. Now put the 2 above the 1 st zero and the remainder of 2 we place next to the 2 nd zero, which will give us 20. Now we divide 4 into 20 which gives us 5, this is written above and we are done. Expressed as a decimal is 0.25 Another Example: Re-write this as: Expressed as a decimal is 0.6 8
Here is one more example: Re-write this as: Expressed as a decimal is 0.125 Recurring Numbers: Sometimes the conversion of a fraction to a decimal can result in what is known as a recurring number, as the following example shows: As a decimal is: In the above situation, as can be seen, the 3 will carry on forever! To show that the three is a recurring number it can be written as: The dot above the 3 means that it is recurring. Here is another example: = 9
Expressed as a decimal is : Another Example: = Expressed as a decimal is : And Finally: = Expressed as a decimal is : 10
Conversion of Decimals to Fractions To convert a decimal to a fraction, you divide the numbers to the right of the decimal point by: 10 if one number is present. 100 if two numbers are present. 1000 if three numbers are present... and so on. The final step is then to cancel the fraction to it s lowest terms. The following three examples explain: (1) Two numbers are present, so we divide by 100 i.e. which, in it s lowest terms is: (2) One number is present, so we divide by 10 i.e. which, in it s lowest terms is: 11
(3) Three numbers are present, so we divide by 1000 i.e. which, in it s lowest terms is: Shortening of Decimal Numbers Decimal Place Sometimes the numbers you obtain from a calculation are longer than is required for a sensible answer. For Example: To make this answer shorter you can round off the numbers to the right of the decimal point, i.e you decrease the number of decimal places, (this is often shortened to d.p) Take a number such as: 12.65456 which has 5 decimal places. This can be shortened so that it has 4,3,2 or 1 decimal place(s). If the number to the right of the decimal place you are shortening to is 5 or greater, then you increase the number in that decimal place by 1; if it is less than 5 it remains the same. i.e. Write 12.65456 correct to 3 decimal places which means 3 numbers to the right of the decimal point. To do this we need to look at the number to the right of the third decimal place: 12.65456 If this number is 5 or greater we need to increase the number in the third decimal place by 1. 12
12.65445 = 12.655 correct to 3 d.p Here are a few examples: Correct 12.65456 to 1 decimal place. We look at the number to the right of the first decimal point, which is 5, so we increase the first decimal place by 1. 12.65456 correct to 1 d.p = 12.7 8.0375 correct to 2 decimal places = 8.04 45.0968 correct to 3 decimal places = 45.097 0.06584 correct to 4 decimal places = 0.0658 the above 4 is less than 5 so the 8 remains the same. Significant Figures You may be asked to write a number or an answer in terms of significant figures. This means that the number is shortened to a number of significant figures, for example: 2 s.f., 3 s.f etc The number of significant figures is always counted from the first significant figure (this is the first non-zero digit). This is not necessarily the first number, especially when dealing with decimal numbers which are less than one (decimal fractions). To write a number in terms of significant figures, the method is similar to decimal place. We look at the number to the right of the number of significant figures, and if it is less than 5 we leave the last number the same, if it is greater than 5 we increase the value by one. Example: Write 42843 to 3 s.f. We look at the 4 th number along, in this case it is a 4. This is less than 5 so the 8 stays the same. 42843 correct to 3 s.f. = 42800 13
Note: You must include the zeros in this case to keep the number value the same i.e. Thousands. Another Example: Write: Correct to 3 s.f. When presented with a decimal number which is less than 1, we always count our number of significant figures from the first non-zero number. In this case the first non-zero number is the 1, so the third significant figure is 8. We now, look at the number to the right of this, a 5 and because this is 5 or above we must add 1 to the third number, which will make it 9. 0.0015852 correct to 3 s.f. = 0.00159 Note: We do not need to include the zeros to the right of the 9 in this case, they are unnecessary as they do not affect the number value of the answer. Another Example: Write 308.8473 correct to 5 s.f. This time we start with the 3 (this is the first significant figure). The 5 th significant figure is the 4 so we look to the right of this, a 7. This is greater than 5 so we must add 1 to the 4, which will make it a 5. 308.8473 correct to 5 s.f. = 308.85 There are a few things to remember when expressing a number in terms of significant figures. They can be tricky, so be careful! (1) The first significant figure is the first non-zero number; (2) Keep the number value the same, by adding zero s if necessary e.g. 75139 to 2 s.f. is 75,000 NOT 75!. (3) Significant figure and decimal point may or may not be the same, it will always depend on the numbers concerned. 14
Example: 0.68446 to 4 s.f. is 0.6845 and 0.68446 to 4 d.p. is 0.6845 These are the same. 0.006538 to 3 s.f. is 0.00654 and 0.006538 to 3 d.p. is 0.007 These are NOT the same. Self Assessment Questions (a) Work out the following problems: (remember to line up the decimal points) 1. 0.425 + 0.308 2. 48.675 + 6.025 3. 12.48 3.75 4. 3.89 0.15 5. 2.6258 1000 6. 28.65 10.2 7. 4.3 16.4 8. 9837208 1000 9. 4.2 2.4 10. 99.99 0.33 (b) Convert the following fractions to decimals: 15
(c) Convert the following decimals to fractions: 1) 0.125 2) 0.85 3) 0.875 4) 0.002 5) 0.25 6) 0.6 7) 0.12 8) 0.3125 Self Assessment Answers (a) 1. 0.425 + 0.308 = 0.733 2. 48.675 + 6.025 = 54.7 16
3. 12.48 3.75 = 8.73 4. 3.89 0.15 = 3.74 5. 2.6258 1000 = 2625.8 6. 28.65 10.2 = 292.23 7. 4.3 16.4 = 70.52 8. 9837208 1000 = 9837.208 9. 4.2 2.4 = 1.75 10. 99.99 0.33 = 303 (b) 17
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