Topic 2: Decimals. Topic 1 Integers. Topic 2 Decimals. Topic 3 Fractions. Topic 4 Ratios. Topic 5 Percentages. Topic 6 Algebra

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1 41 Topic 2: Decimals Topic 1 Integers Topic 2 Decimals Topic 3 Fractions Topic 4 Ratios Duration 1/2 week Content Outline Introduction Addition and Subtraction Multiplying and Dividing by Multiples of Ten Multiplication Division by a Whole Number Division by a Decimal Rounding Decimals Estimation of Decimal Answers Application of Decimals Topic 5 Percentages Topic 6 Algebra Topic 7 Equations and Formulae Topic 8 Measurement TEP023 Foundation Mathematics

2 42 Topic 2: Decimals Introduction The decimal system is based on the number ten. Every number, including fractions, can be shown using ten symbols. For example, would you like to win $30.00 or $3000. Both of them have the same digits but one is a much higher number. This is easy to see and that is the power of the decimal system. How do we know what a digit is worth? This diagram shows the value of each place in a number. Study the diagram below and notice the difference in the value of the digits in the number. These values are called place values Thousands Hundreds Ten thousandths Thousandths Tens Hundredths Units Tenths The places to the right of the decimal point are called decimal places. Each decimal place has a place value one tenth as great as the place on its left, and ten times as great as the place on its right. This continues the grouping process begun with the integers. As you have seen already, the position of each digit tells us its value. For example: = NOTE THAT: The decimals 0.6, 0.60, all represent the same number. You can also express some decimals as fractions e.g. 0.6 six tenths = 6 more on this in Topic 3: Fractions. 10 Study Guide Topic 2 Decimals

3 43 Addition and Subtraction When you are adding and subtracting, it is important to add things of the same place value. Think of money, you need to sure you are adding dollars to dollars and cents to cents. To add and subtract decimals: Write down the numbers, one under the other, with the decimal points lined up. Add zeros so the numbers have the same length Then add normally, remembering to put the decimal point in the answer. Example 1: Calculate The zeros are put in to keep your work lined up so there is less chance of error Step 1: Set your work out as shown, lining up the decimal points and putting in zeros Step 2: Add each column, starting on the right hand side. Don t forget to carry when necessary Example 2: A plumber cut 2.36 meters off the end of a piece of pipe. The piece of pipe was 4 meters long. What is the length of the remaining pipe? Step 1: Write the problem as a mathematical statement: Step 2: Set your work out as shown, lining up the decimal points and putting in zeros Step 3: Subtract the bottom number from the top number starting at the right Step 4: Worded questions need a written answer. The remaining piece of pipe is 1.64 m long. Don t forget to include the units. TEP023 Foundation Mathematics

4 44 Multiplying and Dividing by Multiples of Ten Because decimal notation is just an extension of our base 10 whole number system, decimals are easy to multiply or divide by 10, 100, 1000 and so on. Rule for multiplying by 10, 100, 1000 To multiply any number by a multiple of ten move the decimal point to the right as many places as there are zeros. Example 1: Work out the following using the rule given. a) = Multiplying by 10 moves the decimal point one place to the right. b) = c) = d) = Multiplying by 100 moves the decimal point two places to the right. Remember that the decimal point sits at the end of a whole number, i.e. 4 = 4.0 Multiply by 10 by moving the decimal point 1 place right and then multiply by 2 in your head. = = 121 Study Guide Topic 2 Decimals

5 45 Rule for dividing by 10, 100, 1000 To divide any number by a multiple of ten move the decimal point to the left as many places as there are zeros. Example 2: Work out the following using the rule given. a) = Dividing by 100 moves the decimal point two place to the left. b) = c) = = Dividing by 1000 moves the decimal point three places to the left. Remember that the decimal point sits at the end of a whole number, i.e. 4 = 4.0 = Spaces are filled with zeros. Multiplication Multiplying decimals is similar to multiplying any number but watch out for the decimal point. Be careful where it ends up!! To multiply decimals: Multiply the numbers together IGNORING decimal points. Count the number of digits on the right of the decimal point for each number in the question. Count back that many places in the answer and replace the decimal in your final answer. TEP023 Foundation Mathematics

6 46 Example 1: Calculate Step 1: Multiply the numbers together, ignoring the decimal place. Step 2: Count the number of digits on the right hand side of the decimal place for each number in the question has 2 numbers to the right of the decimal point 2.7 has 1 number to the right of the decimal point That means that altogether there are 3 numbers to the right of the decimal (count back 3 places in the answer) Step 3: Replace the decimal in the final answer = Alternatively, you can estimate your final answer and place the decimal point accordingly Example 2: There are 15 boxes to fit in the shelf. Each one is 5.25cm long. How long will the shelf need to be to fit in all the boxes? Step 1: Write the problem as a mathematical statement: Step 2: Multiply the numbers ignoring the decimal points Step 3: Count the number of digits on the right of the decimal points for each number in the question has 2 numbers to the right of the decimal point 15 has no numbers to the right of the decimal point Altogether there are 2 numbers to the right of the decimal (count back two places) Step 4: Write out the answer: You need a shelf that is cm long. Study Guide Topic 2 Decimals

7 47 Division by a Whole Number You will set out your questions here in the same way as previously explained in Topic 1. To divide a decimal by a whole number: Keep the decimal points in line. The decimal point in the answer should be directly over the decimal point in the question. Example 1: Calculate Step 1: Write out the problem as shown: Place the decimal point in the answer directly above the decimal point in the question Step 2: Divide as shown previously. 4 into 0 goes zero times (write 0 above) Step 3: Write the decimal point in your answer since you are crossing from the units to the decimals. 4 into 1 goes zero times (write 0 above) Step 4: 4 into 14 goes 3 times (write the 3 above the 4) with 2 remainder (write the remainder) Step 5: 4 into 28 goes 7 (write the 7 above the 8) TEP023 Foundation Mathematics

8 48 Example 2: Calculate 7 8 since 8 goes into 7 zero times, write the problem with room for additional working out. Step 1: Write out the problem as shown: Step 2: Divide, keeping the decimal points in line. 8 into 7 goes zero times with a remainder of 7 (write the 0 above the 7). Carry the 7 across Steps 3, 4, 5: 8 into 70 goes 8 times (write 8 above). with 6 remainder (write the remainder). 8 into 60 goes 7 times (write the 7 above). with 4 remainder (write in the remainder). 8 into 40 goes 5 times with no remainder Division by a Decimal What if you want to divide by a number that is a decimal? You need to convert the number you are dividing by to a whole number first, by shifting the decimal point of both numbers to the right: Now you are dividing by a whole number, and can continue as shown previously. Remember: shift the decimal point of both numbers the same number of places to the right. Study Guide Topic 2 Decimals

9 49 Example 1: $100 $50 = 2 $10 $5 = 2 $1 50c = 2 In each of these equations, both the dividend and divisor have been divided by 10 to give you the next expression. In each case the answer is still the same. Example 2: Calculate Step 1: Move the decimal point in the divisor (0.04) to the right until you are dividing by a whole number becomes 4 (move 2 places or multiplied by 100) Step 2: Move the decimal place in the dividend (3.248) the same number of places becomes (2 places) Step 3: Write out the new question: Step 4: Divide as shown previously 4 into 3 goes zero times into 32 goes 8 times (write the 8 above). 4 into 4 goes once (write the 1). Write the decimal point in the answer. 4 into 8 goes twice (write the 2). TEP023 Foundation Mathematics

10 50 Rounding Decimals Numbers can also be rounded to any convenient number of decimal places. Some calculations result in a great many decimal places. However, it is often unnecessary to state all the decimal places in your final answer, so you need to round off to a given number of places after the decimal point. Or you may be required to round an answer to a certain level of precision, to the nearest hundredth, for example. First you need to know if you are rounding to tenths, or hundredths, etc. Or maybe to "so many decimal places". That tells you how much of the number will be left when you finish. Questions involving money will require you to know to round to two decimal places, ie rounding to the nearest cent. The key to rounding is to look at the numbers in the decimal places you are about to remove. Are they big enough that they should be taken notice of? Or are they small numbers that are insignificant and can be just dropped off. Look at the following examples. Example 1: Round 4.62 to one decimal place One decimal place means your answer should have only one number after the decimal point. Step 1: Step 2: Draw a line after the required number of decimal places Look at the size of the number after the line. Since it is smaller than 5, the number is just dropped off. Step 3: 4.62 to 1 d.p. = 4.6 Example 2: Round to the nearest hundredth. Step 1: Step 2: Hundredth means two decimal places. So, draw a line after two decimal places Look at the size of the number after the line. It is larger than 5. This means the number before the line needs to be rounded up (increased) by 1. Step 3: to 2 d.p. = 8.46 Study Guide Topic 2 Decimals

11 51 Estimation of Decimal Answers REMEMBER: Estimating is the practice of making a reasonable guess of an answer. Estimates are easiest to do when you use rounded numbers. In estimation you want to round off the numbers in the question so that you can EASILY work out an approximate answer without having to do any long calculations. Your level of accuracy will depend on your ability to work questions out in your head. You need to understand the place value of each number so that you don t lose important information or retain unnecessary detail. Example 1: Estimate Example 2: Estimate Application of Decimals Example 1: The school fees at college were $62 and were made up of a General Fee of $24.50, Magazines of $3.75 and Sports Fee of $ The Stationary Fee was also included in the $62. How much is the Stationary Fee? Step 1: Write out the problem as a mathematical statement: Stationary Fee = 62 ( ) Step 2: Calculate your answer: 62 ( ) = 22 The stationary fee is $22 TEP023 Foundation Mathematics

12 52 Study Guide Topic 2 Decimals