RATIONAL NUMBERS CHAPTER


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1 RATIONAL NUMBERS CHAPTER
2 70 CHAPTER RATIONAL NUMBERS Section. Recognizing, Reading, Writing and Simplifying Fractions What is a fraction? You have a circle. Cut it into two equal parts. Each part is called a half of a circle. Each part is a fraction of the circle. We can write a half as We now cut a circle into equal parts (fractions). Each part is called one fourth (or one quarter ) of a circle. We write this as If we take one part away, there are now three quarters left. We can write this as MATHEMATICS FOUNDATION
3 RECOGNIZING, READING AND WRITING FRACTIONS 7 A fraction is made of two parts: The numerator tells you how many parts you have. The denominator tells you how many equal parts in total. Note that all of the parts in the fraction must be of equal size. There is part shaded, so the numerator is. There are total parts in the shape, so the denominator is. So the fraction shaded is. There are parts shaded, so the numerator is. There are total parts in the shape, so the denominator is. So the fraction shaded is. Don t forget that the parts must be of equal size!
4 7 CHAPTER RATIONAL NUMBERS Practice What fraction is shaded in the shapes below? b) c) d) e) f) To write a fraction in words, we use numbers for the numerator, and ordinal numbers for the denominator. MATHEMATICS FOUNDATION
5 RECOGNIZING, READING AND WRITING FRACTION 7 Write the fractions in words. b) onethird c) d) 6 twofourths twoquarters Practice Write the fractions in words. b) 8 c) 0 d) e) 9 f) or Exception when the denominator is, we do NOT say second. Instead, we say half (or the plural, halves, if there is more than one). onehalf threehalves
6 7 CHAPTER RATIONAL NUMBERS Write the fractions in words. b) c) twothirds 8 Practice Write the words as fractions sixsevenths b) seventenths c) d) onehalf e) f) three quarters Write the fractions found in the sentences. Seven out of ten people enjoy going swimming. b) Two of the seven Emirates begin with the letter A MATHEMATICS FOUNDATION
7 PROPER FRACTIONS, IMPROPER FRACTIONS AND MIXED NUMBERS 7 Practice Write the fractions found in the sentences. b) Six of the seven Emirates joined at the same time in 97. c) Two of my three brothers like ice cream. d) e) Six of my nine notebooks are blue. = 9 = = 0 = whole 9 0 Practice Write the numerator to make each of the fractions below, equal to whole. 7 There are three different kinds of fractions; proper fractions, improper fractions and mixed numbers. The numerator is less than the denominator. This is called a proper fraction. A proper fraction is less than one whole one.
8 76 CHAPTER RATIONAL NUMBERS 7 The numerator is greater than the denominator. This is called an improper fraction. An improper fraction is greater than one whole one. A proper fraction is less than one whole one. We can also write 7 as is called a mixed number. We have added a whole number to a fraction: + = whole fraction mixed numbers We say this as, one and threequarters. MATHEMATICS FOUNDATION
9 PROPER FRACTIONS, IMPROPER FRACTIONS AND MIXED NUMBERS 77 State whether each of these is a proper fraction, an improper fraction, or as a mixed number. Then write the fraction in words. 9 7 Improper fraction Ninesevenths b) 0 Proper fraction Fivetenths c) Mixed number Two and threequarters Practice 6 State whether each of these is a proper fraction, an improper fraction, or as a mixed number. Then write the fraction in words. 7 6 b) c) d) e) 9 8 6
10 78 CHAPTER RATIONAL NUMBERS Percent means out of % = 00 What percent of each diagram is shaded? Since there are a total of 00 squares, the denominator is 00. There are shaded squares, so the fraction is. 00 This means that % of the diagram is shaded. b) There are shaded squares, so the fraction is. 00 This means that % of the diagram is shaded. MATHEMATICS FOUNDATION
11 PERCENT FRACTIONS 79 Practice 7 What percent of each diagram is shaded? % b) % There are also special percentages and their related fractions and decimals that you should be able to remember: = = 0. = % 0 = = 0. = 0% = = 0.7 = 7% 00 = = 00% (one whole) 00 00
12 80 CHAPTER RATIONAL NUMBERS Compare the diagrams below: We can say that,, and are all the same part of a whole. They are called equivalent fractions because they have the same value. They are equal. We use the sign = for (equal to) or (equivalent to): = or = Finding a fraction that is equal but with smaller numbers is called simplifying a fraction. This is done very easily with a calculator. 8 or 6 = 8 8 On a calculator the fraction button looks like this: a c b MATHEMATICS FOUNDATION
13 SIMPLIFYING FRACTIONS WITH A CALCULATOR 8 Using your calculator, simplify the fractions. 9 Enter a c b b 9 = b) Enter a 0 c 0 = So with our calculator, we found that 9, or = 9, or =. 0 and that 0 Practice 8 Using your calculator, simplify the fractions. 0 b) 0 c) d) 0 e) 8 0 f) 8 9 g) 0 h) 00 0 simplest form? That s okay! Not all fractions will simplify.
14 8 CHAPTER RATIONAL NUMBERS Section. Exercises. Write the fractions in words. b) c) 0 d) e) f) or. Write the words as fractions onethird b) two quarters d) onehalf e) threethirtieths f) seveneighths. Write the fractions found in the sentences. Six out of ten people go to college. b) Two boxes of chocolate are shared by six people. d) My mother had four of her seven brothers and sisters over for lunch. MATHEMATICS FOUNDATION
15 EXERCISES 8. What fraction is shaded in the shapes below? b) c) d). Write the numerator to make each fraction equal to. b) 7 c) 8 d) 6. State whether each of these is a proper fraction, an improper fraction, or as a mixed number. Then write the fraction in words. b) c) 9 8 7
16 8 CHAPTER RATIONAL NUMBERS 7. What percent of each diagram is shaded? % b) % 8. Using your calculator, simplify the fractions. b) 6 60 c) d) e) 8 f) 6 0 g) 7 8 h) 7 MATHEMATICS FOUNDATION
17 READING AND WRITING DECIMALS 8 Section. Reading, Writing, Comparing and Rounding Decimals Do you recall learning about reading and writing whole numbers? Thousands Ones hundred thousands ten thousands thousands hundreds tens ones 6 0 Our table shows that each place gets 0 times bigger as you move to the left. For example, hundred is ten times bigger than ten. thousand is 0 times bigger than hundred, and so on. This is the decimal system. We also use it for numbers that are smaller than one whole. Thousands Ones. Decimal hundred thousands ten thousands thousands hundreds tens ones ttenths hundredths thousandths
18 86 CHAPTER RATIONAL NUMBERS As you move to the right, the tenths place comes after the ones place. 0. = one tenth We use a decimal point (. ) to separate the units and tenths place. Look carefully at the difference between the place names. All place names to the right of the decimal point end with th. tenth hundredth thousandth Thousands Ones. Decimal hundred thousands ten thousands thousands hundreds tens ones ttenths hundredths thousandths Zero point seven two nine. For the number 0.79 above, write the place value of the digit. 7 tenths b) hundredths c) 0 ones d) 9 thousandths MATHEMATICS FOUNDATION
19 READING AND WRITING DECIMALS 87 Practice For the number 0.8, write the place value of the digit. 0 b) c) 8 d) Write the numbers in the correct place on the table. Thousands Hundreds Tens Ones tenths hundredths thousandths.. Practise Write the numbers in the correct place on the table. Thousands Hundreds Tens Ones tenths hundredths thousandths
20 88 CHAPTER RATIONAL NUMBERS Practice Write the place value of the underlined digit. 0.9 b) 7. c).8 d) Each digit after the decimal place is read separately. 0.8 is read as zero point four eight, and NOT zero point fortyeight is read as zero point seven two nine, and NOT zero point seven hundred twentynine. Write these numbers in words..6 three point six one. b).6 Practice Write these numbers in words. 0.8 b) 0. c).6 d) 7.89 There is also another way to read a decimal, by its place value. We use the place value furthest to the right to read the decimal. 0.8 is read as fortyeight hundredths because the digit furthest to the right (8) is in the hundredths place is read as seven hundred twentynine thousandths because the digit furthest to the right (9) is in the thousandths place. MATHEMATICS FOUNDATION
21 WRITING DECIMALS IN WORDS 89 Write these numbers in words reading the place value. 0. four tenths b) 0.6 sixtyone hundredths c) 0.6 two hundred thirtysix thousandths d) eight thousandths Practice Write these numbers in words reading the place value. 0.6 b) b) 0.6 If there is a whole number in front of the decimal point, we read that number and then say and instead of point before reading the decimal part of the number. Write these numbers in words reading the place value.. two and tenths b).9 Practice 6 Write these numbers in words reading the place value..7 b) 0.9 c) 9.6 d) 8.009
22 90 CHAPTER RATIONAL NUMBERS Write these numbers in digits. zero point six two c) two hundred six point two zero six Practice 7 Write these numbers in digits. zero point eight eight b) nine point three six two c) seventyeight point two If a number has more than decimal places, it can be written in groups of threes, just like with whole numbers. Recall: Whole Numbers Decimal Numbers We can read the decimal 0. as three tenths. We also read the fraction So, 0. = 0 as three tenths. 0 MATHEMATICS FOUNDATION
23 COMPARING DECIMALS 9 Write these fractions and mixed numbers as decimals. c) 7 = 0. b) = = 0.06 d) 6 = Practice 8 Write these fractions and mixed numbers as decimals = b) 6 = 000 c) = d) = e) 00 = f) 000 = You have already studied equivalent fractions. For example: 0 = 0 00 = When you write these as decimals you see: = 0. = =
24 9 CHAPTER RATIONAL NUMBERS Write equivalent decimals for these: 0.90, 0.,., tenths hundredths thousandths Practice 9 Write equivalent decimals for these: 0.7,.9, 0.800, 0.0 tenths hundredths thousandths We can also compare decimals, using the signs < or > or = Write the correct symbol, < or > or =, between these decimal numbers: Compare the 0. The tenths place is the 0. Since > we tenths place. same so we move to the have hundredths place. 0. > > 0. MATHEMATICS FOUNDATION
25 COMPARING DECIMALS 9 b) Compare the tenths place. Since 0 < we have 0.09 < < 0.07 c) The tenths place is the same so move to the hundredths place. The hundredths place is the same so look at the thousandths place The thousandths place has no digit. We know the value of this place is 0. The thousandths place has the value of. Since 0 < we have 0.97 < < 0.97 d) The tenths and the hundredths place have the same value. In the second number the thousandths place has no digit. We know the value of this place is = 0.97 These numbers are the same.
26 9 CHAPTER RATIONAL NUMBERS Practice 0 Write the correct symbol, > or < = between these decimal numbers: b) c).87.9 d) e).0.0 f) We round decimals in a very similar way to whole numbers. The only difference when rounding decimals, instead of replacing digits to the right of the given place value with zeros, we remove those digits. Therefore the steps to rounding decimals are as follows: Rules for Rounding Decimals: Step Underline the digit of the given place value. Step Circle the digit to its right. Step If that circled digit is from 0 to, the digit in the given place stays the same. b) If that circled digit is from to 9, add to the digit in the given place. Step Remove all digits to the right of the given place. MATHS FOUNDATION
27 ROUNDING DECIMALS 9 Do you remember your decimal place values?. Decimal tenths hundredths thousandths. 7 9 Let s try an example rounding with decimals. = 0.79 Round 0. to the nearest tenth Step Underline the digit of the given place value (tenths). 0. Step Circle the digit to its right (). 0. Step If that circled digit is from 0 to, the digit in the given place stays the same. b) If that circled digit is from to 9, add to the digit in the given place. Step Remove all digits to the right of the given place value. 0.
28 96 CHAPTER RATIONAL NUMBERS Round 0.76 to the nearest tenth. Step Underline the digit of the given place value (tenths) Step Circle the digit to its right () Step If that circled digit is from 0 to, the digit in the given place stays the same. Step b) If that circled digit is from to 9, add to the digit in the given place. (it is 6, so you add to the underlined digit 7 making it 8) Remove all digits to the right of the given place value. Round to the nearest tenth b) Practice Round to the nearest tenth. 0. b) 0.88 c) 0.7 d) 0.9 e) 0. f) 0.88 Round to the nearest hundredth b) Practice Round to the nearest hundredth. 0. b) 0.76 c) 0.9 d) 0.9 e) 0.77 f) MATHEMATICS FOUNDATION
29 ROUNDING DECIMALS 97 Round to the nearest thousandth b) Practice Round to the nearest thousandth. 0. b) c) d) e) 0.99 f).999 We can also round decimals to a given decimal place. Round 0.8 to one decimal place ( d.p.). is the st decimal place, so it is the same as rounding to the tenth Round 0.87 to two decimal place ( d.p.). 8 is the nd decimal place, so it is the same as rounding to the hundredth Round to three decimal place ( d.p.). is the rd decimal place, so it is the same as rounding to the thousandth
30 98 CHAPTER RATIONAL NUMBERS Practice Round each number as indicated. 0.8 to d.p. b) 0.67 to one decimal place c).6 to one decimal place d) to d.p. e) 0.7 to two decimal places f). to d.p. g) to d.p. h).998 to d.p. i) to three decimal places j) to d.p. is a way of writing very large or very small numbers. A number between and 0. For example,. b) A power with a base of 0, written as x 0 exponent. Look at the table below that uses powers with a base of 0. Can you see the pattern? We say Meaning Decimal Number 0 0 to the exponent to the exponent to the exponent to the exponent to the exponent to the exponent MATHEMATICS FOUNDATION
31 LARGE NUMBERS Move the decimal point to the left until you have a number between and is between and x 0 8 The number of places you moved the decimal point (8) is the exponent of the power of 0. b) Move the decimal point to the left until you have a number between and is between and 0. places.6 x 0 The number of places you moved the decimal point () is the exponent of the power of 0. Practice Write the numbers in x 0 b) c) 8 00 d) e) 80 f)
32 00 CHAPTER RATIONAL NUMBERS Write the numbers in decimal form.. x The exponent () tells you that you moved the decimal point places to the left, so you must move it back to the right b) 8.06 x places to the right Practice 6 Write the numbers in decimal form. x b) 7.7 x 0 c).6 x 0 d). x 0 e). x 0 f).6 x 0 7 MATHEMATICS FOUNDATION
33 LARGE NUMBERS 0 Now look at the table below that uses powers with a base of 0. Can you see the pattern? We say Meaning Fraction Decimal Number X X X X X X X X X X Write the numbers in Move the decimal point to the right until you have a number between and is between and x 0 The number of places you moved the decimal point (8) is the exponent of the power of 0. Since you moved in the opposite direction, the exponent is NEGATIVE. b) Move the decimal point to the right until you have a number between and is between and 0. places 7 x 0 The number of places you moved the decimal point () is the exponent of the power of 0, but NEGATIVE.
34 0 CHAPTER RATIONAL NUMBERS Practice 7 Write the numbers in x 0 b) c) d) e) 0.0 f) 0.9 b) Write the numbers in decimal form.. x 0 The tells you that you moved the decimal point places to the right, so you must move it back to the left b) 8.06 x 0 7 places to the left MATHEMATICS FOUNDATION
35 LARGE NUMBERS 0 Practice 8 Write the numbers in decimal form.6 x b). x 0 c). x 0 d).06 x 0 e) 7.8 x 0 f) 8.0 x 0 calculator. Multiply. 000 x The answer is too large for the calculator, so you see. 0 on your display.. x 0 0 This display actually means You now know that this can also be written in decimal form.
36 0 CHAPTER RATIONAL NUMBERS If a calculator displays.. The answer is too large for the calculator, so you see.0 on your display.. x 0 This display actually means You now know that this can also be written in decimal form. Practice 9 Write the calculator displays in and decimal form... x b) x c).0 d) 6. 0 e) 9. f) MATHEMATICS FOUNDATION
37 EXERCISES 0 Section. Exercises. For the number 0.67, write the place value of the digit. 0 b) c) 6 d) 7. State the place value of the underlined digit. 0.6 b).76 c).99 d) 0.. Write these numbers in words. 0.8 b) 0. c). d) 6.9 e) 0. f) Write these numbers in words using place value. 0.8 b) 0. c). d) 6.9 e) 0. f) 0.990
38 06 CHAPTER RATIONAL NUMBERS. Write these numbers in digits. zero point one seven c) seventy and twentysix hundredths e) six and two thousandths f) two hundred twentyone and three hundred six thousandths 6. Write these fractions and mixed numbers as decimals. c) 7 00 = b) = d) = = 7. Write equivalent decimals for these: , 9., 0.00 tenths hundredths 0.60 thousandths Write the correct symbol, > or < = between these decimal numbers: b) c) d) e).0.0 f) Round the number to the nearest tenth. 0.9 b) 0.8 c) 0.66 d) 0. e) 0.9 f) 0.98 MATHEMATICS FOUNDATION
39 EXERCISES Round to the nearest hundredth. 0.7 b) c) d) 0.8 e) 0.9 f) Round to the nearest thousandth b).9 c).8 d) e) f).999. Round each number as indicated. 0.8 to two decimal places b) 0. to d.p. c).9 to three decimal places d).98 to d.p. e) to d.p. f).99 to one decimal place g) to one decimal place h) to d.p. i).77 to d.p. j) to d.p. 7. b). c).0 d) 6.
40 08 CHAPTER RATIONAL NUMBERS Section. Converting, Comparing and Ordering Decimals, Fractions and Percents How do we compare fractions, decimals and percents to each other? We need to put them all in the same format. You can use your calculator to easily change fractions into decimals. This is done by dividing the numerator by the denominator. Convert the fractions and mixed numbers to decimals. 8 b) c) d) e) 0 keystrokes on the calculator ' 8 ' ' ' ' answer on the calculator If your answer repeats, like in c), we can write this in an easier way. = = 0.6 The bar over top of a digit or digits means that they repeat forever. MATHEMATICS FOUNDATION
41 CONVERTING BETWEEN FRACTIONS AND DECIMALS 09 Practice Convert the fractions to decimals. 7 b) 0 c) 6 d) e) f) 0 9 Recall your decimal place values:. Decimal tenths hundredths thousandths. 7 9 The place value of the digit furthest to the right tells you the denominator of the fraction. Write each decimal number as a fraction. 0. is in the tenths place, so b) 0. is in the hundredths place, so 0 00
42 0 CHAPTER RATIONAL NUMBERS c) 0.07 is in the thousandths place, so If there is a number (other than 0) in front of the decimal point, it simply becomes the whole number part of the mixed number. d).8 8 is in the tenths place, so e) 6. is in the hundredths place, so f).07 7 is in the thousandths place, so Practice Write each decimal number as a fraction. 0. = b) 0.7 = c). = d) 0.88 = e). = f) 0.9 = g). = h) 0.67 = Note: Anytime your answer is a fraction, (remember this?) Again this is easy to do with our calculators. MATHEMATICS FOUNDATION
43 CONVERTING BETWEEN FRACTIONS AND DECIMALS Write the fractions from the last example in simplest form. 0. = 0 = b) 0. = 00 (simplest form) c) 0.07 = = 0 d).8 = e) 6. = 8 = 0 6 = 6 00 f).07 = 07 (simplest form) 000 Practice Write each of your answers from Practice in simplest form (remember, you can use your calculator). = b) = c) = d) = e) = f) = g) = h) =
44 CHAPTER RATIONAL NUMBERS To convert a percent to a decimal, you divide by 00. Convert the percents into decimals. 7% b) % c) % d).% e) 8% Did you notice something? All the answers look similar to the questions. The only difference is that the decimal point has moved places to the left. Practice Convert the percents into decimals. 89% b) 7% c) 7% d) 99% e) 9% f).% g) 8.9% h) 00% i) 8% j) 00% MATHEMATICS FOUNDATION
45 CONVERTING BETWEEN PERCENTS AND DECIMALS To convert a decimal to a percent, you multiply by 00. Convert the decimals into percents x 00 % b) x 00 8% c).7.7 x 00 70% d) x 00 0.% e) x 00 00% Practice Convert the decimals into percents. 0.8 % b) 0.7 % c) 0. % d) 0.87 % e) 0.86 % f) % g).8 % h) 0.00 % i) % j).8 % Remember, percent (%) means out of 00. Write each percent as a fraction. 7% = % = 0 00 % = 00
46 CHAPTER RATIONAL NUMBERS Practice 6 Write each percent as a fraction. 0% = b) 8% = c) 6% = When the percentage is greater than 00, the result is a mixed number. Write each percent as a fraction or mixed number. 0 0% = 00 = 0 0 0% = = 0 0 0% = = 00 Practice 7 Write each percent as a fraction or mixed number. 0% = = b) 0% = = c) % = = d) 97% = = e) % = = f) 0% = = Write the fractions as percents. Step : Convert the fraction into a decimal using your calculator. = 0.6 Step : Convert the decimal into percent ( x00) 0.6 x 00 = 60% b) 8 = x 00 = 6.% MATHEMATICS FOUNDATION
47 CONVERTING BETWEEN PERCENTS AND DECIMALS Practice =.. x 00 = 0% b) c) d) 8 9 = x 00 =.6% e) 0 f) g) h) With the skills learned in this section, we can now convert between all three number forms   fractions, decimals and percents. Fill in the chart below using the skills learned in this section. Fraction Decimal Percent = x 00 = 7% b) 0 =.. x 00 = 0% c) 0.79 = = = %
48 6 CHAPTER RATIONAL NUMBERS Reminder: Don t forget to simply your answers with fractions! Practice 9 Complete the chart below using the skills learned in this section. Fraction or Mixed Number (simplest form) Decimal Percent 0 b) 0. c) 80% d) e). f) 0.% g) h) 0. i) 99.% MATHEMATICS FOUNDATION
49 CONVERTING BETWEEN PERCENTS AND DECIMALSE 7 The easiest way to compare and order fractions, decimals and percents is to convert everything to decimals. You can then use the rules for ordering decimals that you learned in Module. Which number is greater 7 7 or = is greater 7 8 = > 0.7 b) or 0.7 = 0.7 is greater c) or 80% = % is greater 80% = 0.80 Practice 0 Which number is greater? or b) or 0. c) or % d).8 or %
50 8 CHAPTER RATIONAL NUMBERS Order the numbers in ascending order:, %, 0.,, 0%, First, convert each number to decimal form: % 0. 0% Now, order the numbers using the decimal equivalents Finally, write the original numbers that match these decimal equivalents. % 0. 0% Practice Order the numbers in ascending order: 0.88, 0.9,, 90%, 9% 0 b) Order the numbers in ascending order:, 0.7, 7%, 0.66, 8 c) Order the numbers in descending order:.99, 0%, 8, 0 9,. MATHEMATICS FOUNDATION
51 EXERCISES 9 Section. Exercises Remember to simplify all fraction answers in this section and from now on!. Convert the fractions to decimals. b) 7 8 c) 0 d) e) 7 f) 7. Write each decimal number as a fraction 0.8 = b) 0.9 = c).6 = d) 8. = e) 0.07 = f) 0.0= g).00 = h) 0.0 =. Convert the percents into decimals. % b) 89% c) % d).8% e) 6% f).% g) 0.% h) 00%. Convert the decimals into percents. 0. % b) 0.9 % c) 0.9 % d) 0.8 % e) 0.0 % f) %
52 0 CHAPTER RATIONAL NUMBERS. Write each percent as a fraction or mixed number. % = b) 7% = c) % = d) 0% = e) 0% = f) % = g) % = h) 0% = i) 0% = j) 88% = k) 6% = l) 00% = 6. Write the fractions and mixed numbers as percents. b) c) d) 7. Complete the chart below using the skills learned in this section. b) c) Fraction or Mixed Number (simplest form) 8 Decimal 0.0 Percent % d) e) f) 0.9.% MATHEMATICS FOUNDATION
53 EXERCISES 8. Which number is greater? or b) or 0.7 c) or % d).6 or % 9. Put the numbers in ascending order. 8, 0.,, %, % 0 0. Put the numbers in descending order. %, 0 9,., 0%,.9, 8
54 CHAPTER RATIONAL NUMBERS Section. Adding, Subtracting, Multiplying and Dividing Fractions Performing the four arithmetic operations with fractions is very easy when using your module. Your calculator makes adding and subtracting fractions very easy. Add the fractions and mixed numbers. on your calculator 8 + = 8 + = 6 b) = = 6 MATHEMATICS FOUNDATION
55 ADDING, SUBTRACTING, MULTIPLYING AND DIVIDING FRACTIONS Practice Add the fractions and mixed numbers. + = b) = c) + 8 = d) = The fraction button can be used the same way for all four operations (add, subtract, multiply, divide). The way you used your calculator in Section A is the same for all math operations. Just be sure which operation is being asked for in the question! Perform the given operations on the fractions and mixed numbers. 8 b) 8 x 7 on your calculator = 89 = 0 = 8 x 7 = c) 0 0 = 0 0 = 7 8 d) + = = 60
56 CHAPTER RATIONAL NUMBERS Practice Perform the given operations on the fractions and mixed numbers. x = b) 9 = c) 8 = d) = e) 8 6 = f) 6 + = g) 0 7 = h) 8 x = The word of tells you that you must multiply. How much is of 0? Show with a diagram. Of course, this can be done on a calculator, but it is also important that you can see and understand what is being asked for. Remember that means that there are total parts and is shaded. We need to separate 0 into two equal parts, and choose one of those parts. We have now separated the 0 shapes into parts, and chosen of those parts (the shaded circles). Counting the shaded circles, we can see that of 0 = Check on your calculator: x 0 = b) How much is of 6? Show with a diagram. means that there are total parts and are shaded. We need to separate 6 into equal parts, and choose of those parts. Counting the shaded circles, we can see that of 6 = Check on your calculator: x 6 = MATHEMATICS FOUNDATION
57 FINDING A FRACTION OF A NUMBER Practice How much is Diagram: of? Show with a diagram then check with your calculator. Check on your calculator: b) How much is Diagram: of? Show with a diagram then check with your calculator. Check on your calculator: c) How much is Diagram: of? Show with a diagram then check with your calculator. Check on your calculator:
58 6 CHAPTER RATIONAL NUMBERS Remember from the last section that of means to multiply. of 0 = x 0 = 0 percent of a number. You need to remember how to change a percent to a decimal. % = 00 = 0. (remember to simply 00) Find the percent of the numbers below. % of 0 Change the percent to a decimal and multiply. b) 8.% of AED00 Change the percent to a decimal and multiply. 0. x x 00 =. = AED In the last example, notice that a unit was used (AED), so we put the units in the answer. Practice Find the percent of the numbers below. Show how you change the percent to a decimal % of 0 = = b) 8% of AED00 = = c) 8% of 0kg = = d).% of AED0 = = MATHEMATICS FOUNDATION
59 FINDING A PERCENT OF A NUMBER 7 Section. Exercises. Perform the given operations on the fractions and mixed numbers. 9 0 = b) 8 6 = 7 c) + = d) 8 x = e) 7 x = f) = g) 9 x 6 = h) 7 8 =. Draw a diagram and check with your calculator to solve the multiplication questions. of Diagram: Check on your calculator: b) 7 8 of Diagram: Check on your calculator:. Find the percent of the numbers below. Show how you change the percent to a decimal 0% of 0 = = b) % of AED60 = = c) % of AED0 = = d).7% of 0 = =
60 8 CHAPTER RATIONAL NUMBERS Section. Ratios and Proportions Ratio the comparison of two numbers. For example, from the diagram below, the ratio of shaded squares to nonshaded squares is to. This means that there are shaded squares and nonshaded squares. There are three ways that we write a ratio: to : However, all three ways are said the same way, two to three. Use the shapes above to write the ratios. Write the ratios in three different ways. Description Ratio The ratio of shaded circles to nonshaded circles. :, to, b) The ratio of nonshaded squares to nonshaded circles. c) The ratio of shaded circles to all circles. :, to, :7, to 7, 7 MATHEMATICS FOUNDATION
61 EQUIVALENT RATIOS AND PROPORTIONS 9 In example c), the ratio included all circles, which means you must add the shaded circles and nonshaded circles together. Practice Use the shapes above to write the ratios. Write the ratios in three different ways. Description Ratio The ratio of shaded squares to nonshaded squares. b) The ratio of nonshaded squares to all squares. c) The ratio of all squares to all circles. d) The ratio of shaded circles to shaded squares. e) The ratio of all squares to nonshaded circles. The ratio of non shaded squares to shaded squares is to. When we write this ratio in another form, it is. This means that there is nonshaded square for every shaded squares. The picture above still shows a ratio of nonshaded square for every shaded squares. Looking at all the squares together, the ratio of nonshaded squares to shaded squares is to. Since both ratios describe the same thing, they are called equivalent ratios. =
62 0 CHAPTER RATIONAL NUMBERS Write equivalent ratios for the shapes in the diagram above. nonshaded squares to shaded squares = = 6 = 8 Practice Write equivalent ratios for the shapes in the diagram above. moons to stars = = = The order of a ratio is very important! For example, is not the same as. Let s look at the relationship between equivalent ratios. = = 6 = 8 Ratios are equivalent if they have their cross products are equal. 6 x = 6 and x 6 = 6, so these ratios are equivalent (equal). 8 x 8 = 6, and x =, so these ratios are not equivalent. When two or more ratios are equivalent, they are said to be in proportion. = 6 The above ratios are equivalent, so they are in proportion. MATHEMATICS FOUNDATION
63 EQUIVALENT RATIOS AND PROPORTIONS Are the ratios equivalent (in proportion)? Show your work. and 0 x 0 = 0 x = 0 Yes, the ratios are in proportion (equivalent). b) and x = 8 x = 9 No, the ratios are not in proportion (not equivalent). Practice Are the ratios equivalent (in proportion)? Show your work. 6 and 9 b) c) and 0 and 6 d) e) 9 and 6 and
64 CHAPTER RATIONAL NUMBERS Very similar to simplifying fractions, we can simplify ratios. Recall simplifying fractions with your calculator. 0 = 0 and are equivalent (have the same value), but is in simplest form. The same is true for ratios. However be careful, because a ratio must always have parts, or terms! Simplifying the ratios: 6 = Simplifying the ratio is exactly the same as simplifying a fraction. b) 6 = If this were a fraction, it would simplify to. However we need to keep parts or terms, so it must remain as. Practice Simplify the ratios using your calculator. Be sure your answers have terms! 6 8 = b) 9 = c) = d) 0 = e) 0 = f) = 7 g) 6 0 = h) 8 = i) 6 = MATHEMATICS FOUNDATION
65 SOLVING PROPORTIONS We learned that a proportion is when ratios are equal to each other. For example, the ratios 6 and make a proportion because they are equal ( x 9 = 8; x 6 = 8). 9 We write a proportion like this: 6 = 9 6 = Since we know the cross products must be equal, we know that x = 8 x 6 = 8 Therefore, the missing value must be 9 ( x 9 = 8) To complete the proportion, it looks like this: 6 = 9 Complete the proportions. Show your work. = 8 x 8 = 8 x = 8 The missing number must be, because x = 8. The completed proportion is = 8 b) = 6 x 6 = 0 x = 0 The missing number must be, because x = 0. The completed proportion is = 6
66 CHAPTER RATIONAL NUMBERS Practice Complete the proportions. Show your work. = 6 b) 0 = 0 c) 6 = 8 d) = 8 9 e) = 6 MATHEMATICS FOUNDATION
67 EXERCISES Section. Exercises. Use the shapes above to write the ratios. Write the ratios in three different ways. Description Ratio The ratio of shaded circles to nonshaded circles. b) The ratio of all circles to all squares. c) The ratio of shaded squares to shades circles. d) The ratio of shaded squares to all squares. e) The ratio of nonshaded circles to shaded circles.. Write equivalent ratios from the picture above. boys to girls = =
68 6 CHAPTER RATIONAL NUMBERS. Are the ratios equivalent (in proportion)? Show your work using cross products. b) c) d) e) f) 6 and 9 and 0 and 9 and 6 and 7 and 0. Simplify the ratios using your calculator. Be sure your answers have terms! 6 8 = b) 9 = c) = d) 0 = e) 0 = f) = 7 g) 6 8 = h) 0 = i) 6 =. Complete the proportions. Show your work. = b) = 6 c) 8 = 9 d) 7 = 6 e) 6 = MATHEMATICS FOUNDATION
69 SKILL BUILDERS SUBTITLE 7
70 8 CHAPTER RATIONAL NUMBERS SKILL BUILDERS Section. Recognizing, Reading, Writing and Simplifying Fractions. What fraction is shaded in the shapes below? b) c) d). Write the fractions in words. 8 b) c) d) 9 e) f) 0 or MATHEMATICS FOUNDATION
71 SKILL BUILDERS 9. Write the words as fractions onehalf b) three fourths e) ninetwelfths f) onetenth. Write the fractions found in the sentences. One out of every three students go to college. b) Five out of eight people went to Dubai to shop. d) Three of my four sisters are coming home for dinner tonight. e) I scored nine out of ten on the quiz today.. Write the numerator to make each of the fractions below, equal to whole State whether each of these is a proper fraction, an improper fraction, or as a mixed number. Then write the fraction in words. b) 7 0 c)
72 0 CHAPTER RATIONAL NUMBERS 6. What percent of each diagram is shaded? % b) % E. Simplifying Fractions with a Calculator 7. Using your calculator, simplify the fractions. 0 8 b) c) 6 0 d) 0 e) f) 9 0 g) h) 0 7 SKILL BUILDERS Section. Reading, Writing, Comparing and Rounding Decimals. For the number 0.96, write the place value of the digit. b) 9 c) 6 d) 0. State the place value of the underlined digit. 0. b).6 c) 0.96 d) 0.0 MATHEMATICS FOUNDATION
73 SKILL BUILDERS. Write these numbers in words. 0. b) 0.6 c) 6.7 d) 8.89 e) 77.8 f) Write these numbers in words using place value. 0. b) 0.89 c) 7.6 d).0 e) 00.0 f) 0.9. Write these numbers in digits. zero point three b) one point two two six c) six and fourteen hundredths d) nine and eight thousandths
74 CHAPTER RATIONAL NUMBERS 6. Write these fractions and mixed numbers as decimals = b) 000 = c) 6 = d) = Write equivalent decimals for these: , 7., tenths hundredths 0.0 thousandths Write the correct symbol, > or < = between these decimal numbers: b). 0.6 c) d) e).0.0 f) Round the number to the nearest tenth. 0.8 b) 0.7 c).9 d). e) 0.7 f) 0.9 MATHEMATICS FOUNDATION
75 SKILL BUILDERS 0. Round to the nearest hundredth. 0.6 b) 0.9 c) 0.9 d).78 e) 0.99 f) Round to the nearest thousandth b).9 c).8 d) e) 0.9 f) Round each number as indicated. 0.8 to one decimal place b) 0. to d.p. c).9 to two decimal places d).808 to d.p. e) 6. 9 to d.p. f).9 to one decimal place g) to one decimal place h) 0.00 to d.p. i). to d.p. j).0 7 to d.p.. b).7 c). d) 8.
76 CHAPTER RATIONAL NUMBERS SKILL BUILDERS Section. Converting, Comparing and Ordering Decimals, Fractions and Percents. Convert the fractions to decimals. b) 8 c) 0 6 d) e) f) 6. Write each decimal number as a fraction 0. = b) 0. = c) 0.8 = d).7 = e) 0.0 = f) 0.06= g).00 = h) 0.9=. Convert the percents into decimals. % b) % c) 8% d) 7.% e) 99% f).7% g) 0.8% h) 0%. Convert the decimals into percents. 0.0 % b) 0.7 % c) 0.7 % d) 0. % e) 0.00 % f) 0 % MATHEMATICS FOUNDATION
77 SKILL BUILDERS. Write each percent as a fraction or mixed number. 0% = b) 7% = c) 0% = d) 7% = e) 60% = f) 88% = g) 70% = h) 0% = i) 8% = j) 6% = k) 0.8% = l) 000% = 6. Write the fractions and mixed numbers as percents. b) c) 0 d) Complete the chart below using the skills learned in this section. Fraction or Mixed Number (simplest form) Decimal Percent b) 6 0. c) 0% d) 6 e). f).7%
78 6 CHAPTER RATIONAL NUMBERS 8. Which number is greater? b) c) or or 0.7 or % d).6 or % 9. Put the numbers in ascending order. 8, 0.78,, 8%, 9% Put the numbers in descending order. 7%, 0,.9, 00%,., 7 MATHEMATICS FOUNDATION
79 SKILL BUILDERS 7 SKILL BUILDERS Section. Adding, Subtracting, Multiplying and Dividing Fractions. Using your calculator, perform the given operations on the fractions and mixed numbers. 9 0 = b) 8 = c) = d) x = e) 7 8 x = f) 0 = g) 0 9 x = h) 6 9 =. Draw a diagram and check with your calculator to solve the multiplication questions. of Diagram: Check on your calculator: b) of 9 Diagram: Check on your calculator:
80 8 CHAPTER RATIONAL NUMBERS. Find the percent of the numbers below. Show how you change the percent to a decimal 0% of 0 = = b) % of AED00 = = c) % of AED 80 = = d).% of 8 = = SKILL BUILDERS Section. Ratios and Proportions. Use the shapes above to write the ratios. Write the ratios in three different ways. Description Ratio The ratio of all squares to all circles. b) The ratio of shaded squares to all circles. c) The ratio of nonshaded squares to shaded circles. d) The ratio of shaded squares to all squares. e) The ratio of all circles to all squares.. Write equivalent ratios from the picture above. boys to girls = = MATHEMATICS FOUNDATION
81 SKILL BUILDERS 9. Are the ratios equivalent (in proportion)? Show your work using cross products. b) c) d) e) f) 8 and and 8 0 and 0 9 and 6 and 0 and Simplify the ratios using your calculator. Be sure your answers have terms! = b) = c) 0 0 = d) 8 0 = e) = f) 0 = g) 80 9 = h) 00 0 = i) 000 =. Complete the proportions. Show your work. 6 = b) = 0 c) 8 0 = 9 d) = e) 6 = 8
82
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