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4Fractions, decimals TestP1034.1 Equivalent proportionsYou will learn to:CONFIDENCE Convert between fractions, decimals and percentages Compare fractions, decimals and percentages Write a fraction as a decimal.Why learn this?Information aboutproportions can begiven as fractions,decimals orpercentages. Youneed to be able toconvert betweendifferent types.FluencyWhat is the decimalequivalent of1 2 3 4 ExploreCan you write the number π as a fraction?1 10 7 10 Warm upExercise 4.11 Write as a percentage.a 72 out of 100b 25 out of 50c 0.732 Work outa 34 10c 9021 100b 286 10003 Write each decimal as a fraction.a 0.75b 0.3d 1.29c 0.94 Write or between each pair of decimals.b 0.62 u 0.603c 4.049 u 4.5a 0.15 u 0.1725 Sort these into five sets of equivalent 17%70%75%6 Write each fraction as a decimal and as a percentage.1a 20 0.u u%3c 8 0.u u%7e 8 0.u u%79Topic links: Measures3b 20 0.u u%7d 20 0.u u%11f 20 0.u u%Key pointYou can write a fraction as a decimalby dividing the numerator by thedenominator.2 4 2 4 0.5Q5 hintRewrite fractions so they havedenominator 100 to work out thepercentage.Q6 hintUse the S D key on your calculatorto change the fraction into a decimal.

7 Write each fraction as a decimal and as a percentage.32a 40 0.u u%71b 80 0.u u%90c 120 0.u u%105d 150 0.u u%11e 16 0.u u%18f 125 0.u u%8 Problem-solving Which shape has the longer perimeter?BA0.3 m0.1 m0.1 m3 m208 m207 m200.4 m1 m209 Use the decimal equivalent of each fraction to write the sets in order,smallest to largest.1335a 20    5    8 3167b 8    20    40 10 Put these values in order, smallest to largest.13 20    0.62   64.5%   9 out of 16Key pointWorked exampleWrite 0.245 as a fraction.245 0.245 100049 200There are 3 decimal places so245 has been divided by 1000Divide numerator and denominator by 5 to simplify.49 and 200 don’t have any common factors so itcannot be simplified further.11 Write each decimal as a fraction. Simplify where possible.85 a 0.85 100ub 0.375 1000c 0.84e 0.23g 0.19d 0.125f 0.875h 0.444A terminating decimal ends aftera definite number of digits, forexample 0.39 and 1.042.You can write any terminatingdecimal as a fraction.Q11 hintUse the number of decimal placesto decide whether the denominatorshould be 100 or 1000.12 Write each percentage as a fraction. Simplify where possible.uua 35% 0.35 uub 6% c 88%d 5%20100125 e 12.5% 0.125 f 37.5%g 45.8%h 1.2%1000uExplore10013 Explore Can you write π as a fraction?Choose some sensible numbers to help you explore this situation. Thenuse what you’ve learned in this lesson to help you answer the question.Pi 3, Section 4.1Reflect14 Reflect This lesson uses a lot of mathematical terms, such as terminating equivalent percentage common factor.Write down what each of these terms means, in your own words.Which terms are new, and which ones have you met before?Unit 4 Fractions, decimals and percentages80

CheckP93MasterStrengthenP95ExtendP99TestP1034.2 Recurring decimalsYou will learn to:CONFIDENCE Write recurring decimals as fractions.Why learn this?Lots of calculationsin real life have ananswer that is arecurring decimal.FluencyRound each number to2 decimal places. 1.454 6.087 3.3642 10.4985ExploreHow can you tell if a fraction will be aterminating or a non-terminating decimal?Exercise 4.2Warm up1 Work outa 726 62 aiiib iiic iiib 7125 5c 345 4d 842 8Jarred thinks of a number. Half of it is 240. What is 4 of his number?What number did Jarred think of?11Sophie thinks of a number. 6 of it is 12. What is 3 of her number?What number did Sophie think of?11Amee thinks of a number. 10 of it is 250. What is 5 of her number?What number did Amee think of?13 Reasoning a Use your calculator to match each fraction to itsequivalent d example1 9)1. 0000.1111 )1.10101 010 91 0. 1 99 doesn’t go into 1 so write a 0 in the units column.1There are now 10 tenths.9 goes into 10 once with remainder 1.There are now 10 hundredths.Continue like this and the decimal recurs.4 Write each fraction as a decimal.1a 8 1e 6 811b 12 5f 6 5c 12 2g 9 12161613Key pointb Use the decimals to order the fractions in part a, smallest to largest.Write 9 as a decimal.0Q2b hint7d 8 8h 9 A recurring decimal contains adigit, or sequence of digits, whichrepeats itself forever. A dot over thedigit shows it recurs. For example, 0.111 11 . 0. 1

5 Problem-solving / Reasoning Nira works out 12 18 and getsan answer of0.6666667on her calculator.What is the equivalent fraction?Discussion What has the calculator done?6 Use a written method to work out each division. Write your answersas recurring decimals using dot notation.a 823 3b 375 9c 37 564 3d 6385 9e 97 12f 1756 12Investigation Reasoning / Problem-solvingCaroline says, ‘Some fractions can be written as terminating decimals but some are recurring decimals’.1 Write each fraction as a decimal.11111111111 10 11 12 2    3    4    5    6    7    8    9 2 Sort them into terminating decimals and recurring decimals.3 Jack says, ‘If the denominator of a fraction is even, it will be a recurring decimal’.Find an example to show that Jack is wrong.4 Which denominators give terminating decimals?Discussion Caroline says, ‘I think it’s to do with the fact that 2 5 10.’ What do you think she means?5 Investigate what happens if the numerator is not 1.7 a Which bag has the greater proportion of red counters?b What is the proportion of blue counters in each bag?B8 The pie chart shows the first International School Languageslanguage of people workingChinesein a summer school.EnglishWrite the proportion of eachFrench120 150language as a fraction and asRussian60a percentage.30Q8 hintHow many degrees represent thewhole pie chart?9 Explore How can you tell if a fraction will be a terminating or anon‑terminating decimal?Is it easier to explore this question now you have completed the lesson?What further information do you need to be able to answer this?Pi 3, Section 4.2Reflect10 Reflect A shorthand way of writing a recurring decimal is to use adot, or dots, over the numbers that repeat. For example, 0.222 2 iswritten as 0. 2 Write down five other short forms that you use in maths. Do you thinkthese short forms are useful or not?ExploreAUnit 4 Fractions, decimals and percentages82

CheckP93MasterStrengthenP95ExtendP99TestP1034.3 Adding and subtracting fractionsYou will learn to:CONFIDENCE Add and subtract fractions Add and subtract mixed numbers.FluencyWhat is11 2 4 Why learn this?Statisticians addand subtractfractions to workout the probablyof different eventshappening (or nothappening).11 2 – 4 ExploreHow many years ago did people startwriting fractions?33 5 – 10 31 8 2 Warm upExercise 4.31 Write each improper fraction as a mixed number.201412b 8 c 7 a 5 23d 3 2 Write each mixed number as an improper fraction.312b 1 3 c 5 8 a 2 4 c 4 10 3 Work out51a 12 3 12d 3 5 51b 6 – 3 33c 4 – 8 34 Work out each calculation. Give your answer asa mixed number where necessary.5351711127b c 9 – 9 9 a 8 8 8 12 12 12 111d 2 – 3 4 334e 5 10 – 4 241f 3 – 9 6 111b 3 4 5 111c 4 5 6 Q4a hint58185 Work out111a 2 3 4 7 Problem-solving In a clothes shop, 5 of clothes are suits, 3 aretrousers and the rest are tops. What fraction of the clothes are tops?2Worked examplea Work out 3 6 1 4 533 65 1 34 (3 1) ( 65 34 )10 4 ( 9 )12 4 1912 7 4 1 12 7 5 1283121818188 5186 Problem-solving Work out the missing number.3534a 5 4 u 2 b 6 5 – u 111838Add the whole number parts andadd the fraction parts separately.Convert the fractions to equivalentfractions with a common denominator.Change the improper fraction to a mixed numberso you can add the whole number parts.1818181818

b Work out 2 2 – 1 6 51Subtract the whole number partsand the fraction parts separately.2 21 – 1 65 (2 – 1) ( 21 – 65 ) 1 ( 63 – 65 ) 1 (– 62 ) 1 – 31 23 8 Work outa 1 5 2 10 3b 2 4 3 5 c 2 3 2 5 d 3 5 1 3 e 4 3 2 6 f 2 10 3 5 g 1 12 1 6 5h 3 4 1 5 4i 2 9 3 6 a 2 10 – 1 5 2b 2 4 – 1 8 c 4 6 – 3 3 d 3 5 – 1 3 e 4 3 – 2 6 f 3 10 – 3 5 g 4 12 – 2 6 h 3 4 – 1 5 i 3 6 – 2 9 14111121523274559 Work out74311132551323735510 Yazdi uses 2 4 litres of white paint and 2 5 litres of blue paint. How manylitres of paint did he use in total? Give your answer as a mixed number.3111 Peter walked 4 6 km, Brenda walked 3 5 km. How much further did Peterwalk?5412 A farmer needs 3 5 metres of netting for his chickens. He already has111 20 metres. How much more does he need?213 A box of apples weighs 3 7 kg and a box of pears weighs 2 5 kg. Howmuch do they weigh altogether?53Investigation Reasoning1 Write down six fractions that are betweena 0 and 10b 0 and 2 11c 2 and 11212 Write down two fractions that are between31a 4 and 4 32b 8 and 8 87c 10 and 10 34d 5 and 5 14 Explore How many years ago did people start writing fractions?What information do you need to start answering this question?15 Reflect Choose one of the parts of Q9 that you felt confident inanswering. How would you explain the method you used to a classmatewho had missed this lesson?Pi 3, Section 4.3Unit 4 Fractions, decimals and percentagesReflect ExploreDiscussion In how many different ways can you answer the questions in this investigation?84

CheckP93MasterStrengthenP95ExtendP99TestP1034.4 Multiplying fractionsYou will learn to:CONFIDENCE Use strategies for multiplying fractions.FluencyWhat is3 7 of 212 5 of 152 9 of 36Why learn thisYou multiply fractions when working out thedistance you can travel with half a tank of fuel.5 6 of 72ExploreHow many times can you cut a piece ofpaper in half?Exercise 4.41 Work outa 3 4 b 2 7 32c 5 2d 4 9 5e 6 41f 3 61Warm up22 Write each improper fraction as a mixed number in its simplest form.32a 10 22b 4 35c 5 34d 6 Q3 Literacy hint3 What is the inverse operation ofa multiplying by 10An inverse operation is theopposite operation.b dividing by 8?4 Simone says, ‘I’m thinking of a number. I multiply it by 8 and thendivide the answer by 8, and get 5.’ What number was she thinking of?5 Reasoning Work outa 8 1 4b 8 4 c 8 4 1d Explain why the answer is the same in parts a, b and c.1Q6b hint6 Work outa 6 3 b 5 5 2c 3 35d 7 71Work out one-fifth of 5, then multiplyit by 4.4Q7a hint7 Work outa2 5 250b2 3 360c2 3 of3608 Real / Problem-solving A car has 45 litres of fuel in the tank.3The driver uses 5 of the fuel. How many litres of fuel are left?9 Reasoning Which of these products will have an answer less than 1?a 5 3 1b 9 7c 2 15 3d 4 62485Work out one-fifth of 250, thenmultiply by 2.

CheckP93MasterStrengthenP95ExtendP99TestP1034.5 Dividing fractionsYou will learn to:CONFIDENCE Divide by fractions.FluencyHow many 2s are in 8 4s are in 20 1s are in 3 5s are in 100?ExploreDoes division always makesomething smaller?Why learn thisReal-life measurements are not usually whole numbers.You need to be able to calculate with fractions too.Exercise 4.5Warm up1 Write down the common factors of each pair of numbers.a 20 and 35b 16 and 40c 27 and 36d 25 and 552 Work out1a 4 82b 3 1042c 5 3 91d 10 3 3 Write each improper fraction as a mixed number.8a 3 35b 6 28c 3 4 Complete these calculations for each diagram.1a1111   1 4 u  1 u 4   4 u 1444b42131313c1313131212u u u3122 6 31   2 u u1212125 Use your answers to Q4 to work outa how many quarters are in 1c how many halves are in 33 u u  3 u u  u u ub how many thirds are in 2d how many quarters are in 3.6 Copy and complete.a 4 2 b 3 5 11c 2 6 1d 6 4 17 How many 4 litre bottles can be filled from a 3 litre container?18 Katie cuts 4 chocolate cakes into eighths. How many slices are there?9 Reasoning Chris cuts 3 carrot cakes into equal slices. He has 30 slices.What fraction did he cut each cake into?87Q6a hintHow many halves are in 4?

CheckP93MasterStrengthenP95ExtendP99TestP1034.6 Comparing proportionsYou will learn to:CONFIDENCE Use a calculator to work out percentages Compare proportions.FluencyWork out 20% and 5% of 500 360 1800Why learn this?Nutritional information on food packets can begiven as a fraction, a percentage or as a decimal.ExploreHow do calculators show 150%?Exercise 4.61 Work outa 25% of 1200b 15% of 1200c 30% of 1200d 75% of 12002 Write these scores in order, from lowest to highest.Warm up6 out of 10    13 out of 20    14 out of 25    29 out of 503 Write down 6 sets of equivalent fractions, decimals and percentagesfrom .880%34.33.3%4 Write these amounts in order, from smallest to largest.4a 0.409    9     41%    11 out of 202b 0.67    66%    34 out of 50    3 5 Reasoning Which is better value?1a 3 off or a discount of 30%1b A reduction of 15% or a saving of 10 c 20% off or pay 5 of the cost46 Problem-solving A swimming team has 20 members. 11 membersare girls. What percentage of the swimming team are boys?89Topic links: Statistics, Proportion, Pie chartsSubject links: Cookery (Q7), Geography (Q13)

7 Reasoning / Problem-solving Here is someBrand Anutritional information for two similar products.Per 100 ga Write the proportion of fat in Brand A asProteina percentageCarbohydrateb Which brand has a higher proportion of fat?Fatc Write the proportion of protein in Brand A asa percentage.d Which brand has a higher proportion of protein?e Which brand has a higher proportion of carbohydrate?Brand BPercentage content12.5 gProtein15%23 gCarbohydrate22.5%4.5 gFat5%8 Problem-solving The ratio of boys to girls in a class is 11 : 14.What percentage of the class are girls?9 Problem-solving / Reasoning This pie chart shows the languagesstudied by students at a school.Languages being studiedQ9 Strategy hintUse the pie chart to work out thefraction of students who studyFrench.SpanishFrench150 120 Germana 15 people study French. How many people were surveyed?b How many people studyi Germanii Spanish?10 Problem-solving / Reasoning Dave’s driving theory test had20 questions in it. He scored 70%.How many questions did he get wrong?11 Problem-solving / Reasoning 120 students voted in a schoolcouncil election. 56 people voted for Matt, the rest votedfor Hassan. What percentage of votes did Hassan win,to the nearest 1%?Q12 hint12 Max scored 200 out of a possible 250 in his maths test.What was his score as a percentage?200 out of 250 250 20013 Real There are 50 states in the USA. 27 of them have no coast.What percentage of the states have no coast?15 Reflect Q10 and Q11 are tagged as problem-solving questions. Thismeans that there are alternative ways of finding the answer. Whichmethod or methods did you choose? Explain why.Pi 3, Section 4.6Unit 4 Fractions, decimals and percentagesReflect14 Explore How do calculators show 150%?Look back at the maths you have learned in this lesson. How can youuse it to answer this question?Explore 200 25090

CheckP93MasterStrengthenP95ExtendP99TestP1034.7 FINANCE: Percentage changeYou will learn to:CONFIDENCE Work out a percentage increase or decrease.Why learn this?Understandingpercentage changemeans you canmake sure you don’tget overcharged orshort-changed.FluencyWrite 8 as a percentage of 10 16 100 200ExploreWhat does ‘Up to 100% less sugar’actually mean?Exercise 4.7: Percentage changeWarm up1 Work out 30% ofa 70b 25c 102 Write each improper fraction asi a mixed numberii a decimaliii a percentage.150375a 100 b 150 d 4460c 50 e 90225d 50 Key point3 Increase each amount by 20%.a 160 210% 220% 160 An increase of 20% meansoriginal amount (100%) 20% of that amount 120% of the amount. b 3400 210% 220% 3400 c 25 0004 Work outa 180 increased by 20%c 2500 increased by 30%e 3500 increased by 5%b 2600 increased by 15%d 4200 increased by 25%f 4250 increased by 50%5 Decrease each amount by 25%.a 240 4 u25% of 240 u 240 – u ub 320091Q5a hint25% 4 so divide by 4.1c 24 000

MasterP79StrengthenP95CheckExtendP994 Check upLog how you did on yourStudent Progression Chart.Equivalent proportions1 Copy and complete the table showing equivalent fractions, decimalsand percentages.Write fractions in their simplest form.FractionDecimalPercentage0.452 25 5%11 2 0. 3 2 Write 0.82 as a fraction in its simplest form.3 Write each fraction as a decimal.3a 8 1b 9 4 Write or between each pair of fractions.1911a 25 u 15 27b 3 u 10 5 Write these values in order, smallest to largest. 20    0.51   54.5%   8 out of 1511Fraction calculations6 Work out51a 9 6 93b 10 – 4 1 527 Work out the sum of 4 , 6 , and 3 . Write your answer as a mixednumber.8 Work out each calculation. Write your answers in their simplest form.33a 4 5 2 2c ( 5 ) 32b 9 7 9 Work outa 4 5 1 10 1b 1 4 2 5 373c 3 10 – 1 5 9d 2 3 – 1 10 4210 Chantel needs 5 lengths of ribbon. Each length is 4 metres.She says, ‘I know I will need less than 5 metres altogether’.How does she know?311 Work out 13 3 6212 Work outa 7 2 1b 4 5 1c 8 3 213 Write the reciprocal of6a 7 93b 7TestP1032c 3 3

Percentages14 The ratio of boys to girls in a class is 9 : 11. What percentage of theclass are boys?15 A football club has 10 male players and 15 female players.What percentage of the club area maleb female?16 Work outa 460 increased by 25%b 4500 increased by 15%.17 Anna has to pay 20% tax on a laptop. The laptop costs 450 beforetax. How much does Anna pay in total?18 Brian invests 5000 with yearly interest paid at 1%. How muchinterest will he earn after 1 year?19 A flat increased in value by 150%. It was worth 200 000. What is itsnew value?20 Work outa 30 reduced by 25%b 200 reduced by 5%21 Steve’s car has gone down in value by 15%. He paid 8600.What is it worth now?22 20% of an amount is 25. What is the amount?Just guessingFeeling doubtfulReflect23 How sure are you of your answers? Were you mostlyConfidentWhat next? Use your results to decide whether to strengthen orextend your learning.Challenge24 Jarred draws a square.He colours in half red. He colours in half of the rest orange.He continues like this, colouring in half of what’s left using thecolours of the rainbow: red, orange, yellow, green, blue, indigo,violet.a What fraction of the square will be coloured indigo?b What fraction will be coloured violet?1c Freya tries a similar experiment, colouring in 3 of the shape eachtime. Explore the different fractions in Freya’s shape.Unit 4 Fractions, decimals and percentages94

MasterP79STRENGTHENCheckP93ExtendP99TestP1034 StrengthenYou will: Strengthen your understanding with practice.Equivalent proportions1 Write each of these as a decimal number.a 43 100b 17 100c 32 100d 98 100e 52%f 65%Q1e hint52% is the same as 52 out of100 52 1002 Write each decimal as ai percentageii fraction in its simplest form.a 0.13b 0.32Q2b hint0.32 100 32c 0.68Simplify the fraction.3 Write each fraction as a percentage.3a 4 3b 50 7c 20 17d 20 Q3 hint19e 25 Write an equivalent fraction withdenominator 100.4 Match up equivalent fractions, decimals and percentages.1.4220%2 1545140%1 25125%1.2580%0.81 142.25 Write each decimal as a fraction. Simplify where possible.123u a 0.123 b 0.763c 0.442 e 0.988g 0.128d 0.125 f 0.375Q6 Literacy hintA dot above a digit shows it recurs.6. 4 6.444 4 A dot above the beginning and endof a sequence shows the wholesequence recurs.9. 3 9 4 9.394 394 394 394 6 Rewrite each recurring decimal using dot notation.a 0.666 6 b 1.333 3 c 2.151 515 d 3.567 567 567 7 Write these decimals in order, from smallest to largest.a 0.30. 3 0.3290.32b 0. 4 0.450.4390.4 c 0. 5 0.5490.50.56d 0.665 0.680. 6 0.6Q7a hintWhich is larger: 0.3 or 0.333 3 .?Q8a hint8 Use a written method to work out each division.a 235 5b 135 9 5)235c 856 89 Use a written method to write each fraction as a decimal.a957 8 b3 8 c5 16 d3 16 Q9a hint0.uuu8) 7. 70 0 0

10 Which of these fractions will give a recurring decimal?78231956459161237311310Q11 hint11 Write or between each pair of fractions.133a 3 u 10 24b 5 u 3 58c 7 u 9 9d 9 u 11 Write each fraction as a decimal.12 Match each proportion to a bar. Use them to write the proportions inorder, from smallest to largest.A 0.7B 0.52D 3 C 60%3E 4 iiiiiiivv13 Write these proportions in ascending order.4a 5 0.430%0.6 3 2b 95%Q13 Literacy hint53 6 0.9 4 0.85Fraction calculationsAscending means getting larger.Q1a hint1 Write each improper fraction as a mixed number.1111322a 3 b 5 1524d 9 c 4 335133513351232 Write each mixed number as an improper fraction.a 1 5 b 2 4 34c 2 9 Q2a hintd 3 7 843 Work out each calculation. Write your answer as a mixed number.u u527 1 a 8 8 8 847b 9 9 9 342c 5 5 5 5711d 12 12 12 88u55 41 55 5 54 Work out the missing number.53a 9 9 u 1b 1 – 9 – 9 uQ4a hintcdHow many ninths equal 1?eg51 12 12 31 5 10 311– 5 – 10 u 1u 1u2fh4321 – 7 – 7 71 12 6 511 – 8 – 4 uu 1uUnit 4 Fractions, decimals and percentages96

5 Problem-solving What fraction shouldthe third sector on this pie chartbe labelled?12Q5 Strategy hint13Write the question as a subtractioncalculation.6 Work out4114a 2 3 1 5 2 1 3 5 b 4 6 1 5 52u u u3344c 3 5 – 1 4 (3 – 1) ( 5 – 4 ) d 5 3 – 3 4 21u u uQ7a hintFinding a fraction of an amount is thesame as multiplying by the fraction.7 Work outu2 121 a 3 of 2 14b 4 of 5 31c 3 7 31d 4 4 3 2uof12138 Work out35a 4 9 2333b 5 8 52c 9 5 9 Work out341a 3 4 1 1 u334b 3 4 1 3 uc 6 3 Q9a hintHow many quarters in 3?d 6 5 32Percentages1 The price of a computer has increased by 15%. It was originally 450.What is the new price?2 Reasoning Frankie works out 520 reduced by 20% like this:x810% of 520 52x880% of 520 41610% of 520 5220% of 520 104 2 520 – 104 416Whose method do you prefer? Why?3 The value of a car has gone down by 25%. It was originally worth 8000. What is the new value?9710% of 450 u5% of 450 u15% of 450 u u u450 u uQ2 hint100% 20% 80%Rob works it out like this: 2Q1 hint

4 A holiday costs 1200. Booking before the end of themonth saves 15%. How much is the holiday with the saving?5 Reasoning / Problem-solvinga 10% of a number is 25. What is the number?b 25% of a number is 50. What is the number?c 30% of a number is 24. What is the number?d 40% of a number is 320. What is the number?Q5c Strategy hint30% 24434310% u310310100% uEnrichment1 Draw a pie chart to show how you spent the 24 hours in a weekdayduring term-time.What fraction or percentage of your time is spenta sleepingb in lessonsc eating/washing/getting readyd socialising/entertainment/TVe homework?Q1 hintNumber of degrees for each hour 360 24 hours.2 How would your pie chart from Q1 be different for a Saturday inschool holidays?Q3 hintList the letters of the tasks in order.You don’t need to write out thedescriptions.Reflect3 Reflect List these tasks in order from easiest to hardest.A Adding and subtracting fractionsB Multiplying fractionsC Dividing fractionsD Adding and subtracting mixed numbersE Multiplying mixed numbersF Dividing mixed numbersLook at the first task in

Convert between fractions, decimals and percentages Compare fractions, decimals and percentages Write a fraction as a decimal. Master extend P99 test P103 Check P93 strengthen P95 CONFID e NC e 4 Fractions, decimals and percentages Exercise 4.1 1 Write as a percent