Year 7 Fractions, Decimal, Percentages Dr J Frost (jfrost@tiffin.kingston.sch.uk) www.drfrostmaths.com Objectives: Convert between fractions, decimals and percentages, including fractions to recurring decimals. Be able to order fractions. Last modified: 2nd February 2016

RECAP :: Basic Decimal-Fraction conversions Percentage 1 2 0.5 50% 1 4 0.25 25% 3 4 0.75 75% 2 5 0.4 40% 7 10 0.7 70% 1 8 0.125 12.5% 7 8 0.875 87.5% 7 20 0.35 35% Fill in the table with the missing decimals/fractions/%s, and place the fractions all on a single number line as pictured. (Copying note: don’t waste time drawing lots of lines for your table!) ? ? ? ? ? ? ? ? ? 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ? ? 1 8 1 4 7 20 2 5 1 2 7 10 3 4 7 8 ? ? ? ?

A B C D E A B C D E A B C D E Ordering Bro Tip: Picture a number line in your head. [JMC 2002 Q15] In which of the following lists are the terms not increasing? A 1 5 , 0.25, 3 10 , 0.5 B 3 5 , 0.7, 4 5 , 1.5 C 2 5 , 0.5, 7 10 , 0.9 D 3 5 , 0.5, 4 5 , 0.9 E 2 5 , 1.5, 10 5 , 2.3 A B C D E [JMC 1998 Q11] Which is the smallest of these fractions? A 5 8 B 6 13 C 7 12 D 9 17 E 10 19 A B C D E [JMC 2005 Q14] If the following fractions are arranged in increasing order of size, which one is in the middle? A 1 2 B 2 3 C 3 5 D 4 7 E 5 9 A B C D E

Decimals → Fractions 0.255= 255 1000 = 51 200 0.45= 45 100 = 9 20 Is there a method for converting any arbitrary decimal to a fraction? 0.255= 255 1000 = 51 200 ? ? 0.45= 45 100 = 9 20 ? ? 0.85= 85 100 = 17 20 We used hundred because the last digit was the hundredths digit. ? ? 0.126= 126 1000 = 63 500 ? ? 0.612= 612 1000 = 153 250 ? ? 0.16= 16 100 = 4 25 ? ? 0.0005= 5 10000 = 1 2000 ? ?

Check Your Understanding 0.03= 𝟑 𝟏𝟎𝟎 0.007= 𝟕 𝟏𝟎𝟎𝟎 0.08= 𝟖 𝟏𝟎𝟎 = 𝟐 𝟐𝟓 0.015= 𝟏𝟓 𝟏𝟎𝟎𝟎 = 𝟑 𝟐𝟎𝟎 ? ? ? ? If that’s too easy: 0.𝑎0𝑏= 𝟏𝟎𝟎𝒂+𝒃 𝟏𝟎𝟎𝟎 N ?

Fractions → Decimals 3 8 just means 3÷8. So we could use long division to convert it to a decimal. . 3 7 5 8 3 . 30 60 40 Uh oh. We’ve run out of digits and hence have nowhere to put the remainder. What can we do to the 3 without changing its value?

Fractions → Decimals 4 11 =0. 3 6 . 3 6 3 6 11 4 . 40 70 40 70

Use of recurring dot What do the following represent? 0. 2 =0.22222222… 0.31 5 =0.315555555… 0. 3 1 5 =0.315315315… 0.3 1 5 =0.315151515… ? ? ? ?

Check Your Understanding 7 15 =0.4 6 ? 5 9 =0. 5 ? 6 7 =0. 8 5714 2 ?

Exercise 1 What are the following decimals as fractions in their simplest form? 0.32= 𝟖 𝟐𝟓 0.9= 𝟗 𝟏𝟎 0.65= 𝟏𝟑 𝟐𝟎 0.04= 𝟏 𝟐𝟓 0.312= 𝟑𝟗 𝟏𝟐𝟓 0.888= 𝟏𝟏𝟏 𝟏𝟐𝟓 0.325= 𝟏𝟑 𝟒𝟎 0.998= 𝟒𝟗𝟗 𝟓𝟎𝟎 Put the following numbers in ascending order: 7 8 , 0.85, 9 10 1 5 , 3 20 , 0.12, 1 9 3 4 , 0.715, 29 40 1 3 , 0.33, 3 10 , 3 8 Convert the fractions to (potentially recurring) decimals. 1 11 =𝟎. 𝟎 𝟗 2 9 =𝟎. 𝟐 3 16 =𝟎.𝟏𝟖𝟕𝟓 2 7 =𝟎. 𝟐 𝟖𝟓𝟕𝟏 𝟒 33 13 =𝟐. 𝟓 𝟑𝟖𝟒𝟔 𝟏 102 101 =𝟏. 𝟎 𝟎𝟗 𝟗 [JMC 2009 Q9] How many different digits appear when 20 11 is written as a recurring decimal? A 2 B 3 C 4 D 5 E 6 Solution: A [JMO 2001 A5] Find the 100th digit after the decimal point in the decimal representation of 3 7 . Solution: 5 [IMC 2008 Q18] When the following values are put in ascending order, which is in the middle? A 0.200 8 B 0.2008 C 0.20 0 8 D 0.2 0 0 8 E 0. 2 00 8 Solution: C [Kangaroo Pink 2009 Q15] Which of these decimals is less than 2009 2008 but greater than 20009 20008 ? A 1.01 B 1.001 C 1.0001 D 1.00001 E 1.000001 Solution: C. 𝟐𝟎𝟎𝟗 𝟐𝟎𝟎𝟖 =𝟏 𝟏 𝟐𝟎𝟎𝟖 . Since 2008 > 2000, this is slightly less than 1.0005. Similarly 𝟐𝟎𝟎𝟎𝟗 𝟐𝟎𝟎𝟎𝟖 =𝟏 𝟏 𝟐𝟎𝟎𝟎𝟖 . 1 𝟏 𝟐𝟎𝟎𝟎𝟎 =𝟏.𝟎𝟎𝟎𝟎𝟓. 1 4 a ? e ? ? b ? f ? c ? 5 g ? d ? h ? ? 2 6 a c ? b d 7 3 a ? d ? b ? e ? ? c ? ? f