1. Year 7 Fractions, Decimal, Percentages
Dr J Frost
Objectives: Convert between fractions, decimals and percentages, including fractions to recurring decimals. Be able to order fractions.
Last modified: 2nd February 2016

2. 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!) ? ? ? ? ? ? ? ? ? ? ?
1 8
1 4
7 20
2 5
1 2
7 10
3 4
7 8 ? ? ? ?

3. 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 , 0.25, 3 10 , 0.5 B , 0.7, 4 5 , 1.5 C , 0.5, 7 10 , 0.9
D , 0.5, 4 5 , 0.9 E , 1.5, 10 5 , 2.3 A B C D E
[JMC 1998 Q11] Which is the smallest of these fractions? A B C D E 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 B C D E A B C D E

4. 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= = ? ?
0.45= = 9 20 ? ?
0.85= = 17 20
We used hundred because the last digit was the hundredths digit. ? ?
0.126= = ? ?
0.612= = ? ?
0.16= = 4 25 ? ?
0.0005= = ? ?

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

6. 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?

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

8. Use of recurring dot
What do the following represent?
0. 2 = … = … = … = … ? ? ? ?

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

10. 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 = 𝟏𝟏𝟏 𝟏𝟐𝟓
0.325= 𝟏𝟑 𝟒𝟎 = 𝟒𝟗𝟗 𝟓𝟎𝟎
Put the following numbers in ascending order:
7 8 , 0.85, , 3 20 , 0.12, 1 9
3 4 , 0.715, , 0.33, 3 10 , 3 8
Convert the fractions to (potentially recurring) decimals.
1 11 =𝟎. 𝟎 𝟗 =𝟎. 𝟐
3 16 =𝟎.𝟏𝟖𝟕𝟓 2 7 =𝟎. 𝟐 𝟖𝟓𝟕𝟏 𝟒
33 13 =𝟐. 𝟓 𝟑𝟖𝟒𝟔 𝟏 =𝟏. 𝟎 𝟎𝟗 𝟗
[JMC 2009 Q9] How many different digits appear when 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 Solution: 5
[IMC 2008 Q18] When the following values are put in ascending order, which is in the middle? A B C D E Solution: C
[Kangaroo Pink 2009 Q15] Which of these decimals is less than but greater than ? A B C
D E Solution: C. 𝟐𝟎𝟎𝟗 𝟐𝟎𝟎𝟖 =𝟏 𝟏 𝟐𝟎𝟎𝟖 . Since 2008 > 2000, this is slightly less than 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