Decimals

Overview


It’s easy to get fazed by dec­i­mal points. There are few sim­ple tech­niques that make sums involv­ing dec­i­mal points easy to do. Also I think there are poten­tial rel­a­tively easy Grade A ques­tions about con­vert­ing recur­ring dec­i­mals to frac­tions (sounds scary but there is a neat trick to make it straightforward).

Dec­i­mals Questions

1. Van Per­sie played 44 games and aver­aged 0.75 goals per game. Wal­cott played 64 games and aver­aged 0.5625 goals per game. Who scored the most goals?

2. A local news­pa­per charges 10.6p per word to place an advert. John’s advert costs £6.89. How many words are in John’s advert?

3. Her­nan­dez aver­ages 0.72 goals per game. In what frac­tion of games played does Her­nan­dez score?

4. Con­vert these frac­tions to decimals:-

a) 2/9

b) 11/18

c) 4/7

5. Con­vert the­ses recur­ring dec­i­mals to frac­tions in their sim­plest forms;

a) 0.77777…

b) 0.030303…

c) 0.58555…

d) 0.73929292…  NB this involves a lot of arith­metic– I’m not sure actual exam ques­tion would be this awkward.

Dec­i­mals Approach

1. To mul­ti­ply by dec­i­mal numbers

a) Ignore the dec­i­mal points and mul­ti­ply the numbers.

b) Add the dec­i­mal places in all the numbers

c) Put the num­ber of dec­i­mal places found in b) in your answer.

So, for exam­ple, if you are mul­ti­ply­ing 2 x 0.5:

a) Ignore the dec­i­mal points and mul­ti­ply = 2 x 5 = 10

b) Add the dec­i­mal places = 1 (just o.5)

c) Put 1 dec­i­mal place in the answer = 1.0

Another exam­ple 0.5 x 0.5

a) Ignore the dec­i­mal points and mul­ti­ply = 5 x 5 = 25

b) Add the dec­i­mal places = 2 (o.5 and o.5)

c) Put 2 dec­i­mal places in the answer = 0.25

2. When divid­ing by dec­i­mal points. Mul­ti­ply both num­bers by a fac­tor of 10 to elim­i­nate the dec­i­mal point. Then just use nor­mal divi­sion rules to com­plete the sum.

For exam­ple 12/0.4 = 120/4 (mul­ti­ply both num­bers by 10) = 30.

3. To con­vert a dec­i­mal to a frac­tion remem­ber that

One dec­i­mal place will con­vert to tenths

Two dec­i­mal places will con­vert to hundredths

Three dec­i­mal places will con­vert to thou­sandths etc etc

Once you have con­verted to tenths or hun­dredths or thou­sandths… then sim­plify the frac­tion as much as pos­si­ble using your knowl­edge of frac­tions and fac­tors.

4. To con­vert a frac­tion into a dec­i­mal divide the top (numer­a­tor) of the frac­tion by the bot­tom (denom­i­na­tor) of the frac­tion. The result­ing answer could be a recur­ring dec­i­mal or a ter­mi­nat­ing dec­i­mal. You can pre­dict hat type of dec­i­mal you will get:-

Denom­i­na­tor (bot­tom) has ONLY prime fac­tors of 2 and/or 5, answer = Terminating

Denom­i­na­tor (top) has ANY prime fac­tors other than 2 and 5 = Recurring

5. Recur­ring dec­i­mals can be con­verted to frac­tions using the method out­lined in this example:

Con­vert 0.333333.. to a fraction.

10 x 0.333333… =                     3.33333333

Sub­tract 1 x 0.3333…               0.33333333

Gives 9 x o.33333 =                  3.0000000

There­fore 0.33333 = 3/9 = 1/3.

You can use a sim­i­lar method for any recur­ring frac­tion. This was a sim­ple exam­ple involv­ing one recur­ring digit. For 2 recur­ring dig­its you would mul­ti­ply by 100 and 3 recur­ring dig­its you would mul­ti­ply by 1,000 etc etc.

Hav­ing done a few of these, I can see why they are Grade A. It seems that cal­cu­lat­ing the first part is ok if you remem­ber the method out­lined above. How­ever you often left with frac­tions with 3 or 4 dig­its which you then have to sim­plify. This needs to be done with care and you often have to use fac­tor trees to find com­mon fac­tors to sim­plify the ini­tial frac­tion. Def­i­nitely requires prac­tice (well it does for me!)

Dec­i­mals Answers

1. Van Per­sie = 44 x 0.75 = (44 x 75)/100 = 33.

Wal­cott = 64 x .5625 = (64 x 5625)/10000 = 35.

Wal­cott scored the most goals.

2. £6.89/10.6p = 6.89÷0.106 = 6890/106 = 65 words in the advert.

3. 0.72 = 72/100 = 36/50 = 18/25

4. 2÷9− This will be recur­ring as denom­i­na­tor has a prime fac­tor that is not 2 or 5.

= 0.22222…

11/18 will also be recurring

= 0.61111.…..

4/7 = 0.57142857142857…

5 a)  0.7777 x 10 = 7.77777

0.7777 x  1  = 0.77777

9 x 0.777777   = 7.000

0.77777 = 7/9

5 b)  0.030303  x 100 = 3.030303

0.030303   x     1  = 0.030303

99 x 0.030303      = 3.000

0.0303030 =  3/99 = 1/33

5 c)   0.585555 x 1,000 =  585.55555

0.5855555  x 1      =      0.58555

999 x 0.58555       =  584.970

0.585555÷999 =   584.970 = 58497/99900 = 19499/33300

Using fac­tor tree 33,300 = 2² x 3² x 5² x 37 and 37 is a fac­tor of 19499.

There­fore 0.585555… = 527/2700

5 d) 0.73929292.. x 10,000 = 7392.929292

0.73929292.. x     1        =        0.739292

0.73929292..x   9,999  =     7392.19000

0.73929292         =       7392.19000÷9999 =   739219/999900 = 7319/9900

 

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