1. What is $332_4-212_3$ when expressed in base 10?

2. The fraction $\frac{a}{a+27}$, where $a$ is a positive integer, equals $0.865$. What is the value of $a$?

3. If $\frac{5}{33}$ is expressed in decimal form, what digit is in the 92nd place to the right of the decimal point?

Mellie
May 24, 2015

#2**+10 **

Hi Mellie,

1)

$$\\332_4-212_3\\

=(3*4^2+3*4+2)-(2*3^2+1*3+2)\\

=(3*16+12+2)-(2*9+3+2)\\

=62-23\\

=39$$

2)

$$\\\frac{a}{a+27}=\frac{865}{1000}\\\\

\frac{a+27}{a}=\frac{1000}{865}\\\\

1+\frac{27}{a}=\frac{1000}{865}\\\\

\frac{27}{a}=\frac{1000-865}{865}\\\\

\frac{27}{a}=\frac{27}{173}\\\\

a=173$$

3)

$${\frac{{\mathtt{5}}}{{\mathtt{33}}}} = {\mathtt{0.151\: \!515\: \!151\: \!515\: \!151\: \!5}}$$

2nd place is 5, 4th place is 5 all the even places are 5 SO the 92nd place is 5

Melody
May 24, 2015

#2**+10 **

Best Answer

Hi Mellie,

1)

$$\\332_4-212_3\\

=(3*4^2+3*4+2)-(2*3^2+1*3+2)\\

=(3*16+12+2)-(2*9+3+2)\\

=62-23\\

=39$$

2)

$$\\\frac{a}{a+27}=\frac{865}{1000}\\\\

\frac{a+27}{a}=\frac{1000}{865}\\\\

1+\frac{27}{a}=\frac{1000}{865}\\\\

\frac{27}{a}=\frac{1000-865}{865}\\\\

\frac{27}{a}=\frac{27}{173}\\\\

a=173$$

3)

$${\frac{{\mathtt{5}}}{{\mathtt{33}}}} = {\mathtt{0.151\: \!515\: \!151\: \!515\: \!151\: \!5}}$$

2nd place is 5, 4th place is 5 all the even places are 5 SO the 92nd place is 5

Melody
May 24, 2015

#4**+8 **

Notice, rosala.....if we were to write 332 ....in base 10....it would just be

3(10)^2 + 3(10)^2 + 2(10)^0 [ prove this for yourself]

So.......writing 332 in base 4 is similar.....except that we're using....well......base 4 !!!!

So we have

3(4)^2 + 3(4)^1 + 2(4)^0 = 3*16 + 3*4 + 2(1) = 48 + 12 + 2 = 62 (in base 10)

Not too tricky, huh???

Using this example.....see if you can do 212 in base 3 and then covert it to base 10 just as I did wth the last one.....

CPhill
May 24, 2015

#5**+8 **

I can try :)

Our numbers are in base 10. That means we use 10 digits 0,1,2,3,4,5,6,7,8 and 9

If I write this number

3658 it is the number $$3*10^3+6*10^2+5*10+8*1=3000+600+50+8$$

I can also write it like this $$3658_{10}$$ The little 10 means that it is base 10.

NOW

Mellie's numbers are not base 10

she has 332 base 4 and 212 base 3

$$\\332_4\\

$if I convert this to base 10 I get$\\

=3*4^2+3*4+2\\

=3*16+12+2\\

=48+12+2\\

=62\\\\\\

212_3\\

$if I convert this to base 10 I get$\\

=2*3^2+1*3+2*1\\

=2*9+3+2\\

=18+3+2\\

=23$$

The i just had to do the subtraction :)

Do you get it ?

Melody
May 24, 2015

#6**+5 **

But CLhill in the first one you write 2(10)^2 and in the second one you wrote 3(4)^1......see your answer again...you'll know what I mean! This is confusing!

rosala
May 24, 2015

#8**0 **

Oops!!!....sorry for the mis-type.....!!!.....I've corrected it....but....I think you get the idea....!!!!

CPhill
May 24, 2015

#9**+5 **

Yes Chris made a boo boo. He's all tuckered out. He has been out mowing his lawn. That's the problem :)

It was really slow going too because the grass was really long and he was frightened he'd go over the top of the Roman Zero and chop it into little pieces and then he would never be able to get his problems correct.

He'd be in big trouble with the Gods too....

Melody
May 24, 2015