Can anyone help me with these two algebra problems?

Question

First one:

Simplify:
1+ 2/3 divided by 2 1/2


Second one:

Simplify:

3 / x-5 +1 divided by 1 - 4/ x-5

Answer 1

(10 1+2/3 divided by 2+1/2

First convert into improper fractions:

1 + 2/3 = 3/3 + 2/3 = (3+2)/3 = 5/3
2 + 1/2 = 4/2 + 1/2 = (4+1)/2 = 5/2

Then you can divide, which is the same as multiplying by the reciprocal of the denominator:

(5/3) / (5/2) =
(5/3) * (2/5) =
(5*2)/(3*5) =
2/3

(2) 3 / x-5 +1 divided by 1 - 4/ x-5

Assuming that there are parentheses like so:

{ 3/(x-5) + 1 } / { 1 - 4/(x-5) }

First work on x-5 as the common denominator in each of the numerator and denominator:

{ 3/(x-5) + 1 } / { 1 - 4/(x-5) }
{ (3 + (x-5))/(x-5) } / { ((x - 5) - 4)/(x-5) }
{ (x - 2)/(x - 5) } / { (x - 9)/(x - 5) }

The x-5 both cancel out, leaving:

(x - 2) / (x - 9)

Answer 2

I agree

if we can tackle the 2nd, more difficult, example...
then the 1st problem will be easy

PROBLEM TWO

3/(x-5) + 1 divided by 1 - 4/(x-5)

anything divided by itself is 1 (one)
so,
1 = (x-5)/(x-5)

rewritten
3/(x-5) + (x-5)/(x-5) divided by (x-5)/(x-5) - 4/(x-5)
just so happens (x-5) is already a common denominator
let's combine stuff using (x-5)

[3 + (x-5)]/(x-5) divided by [(x-5) -4(x-5)]/(x-5)
when you divide, you upend the thing you're dividing by
just so happens (x-5) goes away...

3+x-5 divided by x-5-4
3-5+x divided by x-9
rewrite you get x-2 divided by x-9

now
PROBLEM ONE
1 + 2/3 divided by 2-1/2
we don't need a common denominator, but we do need to use the fact that anything divided by itself is one and, for 2, it's twice that

(2/3)/(2/3) + 2/3 divided by 4/2 + 1/2
5/3 divided by 5/2
upend the thing you're dividing by
5/3 x 2/5
the 5s cancel each other
and you're left with
2/3

Answer 3

First one:
(1 + 2/3) / (3 1/2)
(1 2/3) / (3 1/2)
(5/3) / (7/2)
5/3 * 2/7
10/21

Answer 4

1+2/3=3/3+2/3=5/3
2 1/2=5/2
(5/3)/(5/2)=(5/3)(2/5)=2/3

second:
3/(x-5)+1=3/(x-5)+1=[3+(x-5)]/(x-5)] -- (1)
[1-4/(x-5)]=[(x-5)-4)]/(x-5) -- (2)
(1)/(2)
Divide (1) by (2)
[(x-2)/(x-5)]/[(x-9)/(x-5)]=(x-2)/(x-9)