1. ## Set Theory

In a certain city, one-seventh of the population who are left-handed are also short-sighted, and two-ninths who are short-sighted are also left-handed.
One twelfth are neither left-handed nor short-sighted.
Given that the number of people who are left-handed and short-sighted is $2x$, draw a Venn diagram to illustrate this information.
(The Venn diagram should show the number of elements, in terms of $x$, in each area.)

$\color{red}\text{Find the fraction of the population who are}$
$\color{red}\text{a)left-handed but not short-sighted,}$
$\color{red}\text{b)either left-handed or short-sighted, but not both.}$
$\color{red}\text{If there are 154000 people who are left-handed, find the total population.}$

2. Originally Posted by SengNee
In a certain city, one-seventh of the population who are left-handed are also short-sighted, and two-ninths who are short-sighted are also left-handed.

One twelfth are neither left-handed nor short-sighted. Mr F asks: 1/12 of what? The total population?

Given that the number of people who are left-handed and short-sighted is $2x$, draw a Venn diagram to illustrate this information.
(The Venn diagram should show the number of elements, in terms of $x$, in each area.)

Find the fraction of the population who are
a)left-handed but not short-sighted,
b)either left-handed or short-sighted, but not both.
If there are 154000 people who are left-handed, find the total population.
Attached is what the Venn diagram looks like.

From the given info: (a + 2x)/7 = 2x => a = 12x.

2(2x + b)/9 = a + 2x = 14x => b = 61x.

c = (a + 2x + b + c)/12 = (75x + c)/12 => c = 75x/11.

3. Originally Posted by mr fantastic
Attached is what the Venn diagram looks like.

From the given info: (a + 2x)/7 = 2x => a = 12x.

2(2x + b)/9 = a + 2x = 14x => b = 61x.

c = (a + 2x + b + c)/12 = (75x + c)/12 => c = 75x/11.
I think you are wrong.

L=Left-handed
S=Short-sighted

White=Right-handed and Not short-sighted
Red= Left-handed and Not short-sighted
Purple=Left-handed and Short-sighted
Blue=Right-handed and Short-sighted

Are you agree??

4. Originally Posted by SengNee
I think you are wrong.

L=Left-handed
S=Short-sighted

White=Right-handed and Not short-sighted
Red= Left-handed and Not short-sighted
Purple=Left-handed and Short-sighted
Blue=Right-handed and Short-sighted

Are you agree??
So where do you think my Venn diagram is wrong? On your one, isn't purple the intersection of the red and blue .....? That doesn't make sense to me.

5. Originally Posted by mr fantastic
So where do you think my Venn diagram is wrong? On your one, isn't purple the intersection of the red and blue .....? That doesn't make sense to me.
I see, pardon me.

Can you answer the question completely?

6. Originally Posted by SengNee
I see, pardon me.

Can you answer the question completely?
Well, I've already given you the Venn diagram and found in terms of x the number of elements in each subset. Where are you stuck?

7. Originally Posted by mr fantastic
Well, I've already given you the Venn diagram and found in terms of x the number of elements in each subset. Where are you stuck?
The red part.

8. Originally Posted by SengNee
The red part.
All of the questions ....??
Originally Posted by SengNee
In a certain city, one-seventh of the population who are left-handed are also short-sighted, and two-ninths who are short-sighted are also left-handed.
One twelfth are neither left-handed nor short-sighted.
Given that the number of people who are left-handed and short-sighted is $2x$, draw a Venn diagram to illustrate this information.
(The Venn diagram should show the number of elements, in terms of $x$, in each area.)

$\color{red}\text{Find the fraction of the population who are}$
$\color{red}\text{a)left-handed but not short-sighted,}$
$\color{red}\text{b)either left-handed or short-sighted, but not both.}$
$\color{red}\text{If there are 154000 people who are left-handed, find the total population.}$
If everything got answered, there'd be nothing left for you to do ..... What have you tried?

The total population is P = a + 2x + b + c = ......

1) According to the Venn diagram, how many are left handed but not short-sighted? Divide that by P. There's the answer.

2) According to the Venn diagram, how many are left handed or short-sighted but not both? Divide that by P. There's the answer.

3) According to the Venn diagram, how many are left handed? Equate that to 154000 and solve for x. Sub x into P. There's the answer.

Please be specific on exactly where you're stuck.

9. Originally Posted by mr fantastic
All of the questions ....??

If everything got answered, there'd be nothing left for you to do ..... What have you tried?

The total population is P = a + 2x + b + c = ......

1) According to the Venn diagram, how many are left handed but not short-sighted? Divide that by P. There's the answer.

2) According to the Venn diagram, how many are left handed or short-sighted but not both? Divide that by P. There's the answer.

3) According to the Venn diagram, how many are left handed? Equate that to 154000 and solve for x. Sub x into P. There's the answer.

Please be specific on exactly where you're stuck.