Question

A survey 1500 commuters in New York City showed that 1150 take the subway 680 take the bus and 170 do not take either A) how many take the bus and subway B) how many only take the subway

Answer 1

EXPLANATION:

Given;

We are given the following information;

$\begin{gathered} All\text{ }commuters=1500 \\ Subway=1150 \\ Bus=680 \\ Neither=170 \end{gathered}$

Required;

We are required to find how many commuters take the bus and subway. Also how many take the subwa only.

Step-by-step solution;

Since we have a survey of 1500 commuters, the universal set will be 1500, that is

$U=1500$

Then the set that contains tiose who take the subway is;

$w=1150$

The set that contains those who take the bus is;

$b=680$

Also we are told that 170 take neither the subway nor the bus, the this set is;

$n=170$

Now we know that 170 who take neither the subway nor the bus belongs to the universal set but not to either of w or b.

This means;

$\begin{gathered} (n)=(U)-(w\cup b) \\ 170=(U)-(w\cup b) \\ 170=1500-(w\cup b) \end{gathered}$

We can collect like terms;

$\begin{gathered} (w\cup b)=1500-170 \\ (w\cup b)=1330 \end{gathered}$

This means there is a total of 1330 commuters who travel by both subway and by bus.

To calculate commuters who take both the subway and the bus;

$\begin{gathered} (w\cup b)=(w)+(b)-(w\cap b) \\ 1330=1150+680-(w\cap b) \\ 1330=1830-(w\cap b) \end{gathered}$$\begin{gathered} (w\cap b)=1830-1330 \\ (w\cap b)=500 \end{gathered}$

This means there are 500 commuters who travel by both the subway and the bus.

Commuters who take only the subway is derived by removing those who take both subway and bus from those who take the subway. That is;

$Number\text{ }who\text{ }take\text{ }only\text{ }subway=(w_1)$$\begin{gathered} (w_1)=(w)-(w\cap b) \\ (w_1)=1150-500 \\ (w_1)=650 \end{gathered}$

Therefore,

ANSWER:

$\begin{gathered} (a)\text{ }500\text{ }take\text{ }the\text{ }bus\text{ }and\text{ }subway \\ (b)650\text{ }take\text{ }only\text{ }the\text{ }subway \end{gathered}$