Survey of math question

Question

​Ms. Teller surveyed the 35 students in her class about their families. 27 students have a brother or a sister, 16 students have a brother, and 8 students have both a brother and a sister. How many students have only a sister?

Answer 1

Let's start by making a 2-way table. Horizontally, we will look at brothers and vertically, we will look at sisters. When it comes to brothers, there are two options, either you have a brother - that's possibility 1 or you don't have a brother - possibility 2. Each possibility becomes its own row.

The same can be said for sisters. Having a sister is a possibility, so it becomes a column. The other possibility is not having a sister, so it becomes a column as well. To help keep things organized, I like to add a row at the bottom for totals and a column at the far right side for totals as well. Ultimately, the table would look like this:

(It may help to rotate your screen to read the tables. I tried my best to format it!)

_______________________________________________

| Have a Sister | No Sister | Total |

_______________|______________|__________|______|

Have a Brother | | | |

_______________|______________|__________|______|

No Brother | | | |

_______________|______________|__________|______|

Total | | | |

_______________|______________|__________|______|

We know that in Total, 35 students were surveyed total. So that can go in the cell where the Total row intersects the Total column, the bottom far right cell. **

We also know 8 students have a brother AND a sister. So we go to the "Has a Brother" row and find where the row intersects the "Has a Sister" column. In the cell where this intersection occurs, we put an 8. **

(**I used dashed lines in the table below to highlight the cell where the intersection occurs to help visualize the verbal explanation since people typically understand pictures and graphs better than words.)

________________________________________________

| Have a Sister | No Sister | Total |

_______________|______|_______|___________|___|__|

Have a Brother -------- 8 | | | |

_______________|______________|___________|__|___|

No Brother | | | | |

_______________|______________|___________|__|___|

Total --------------------------------------------------------------35 |

_______________|______________|___________|_____|

Now, we know in Total, 16 students Have a Brother. So we can put 16 in the cell where the Have a Brother row intersects the Total column.

________________________________________________

| Have a Sister | No Sister | Total |

_______________|______________|___________|___|__|

Have a Brother -----------8---------------------------------16 |

_______________|______________|___________|_____ |

No Brother | | | |

_______________|______________|___________|_____|

Total | | | 35 |

_______________|______________|___________|_____|

Now the total number of students who have a brother (16) is equivalent to the sum of the number of students who have both a brother and a sister (8) and the number of students who have a brother, but not a sister (x).

________________________________________________

| Have a Sister | No Sister | Total |

_______________|______________|__________|______|

Have a Brother | 8 + x = 16 |

_______________|______________|__________|______|

No Brother | | | |

_______________|______________|__________|______|

Total | | | 35 |

_______________|______________|__________|______|

Giving us 8 + x = 16

Subtracting 8 from both sides leaves us with

x = 8

Therefore, there are 8 students who have a brother, but don't have a sister.

If we add up the total number of students that have a brother (16) and the total number of students that don't have a brother (y), we will get the total number of students surveyed (35).

________________________________________________

| Have a Sister | No Sister |Total |

_______________|______________|___________|_____|

Have a Brother | 8 | 8 | 16 |

_______________|______________|___________|__+__|

No Brother | | | y |

_______________|______________|___________|__=__|

Total | | | 35 |

_______________|______________|___________|_____|

16 + y = 35

Subtracting 16 from both sides gives us

y = 19

Therefore, there are a total of 19 students who don't have a brother.

We were told 27 students have brothers and sisters. We can set this equal to the number of students who have both a brother and a sister (8) added to the number of students who have a brother, but no sister (8), plus the number of students who have a sister, but no brother (z).

________________________________________________

| Have a Sister | No Sister | Total |

_______________|______________|___________|_____|

Have a Brother | 8 | 8 | 16 |

_______________|______________|___________|_____|

No Brother | z | | 19 |

_______________|______________|___________|_____|

Total | | | 35 |

_______________|______________|___________|_____|

The equation we can create is:

27 = 8 + 8 + z

27 = 16 + z

z = 11

Therefore, the number of students with a sister, but no brother (aka have only a sister) is 11.