Word problem fun

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Momjam

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Jan 29, 2013
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Hello. My 6th grader brought this one home tonight and the whole family is stumped.
Any help would be appreciated.

Of 40 people asked, 27 have older siblings, 20 have younger siblings and 4 have no siblings.
How many have:
only older siblings?
only younger siblings?
both younger and older siblings?

We've gotten so far as only 36 have siblings and 9 possibly have younger or no siblings. We can't wrap our heads around it to figure out the rest.
Thanks again.
 
Hello, Momjam!

Of 40 people asked:
. . 27 have older siblings,
. . 20 have younger siblings, and
. . 4 have no siblings.

How many have:
. . (a) only older siblings?
. . (b) only younger siblings?
. . (c) both younger and older siblings?
Click to expand...

Make a Venn diagram of the situation.

Code: 
      *-------------------------------------------*
      |                                           |
      |       *-----------*   *-----------*       |
      |      /             \ /             \      |
      |     /     Older     X    Younger    \     |
      |    /               / \               \    |
      |   *               *   *               *   |
      |   |               |   |               |   |
      |   |        16     | 11|     9         |   |
      |   |               |   |               |   |
      |   *               *   *               *   |
      |    \               \ /               /    |
      |     \               X               /     |
      |      \             / \             /      |
      |   4   *-----------*   *-----------*       |
      |                                           |
      *-------------------------------------------*
Now you can answer the questions.
 
Momjam said:
Hello. My 6th grader brought this one home tonight and the whole family is stumped.
Any help would be appreciated.

Of 40 people asked, 27 have older siblings, 20 have younger siblings and 4 have no siblings.
How many have:
only older siblings?
only younger siblings?
both younger and older siblings?

We've gotten so far as only 36 have siblings and 9 possibly have younger or no siblings. We can't wrap our heads around it to figure out the rest.
Thanks again.
Click to expand...

Soroban drew the Venn diagram for you - did you figure out how those numbers (16, 11, 9) were obtained?
 
A Venn diagram is the simplest way to visualize it, but you can do this problem through careful counting.

There are 40 people. That is the big box in soroban's Venn diagram

4 have no siblings, which means that 36 do have siblings.

The two circles represent the 36 with siblings, but outside the circles are those without siblings. You can see that soroban has filled in 4 for that part of the rectangle.

Now let's think about the 36 that have siblings. 27 have older siblings, and 20 have younger siblings.

But 20 + 27 = 47 > 36. What happened?

We counted those with at least one older sibling and at least one younger sibling twice, once in the group with older siblings and once in the group with younger siblings. We need to break our analysis down more finely.

47 - 36 = 11. There were 11 we counted twice. You still with me?

So there are 11 who have both older and younger siblings.

So of the 27 who have older siblings, 11 also have younger siblings, and 16 have ONLY older siblings. In soroban's diagram, those with
older siblings are in the circle to the left and are divided into those without younger siblings in the left portion of the left circle and those with both older and younger siblings in the right portion of the left circle.

Of the 20 who have younger siblings, 11 also have older siblings and 9 have ONLY younger siblings. In soroban's diagram, those with
younger siblings are in the circle to the right and are divided into those without older siblings in the right portion of the right circle and those with both older and younger siblings in the left portion of the right circle.

Notice that the 11 with both older and younger siblings is in BOTH circles.

So

\(\displaystyle \ \ 4 = number\ without\ siblings.\)
\(\displaystyle 16 = number\ with\ only\ older\ siblings.\)
\(\displaystyle 11 = number\ with\ older\ and\ younger\ siblings.\)
\(\displaystyle \ \ 9 = number\ with\ only\ younger\ siblings.\)

\(\displaystyle 40 = total\ number.\)

\(\displaystyle 16 = number\ with\ only\ older\ siblings.\)
\(\displaystyle 11 = number\ with\ older\ and\ younger\ siblings.\)

\(\displaystyle 27 = number\ with\ older\ siblings.\)

\(\displaystyle \ \ 9 = number\ with\ only\ younger\ siblings.\)
\(\displaystyle 11 = number\ with\ older\ and\ younger\ siblings.\)
\(\displaystyle 20 = number\ with\ younger\ siblings.\)

Soroban's Venn diagram makes the whole logic visual. The 11 are in both circles.

Does this help?
 
Last edited:
Great explanation!
We actually came up with the 11 but didn't know how it fit in.
It makes senses now.
Thank you for your time.
 
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