Among the $900$ residents of Aimeville, there are $195$ who own a diamond ring, $367$ who own a set of golf clubs, and $562$ who own a garden spade. In addition, each of the $900$ residents owns a bag of candy hearts. There are $437$ residents who own exactly two of these things, and $234$ residents who own exactly three of these things. Find the number of residents of Aimeville who own all four of these things.

View step-by-step help

Home

Math Problems

Grade 6

Unit prices with fractions and decimals

Full solution

Q. Among the

900

residents of Aimeville, there are

195

who own a diamond ring,

367

who own a set of golf clubs, and

562

who own a garden spade. In addition, each of the

900

residents owns a bag of candy hearts. There are

437

residents who own exactly two of these things, and

234

residents who own exactly three of these things. Find the number of residents of Aimeville who own all four of these things.

Denote Residents: Let's denote the number of residents who own all four items as $x$ . According to the problem, there are $437$ residents who own exactly two of these things and $234$ residents who own exactly three of these things. Since every resident owns a bag of candy hearts, we only need to consider the other three items when determining the overlaps.
Use Inclusion-Exclusion Principle: We can use the principle of inclusion-exclusion to find the number of residents who own all four items. The formula for three sets $A$ , $B$ , and $C$ is given by: $\newline$ $|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|$ $\newline$ In this context, $|A \cup B \cup C|$ is the total number of residents who own at least one of the three items (diamond ring, golf clubs, garden spade), $|A|$ , $|B|$ , and $|C|$ are the numbers of residents who own each item individually, $|A \cap B|$ , $|A \cap C|$ , and $|B \cap C|$ are the numbers of residents who own exactly two of the items, and $B$ $0$ is the number we're looking for, $B$ $1$ .
Calculate Total Residents: First, we calculate the total number of residents who own at least one of the three items (diamond ring, golf clubs, garden spade). Since every resident owns a bag of candy hearts, we can ignore it for this calculation. The total number of residents is $900$ , so: $\newline$ $|A \cup B \cup C| = 900$
Plug in Values: Now we plug in the values we know into the inclusion-exclusion formula: $\newline$ $900 = 195 + 367 + 562 - |A \cap B| - |A \cap C| - |B \cap C| + x$ $\newline$ We know that $|A \cap B| + |A \cap C| + |B \cap C|$ includes the residents who own exactly two items and those who own all three items. Since there are $437$ residents who own exactly two items and $234$ who own exactly three, we can express this as: $\newline$ $|A \cap B| + |A \cap C| + |B \cap C| = 437 + 234$
Find Value of x: Substituting the value of $|A \cap B| + |A \cap C| + |B \cap C|$ into the inclusion-exclusion formula, we get: $\newline$ $900 = 195 + 367 + 562 - (437 + 234) + x$ $\newline$ Now we perform the calculations: $\newline$ $900 = 1124 - 671 + x$ $\newline$ $900 = 453 + x$
Find Value of x: Substituting the value of $|A \cap B| + |A \cap C| + |B \cap C|$ into the inclusion-exclusion formula, we get: $\newline$ $900 = 195 + 367 + 562 - (437 + 234) + x$ $\newline$ Now we perform the calculations: $\newline$ $900 = 1124 - 671 + x$ $\newline$ $900 = 453 + x$ To find the value of x, we subtract $453$ from both sides of the equation: $\newline$ $x = 900 - 453$ $\newline$ $x = 447$