Free math worksheets

- Describe the location of each letter relative to both circles. - Begin with the region that is clearly outside both circles. - Use the rectangle as the boundary of the full sample space.

1. Point \(P\) lies inside \(A\) but outside \(B\), so it represents only \(A\). 2. Point \(Q\) lies in the overlap, so it represents \(A\cap B\). 3. Point \(R\) lies inside \(B\) but outside \(A\), so it represents only \(B\). 4. Point \(S\) lies inside the universal-set rectangle but outside both circles, so it represents the region outside \(A\cup B\).

\(P\): only \(A\); \(Q\): \(A\cap B\); \(R\): only \(B\); \(S\): outside \(A\cup B\).

- Does the shading include the overlap of the two circles? - Decide whether the diagram represents “and” or “or.”

1. Every point inside \(A\) is shaded, including the overlap. 2. Every point inside \(B\) is also shaded. 3. Therefore, the shaded event consists of outcomes in \(A\) or \(B\), which is \(A\cup B\).

\(A\cup B\).

- Try organizing the information in a Venn diagram with two overlapping circles. - What happens if you add the French and Italian totals directly? Why would some students be counted twice? - After finding how many students take at least one language, compare that number with the total enrollment.

1. Find the number of students who take at least one of the two languages: \(94 + 66 - 25 = 135\). 2. Subtract from the total number of students: \(255 - 135 = 120\).

\(120\) students take neither language elective.

- Use the visible rectangle as the full sample space when interpreting complements. - For an intersection, identify the region that belongs to both specified parts. - For a union, combine every region named by either set. - Describe each expression in words before deciding which part of its panel is included.

1. For \(A^c\cap B\), shade the part of \(B\) that lies outside \(A\), excluding the overlap. 2. For \(A\cup B^c\), shade all of \(A\) together with everything outside \(B\). The only unshaded region is the part of \(B\) outside \(A\). 3. For \((A\cup B)^c\), shade the part of \(S\) outside both circles.

a) The part of \(B\) that does not overlap \(A\). b) Every region except the part of \(B\) outside \(A\). c) The region inside \(S\) but outside both \(A\) and \(B\).

- Match each region with whether it is inside or outside each circle. - Use the rectangle to identify the region for students in neither group. - To find the total, count each nonoverlapping region exactly once.

1. Region \(P\) contains \(11\) students who sing only in the choir. 2. Region \(Q\) contains \(4\) students who sing in the choir and play in the school band. 3. Region \(R\) contains \(7\) students who play only in the school band. 4. Region \(S\) contains \(3\) students who participate in neither group; the universal-set rectangle makes this outside-both region explicit. 5. Add the four nonoverlapping regions: \(11+4+7+3=25\).

Only choir: \(11\); both groups: \(4\); only school band: \(7\); neither group: \(3\). The total is \(25\) students.

- Remember that the overlap belongs to both circles. - For each requested set, include only its corresponding regions.

1. Set \(A\) includes its exclusive region and the overlap: \(n(A)=18+6=24\). 2. Set \(B\) includes its exclusive region and the overlap: \(n(B)=9+6=15\). 3. The union includes all three regions inside the circles: \(n(A\cup B)=18+6+9=33\). 4. The sample space also includes the outside region: \(n(S)=18+6+9+5=38\).

\(n(A)=24\), \(n(B)=15\), \(n(A\cup B)=33\), and \(n(S)=38\).

- Subtract the overlap from each circle total. - Assign each given or calculated value to exactly one region.

1. The number only in \(A\) is \(26-7=19\). 2. The number only in \(B\) is \(19-7=12\). 3. The overlap contains \(7\) elements. 4. The region outside both circles contains \(4\) elements.

Only \(A\): \(19\); both: \(7\); only \(B\): \(12\); outside both: \(4\).

- Which diagram regions lie completely inside circle \(A\)? - Check whether the overlap has been counted.

1. Set \(A\) includes both the “only \(A\)” region and the overlap. 2. Tarek failed to count the \(6\) students in \(A\cap B\). 3. The correct total is \(n(A)=14+6=20\).

Tarek omitted the overlap. The correct value is \(n(A)=20\).

- Read each circle as the combination of its exclusive region and the overlap. - The person outside both circles belongs to neither set.

1. Set \(A\) contains the “only \(A\)” region and the overlap: \(A=\{\text{Lea},\text{Noah},\text{Mina}\}\). 2. Set \(B\) contains the “only \(B\)” region and the overlap: \(B=\{\text{Mina},\text{Emil},\text{Sara}\}\). 3. The union contains everyone inside at least one circle: \(A\cup B=\{\text{Lea},\text{Noah},\text{Mina},\text{Emil},\text{Sara}\}\). 4. The intersection contains only Mina: \(A\cap B=\{\text{Mina}\}\).

\(A=\{\text{Lea},\text{Noah},\text{Mina}\}\), \(B=\{\text{Mina},\text{Emil},\text{Sara}\}\), \(A\cup B=\{\text{Lea},\text{Noah},\text{Mina},\text{Emil},\text{Sara}\}\), and \(A\cap B=\{\text{Mina}\}\).

- Include the overlap in each individual event. - For the union, add every region inside at least one circle.

1. Event \(A\) includes the “only \(A\)” region and the overlap: \(P(A)=0.31+0.18=0.49\). 2. Event \(B\) includes the “only \(B\)” region and the overlap: \(P(B)=0.27+0.18=0.45\). 3. The union includes all three regions inside at least one circle: \(P(A\cup B)=0.31+0.18+0.27=0.76\).

\(P(A)=0.49\), \(P(B)=0.45\), and \(P(A\cup B)=0.76\).

- Each circle total already includes the shared group. - Subtract the shared group to find each exclusive region.

1. The number who like only puzzles is \(15-5=10\). 2. The overlap contains \(5\) people. 3. The number who like only logic games is \(12-5=7\). 4. The outside region contains \(2\) people.

Only \(A\): \(10\); both: \(5\); only \(B\): \(7\); neither: \(2\).

- Add all four nonoverlapping regions. - Use the total number of elements to determine the incorrect region.

1. The four displayed values sum to \(9+8+10+5=32\), not \(30\). 2. The three correct regions inside the circles contain \(9+8+10=27\) elements. 3. Therefore, the outside region must contain \(30-27=3\) elements. 4. Replace \(5\) with \(3\).

The outside value is incorrect. It should be \(3\), not \(5\).

- Separate the union into its three nonoverlapping regions. - Find the missing part of the total inside the circles.

1. The union consists of three nonoverlapping regions: only \(A\), both sets, and only \(B\). 2. Subtract the two exclusive regions from the union total: \(41-17-14=10\). 3. Therefore, \(n(A\cap B)=10\).

The overlap contains \(10\) elements.

- First add the three regions inside the circles. - Then find the remaining part of the full \(100\%\).

1. The percentage inside at least one circle is \(23\%+17\%+28\%=68\%\). 2. Subtract from the whole: \(100\%-68\%=32\%\).

\(32\%\) lies outside both sets.

- Identify the regions whose elements have exactly one property. - Decide whether the overlap matches the phrase “exactly one.”

1. “Exactly one event” includes only the two nonoverlapping parts of the circles. 2. The \(7\) elements in the intersection satisfy both events and must not be counted. 3. The correct number is \(12+15=27\).

The correct number is \(27\); the \(7\) elements in the overlap are excluded.

- In Diagram I, identify which value is only in \(A\) and which is only in \(B\). - Which regions depend on the name of one particular circle? - Which regions are unchanged when the two circle names are swapped?

1. Switching the circle labels changes the “only \(A\)” region into the “only \(B\)” region and vice versa. 2. Therefore, \(12\), which is only in \(A\), and \(9\), which is only in \(B\), must switch sides. 3. The intersection is still the intersection of the two sets, so \(7\) can remain in the overlap. 4. The outside-both region is unchanged, so \(4\) can remain outside both circles.

\(12\) and \(9\) must switch sides; \(7\) in the overlap and \(4\) outside both circles can remain in the same relative regions.

- Compare the meanings of “in both” and “in at least one.” - Name the relevant diagram regions explicitly.

1. The overlap \(Q\) represents the intersection \(A\cap B\). 2. The union includes every outcome that lies in at least one circle. 3. Therefore, regions \(P\), \(Q\), and \(R\) belong to \(A\cup B\). 4. Only region \(S\) lies outside the union.

Region \(Q\) is the intersection. The union consists of regions \(P\), \(Q\), and \(R\).

- Match each question to the appropriate Venn-diagram regions. - Use the total number of orders to find the outside region.

1. Exactly one item appears in \(19+13=32\) orders. 2. At least one item appears in \(19+8+13=40\) orders. 3. Neither item appears in \(48-40=8\) orders.

Exactly one: \(32\) orders At least one: \(40\) orders Neither: \(8\) orders

- Identify which circle contains the shaded region. - Check whether the overlap is shaded or excluded. - Think about how “in one event but not the other” is written symbolically.

1. The shaded region is inside \(A\) but excludes every point that is also in \(B\). 2. That region is \(A\setminus B\), equivalently \(A\cap B^c\).

a) \(A\setminus B\), equivalently \(A\cap B^c\). b) Outcomes that are in \(A\) but not in \(B\).

- First name the region consisting of everything inside either circle. - The shading is the complement of that entire region. - How does De Morgan's law rewrite the complement of a union?

1. The shaded outcomes are not in \(A\) or \(B\), so they are outside \(A\cup B\). 2. Therefore, the event is \((A\cup B)^c\). 3. By De Morgan's law, the same event is \(A^c\cap B^c\).

\((A\cup B)^c=A^c\cap B^c\).

- What does it mean that one circle lies completely inside the other? - For the intersection, keep only outcomes in both sets. - For the set difference, ask whether the inner circle has any part outside the outer circle.

1. The diagram shows that every outcome in \(B\) is also in \(A\), so \(B\subseteq A\). 2. Therefore, \(A\cap B=B\). 3. Since \(B\) adds no outcomes outside \(A\), \(A\cup B=A\). 4. There are no outcomes in \(B\) that lie outside \(A\), so \(B\setminus A=\varnothing\).

a) \(B\) b) \(A\) c) \(\varnothing\)

- Organize the information in a two-way table or a Venn diagram. - Think about how students who do both activities affect the two group totals. - Compare the number in at least one activity with the total class size.

1. Let \(I\) be the set of students who play an instrument and \(S\) be the set of students who participate in a school sport. There are \(24\) students total, with \(|I| = 14\), \(|S| = 10\), and \(|I \cap S| = 6\). 2. The number who do at least one activity is \(|I \cup S| = |I| + |S| - |I \cap S| = 14 + 10 - 6 = 18\). Therefore, \(24 - 18 = 6\) students do neither activity, so \(P = \frac{6}{24} = \frac{1}{4} = 0.25\). 3. The number who play an instrument but do not participate in a school sport is \(14 - 6 = 8\). Therefore, \(P = \frac{8}{24} = \frac{1}{3} \approx 0.333\). 4. Elif's reasoning is incorrect because adding \(14 + 10\) counts the \(6\) students who do both activities twice. There are only \(18\) different students who do at least one activity, leaving \(6\) students who do neither.

a) \(P = \frac{1}{4} = 0.25\) b) \(P = \frac{1}{3} \approx 0.333\) c) The \(6\) students who do both activities are counted twice in \(14 + 10\). Only \(18\) different students do at least one activity, so \(6\) students do neither.

- Which set expression represents students who participate in at least one activity? - Why does adding the two club totals count some students twice? - How can the choir set be divided into two nonoverlapping parts?

1. Use the cardinality rule for a union: \(n(A\cup B)=n(A)+n(B)-n(A\cap B)\). 2. Solve for the intersection: \(n(A\cap B)=n(A)+n(B)-n(A\cup B)=45+32-60=17\). 3. Choir members are divided into those in both activities and those in choir only, so \(n(B\setminus A)=n(B)-n(A\cap B)\). 4. Substitute the values: \(n(B\setminus A)=32-17=15\).

\(n(A\cap B)=17\), and \(n(B\setminus A)=15\).

- Use each letter's location rather than relying only on memorized definitions. - For every part, check the letter's position relative to both circles. - For complements, remember that the rectangle shows the full sample space, including the region outside both circles.

1. The regions inside at least one circle are \(P\), \(Q\), and \(R\), so they represent \(A\cup B\). 2. Only \(Q\) lies in both circles, so it represents \(A\cap B\). 3. Only \(P\) lies in \(A\) but outside \(B\), so it represents \(A\cap B^c\). 4. Only \(S\) lies inside the universal-set rectangle but outside both circles, so it represents \(A^c\cap B^c\).

a) \(P,Q,R\) b) \(Q\) c) \(P\) d) \(S\)

- First use the sum of all four nonoverlapping regions. - Then identify which regions make up circle \(A\).

1. Add the four nonoverlapping regions: \(x+6+9+4=31\). 2. Solve for \(x\): \(x=31-19=12\). 3. Set \(A\) contains the “only \(A\)” region and the overlap. 4. Therefore, \(n(A)=12+6=18\).

\(x=12\), and \(n(A)=18\).

- First check whether the four regions make up the entire sample space. - Then match each event expression to its diagram region.

1. The four nonoverlapping proportions sum to \(0.22+0.31+0.29+0.18=1\), so the assignment is possible. 2. The intersection is the overlap, so \(P(A\cap B)=0.31\). 3. The region outside both circles represents \(A^c\cap B^c\), so \(P(A^c\cap B^c)=0.18\).

The assignment is possible. \(P(A\cap B)=0.31\), and \(P(A^c\cap B^c)=0.18\).

- Each course total already includes the students who take both courses. - Find the outside region only after determining the three regions inside the circles.

1. The number who take only course \(A\) is \(21-9=12\). 2. The overlap contains \(9\) students. 3. The number who take only course \(B\) is \(26-9=17\). 4. The circles contain \(12+9+17=38\) students. 5. The number outside both circles is \(52-38=14\).

Only \(A\): \(12\); both courses: \(9\); only \(B\): \(17\); neither course: \(14\).

- Use the parts of circle \(A\) to find the overlap. - Then build the other requested totals from the diagram regions.

1. The overlap is the part of \(A\) not included in the “only \(A\)” region: \(34-23=11\). 2. Set \(B\) contains its exclusive region and the overlap: \(n(B)=16+11=27\). 3. The union contains \(23+11+16=50\) elements. 4. Including the outside region gives \(n(S)=50+8=58\).

The overlap contains \(11\) elements, \(n(B)=27\), and \(n(S)=58\).

- Determine whether one divisibility condition always implies the other. - Then test each number in the sample space.

1. Every number divisible by \(4\) is also divisible by \(2\), so circle \(B\) must lie entirely inside circle \(A\). 2. The elements in \(A\) but not \(B\) are \(2\) and \(6\). 3. The elements in \(B\) are \(4\) and \(8\). 4. The elements outside \(A\) are \(1, 3, 5, 7\). The region in \(B\) but not \(A\) is empty.

Circle \(B\) lies entirely inside circle \(A\). \(A\) only: \(\{2, 6\}\); \(B\): \(\{4, 8\}\); outside both: \(\{1, 3, 5, 7\}\); \(B\) only: \(\varnothing\).

- Subtract the overlap from each individual total. - Find the outside region by subtracting the union count from the total.

1. The canoeing-only count is \(29-17=12\). 2. The overlap count is \(17\). 3. The night-hike-only count is \(36-17=19\). 4. The number choosing at least one activity is \(12+17+19=48\). 5. The number choosing neither is \(64-48=16\).

Canoeing only: \(12\) Both: \(17\) Night hike only: \(19\) Neither: \(16\)

- Find each nonoverlapping region separately. - Divide their combined count by the total number of passes.

1. The number valid only on buses is \(420-260=160\). 2. The number valid only on commuter trains is \(310-260=50\). 3. The number valid on exactly one is \(160+50=210\). 4. The percentage is \(\frac{210}{500}\cdot100\%=42\%\).

\(210\) passes, or \(42\%\), are valid on exactly one form of transportation.

- Add the three regions inside the two sets. - Subtract from the total to find the outside region.

1. The number with at least one defect is \(28+19+11=58\). 2. The number with neither defect is \(75-58=17\). 3. The probability is \(\frac{17}{75}\approx0.2267\). 4. As a percent rounded to the nearest tenth, this is \(22.7\%\).

\(\frac{17}{75}\approx22.7\%\)

- With a total of \(100\), convert each probability directly to a count. - Subtract the overlap from each individual event count.

1. Because there are \(100\) projects, the probabilities correspond directly to counts: \(58\) use a microcontroller, \(46\) use a web interface, and \(27\) use both. 2. The count in \(A\) only is \(58-27=31\). 3. The count in \(B\) only is \(46-27=19\). 4. The union count is \(31+27+19=77\). 5. The count in neither event is \(100-77=23\).

\(A\) only: \(31\) Both: \(27\) \(B\) only: \(19\) Neither: \(23\)

- Use the individual counts and union count to find the overlap. - Then find each remaining region by subtraction.

1. The overlap count is \(n(A\cap B)=74+58-103=29\). 2. The count in \(A\) only is \(74-29=45\). 3. The count in \(B\) only is \(58-29=29\). 4. The count in neither event is \(160-103=57\).

\(A\) only: \(45\) Both: \(29\) \(B\) only: \(29\) Neither: \(57\)

- Which two regions together make event \(A\)? - After finding the overlap, include every region once to obtain \(n(S)\). - For the claim about \(B\), use only the regions inside circle \(B\).

1. Event \(A\) consists of its only-\(A\) region and the overlap, so \(12+x=25\). Thus, \(x=13\). 2. The sample-space total is \(12+13+17+8=50\). 3. Event \(B\) contains the overlap and the \(B\)-only region, so \(n(B)=13+17=30\). Therefore, the claim \(n(B)=31\) is inconsistent with the diagram.

a) \(x=13\) b) \(n(S)=50\) c) No. The diagram gives \(n(B)=30\), not \(31\).