Question

1. there are 88 seats in a theater. the seating in the theater is split into 4 identical sections. each section has 14 red seats and some blue seats.

a) draw a tape diagram to represent the situation.
b) what unknown amounts can be found by using the diagram or reasoning about the situation?

Explanation:

Part (a) - Tape Diagram Explanation:

To draw a tape diagram for this situation:

1. First, represent the total number of seats (88) with a long tape.
2. Divide this tape into 4 equal parts (since there are 4 identical sections). Each part represents one section.
3. Within each of these 4 parts, split it into two smaller parts: one part to represent the 14 red seats and another part (unknown length) to represent the blue seats in that section.

Visually, it would look like:

• A large rectangle (tape) labeled "Total Seats: 88".
• This rectangle is divided into 4 equal smaller rectangles (each labeled "Section").
• Each "Section" rectangle is further divided into two: one labeled "Red: 14" and the other labeled "Blue:?".

Part (b) - Unknown Amounts Explanation:

To find the unknown amounts, we can use the given information:

1. Let \( x \) be the number of blue seats in one section.
2. The total number of seats in one section is the sum of red and blue seats, so it is \( 14 + x \).
3. Since there are 4 identical sections, the total number of seats in the theater is \( 4\times(14 + x) \), and we know this equals 88. So we can set up the equation \( 4\times(14 + x)=88 \).
4. First, find the number of seats in one section: divide the total seats by the number of sections. So \( \frac{88}{4} = 22 \) seats per section.
5. Then, to find the number of blue seats in one section, subtract the number of red seats from the total seats in one section: \( 22 - 14 = 8 \) blue seats per section.
6. We can also find the total number of red seats in the theater: \( 4\times14 = 56 \) red seats.
7. And the total number of blue seats in the theater: \( 4\times8 = 32 \) blue seats (or alternatively, total seats - total red seats: \( 88 - 56 = 32 \)).

So the unknown amounts we can find are:

• The number of blue seats in one section (8).
• The total number of red seats in the theater (56).
• The total number of blue seats in the theater (32).
• The total number of seats in one section (22).

Final Answers (for part b, key unknowns found):

• Number of blue seats in one section: \( \boldsymbol{8} \)
• Total number of red seats: \( \boldsymbol{56} \)
• Total number of blue seats: \( \boldsymbol{32} \)
• Number of seats in one section: \( \boldsymbol{22} \)

Answer:

Part (a) - Tape Diagram Explanation:

To draw a tape diagram for this situation:

1. First, represent the total number of seats (88) with a long tape.
2. Divide this tape into 4 equal parts (since there are 4 identical sections). Each part represents one section.
3. Within each of these 4 parts, split it into two smaller parts: one part to represent the 14 red seats and another part (unknown length) to represent the blue seats in that section.

Visually, it would look like:

• A large rectangle (tape) labeled "Total Seats: 88".
• This rectangle is divided into 4 equal smaller rectangles (each labeled "Section").
• Each "Section" rectangle is further divided into two: one labeled "Red: 14" and the other labeled "Blue:?".

Part (b) - Unknown Amounts Explanation:

To find the unknown amounts, we can use the given information:

1. Let \( x \) be the number of blue seats in one section.
2. The total number of seats in one section is the sum of red and blue seats, so it is \( 14 + x \).
3. Since there are 4 identical sections, the total number of seats in the theater is \( 4\times(14 + x) \), and we know this equals 88. So we can set up the equation \( 4\times(14 + x)=88 \).
4. First, find the number of seats in one section: divide the total seats by the number of sections. So \( \frac{88}{4} = 22 \) seats per section.
5. Then, to find the number of blue seats in one section, subtract the number of red seats from the total seats in one section: \( 22 - 14 = 8 \) blue seats per section.
6. We can also find the total number of red seats in the theater: \( 4\times14 = 56 \) red seats.
7. And the total number of blue seats in the theater: \( 4\times8 = 32 \) blue seats (or alternatively, total seats - total red seats: \( 88 - 56 = 32 \)).

So the unknown amounts we can find are:

• The number of blue seats in one section (8).
• The total number of red seats in the theater (56).
• The total number of blue seats in the theater (32).
• The total number of seats in one section (22).

Final Answers (for part b, key unknowns found):

• Number of blue seats in one section: \( \boldsymbol{8} \)
• Total number of red seats: \( \boldsymbol{56} \)
• Total number of blue seats: \( \boldsymbol{32} \)
• Number of seats in one section: \( \boldsymbol{22} \)