Linear Patterns, Formula, Tables and Graphs - Starting Points Maths - Free Printable

Let’s solve this step by step.

We are given diagrams of square tables with circular chairs around them.

Looking at the diagrams:

- 1 table: There are 4 chairs (one on each side).
- 2 tables pushed together: The two inner sides touch, so we lose 2 chairs (one from each table that’s now touching). So instead of 4 + 4 = 8, we have 8 - 2 = 6 chairs.
- 3 tables in a row: Each time we add a table, we only add 2 new chairs (because one side is attached to the previous table). So for 3 tables: 6 + 2 = 8 chairs? Wait — let’s count from the diagram.

Actually, looking carefully at the diagrams provided:

→ For 1 table: 4 chairs
→ For 2 tables: 6 chairs
→ For 3 tables: 8 chairs

So every time we add a table, we add 2 more chairs.

That means:

- Tables: 1 → Chairs: 4
- Tables: 2 → Chairs: 6
- Tables: 3 → Chairs: 8
- Tables: 4 → Chairs: 10
- Tables: 5 → Chairs: 12

This is an arithmetic pattern: starts at 4, increases by 2 each time.

---

(a) Complete the table:

| Tables | Chairs Needed |
|--------|---------------|
| 1 | 4 |
| 2 | 6 |
| 3 | 8 |
| 4 | 10 |
| 5 | 12 |

---

(b) Complete the formula:

We want:
Chairs Needed = ___ × Tables – ___

From the pattern:

When Tables = 1 → Chairs = 4
If we try: Chairs = 2 × Tables + 2 → 2×1 + 2 = 4 ✔️
But the formula says “minus”, not “plus”.

Wait — maybe rearrange:

Chairs = 2 × Tables + 2
= 2 × Tables – (-2) → but that’s awkward.

Alternatively, think:
Each table adds 2 chairs, but the first table has 4, which is 2 extra.

Another way:
Imagine if you had 0 tables → how many chairs? Probably 0, but let’s see what the formula gives.

Try fitting: Chairs = m × Tables + b

We know:
At T=1, C=4 → 4 = m(1) + b
At T=2, C=6 → 6 = m(2) + b

Subtract:
6 - 4 = 2m → m = 2
Then 4 = 2(1) + b → b = 2

So: Chairs = 2 × Tables + 2

But the question says:
“Chairs Needed = _____ x Tables - _____”

So we can write:
Chairs = 2 × Tables + 2 = 2 × Tables - (-2)

But negative numbers might be confusing here. Maybe they expect us to think differently?

Wait — perhaps another approach.

Notice:
For n tables, chairs = 2n + 2

But if we force it into “minus” form:

Chairs = 2 × Tables - (-2) → still messy.

Alternatively, maybe the problem expects:

Look at the difference between chairs and twice the tables:

Tables | Chairs | 2×Tables | Difference
1 | 4 | 2 | +2
2 | 6 | 4 | +2
3 | 8 | 6 | +2

So always 2 more than 2×tables.

But again, the blank says “minus”.

Perhaps there's a mistake in interpretation?

Wait — let me re-read the question:

> Chairs Needed = _____ x Tables - _____

Maybe they want us to express it as:

Chairs = 2 × Tables + 2 = 2 × Tables - (-2)

But since it says “minus”, and blanks are for numbers, probably they accept:

Chairs Needed = 2 x Tables - (-2) — but that’s unusual.

Alternatively, maybe I miscounted?

Wait — look back at the diagrams.

In the first diagram (1 table): 4 chairs — correct.

Second diagram (2 tables): Let’s count — left end: 1 chair, right end: 1 chair, top: 2 chairs (one per table), bottom: 2 chairs → total 6 — correct.

Third diagram (3 tables): left:1, right:1, top:3, bottom:3 → total 8 — correct.

So yes, chairs = 2n + 2.

But the formula format is “A x Tables - B”

So:

2n + 2 = A*n - B

Then A=2, and -B = +2 → B = -2

So:

Chairs Needed = 2 x Tables - (-2)

But writing “- (-2)” is odd. Maybe the problem allows positive number in the blank even if it’s subtracting a negative? Or perhaps I made a mistake.

Alternative idea: Maybe they consider the ends differently.

Another common way: When tables are joined, each additional table adds 2 chairs, but the first table has 4.

So general formula: Chairs = 2*(number of tables) + 2

To fit “minus”, perhaps:

Chairs = 2*Tables + 2 = 2*Tables - (-2)

I think we have to go with that.

But let’s check part (d): Use formula for 0 tables.

If Chairs = 2*0 + 2 = 2 chairs for 0 tables? That doesn’t make sense — no tables, no chairs needed.

Ah! Here’s the issue.

If we use Chairs = 2n + 2, then for n=0, chairs=2 — which is wrong.

But in reality, for 0 tables, chairs should be 0.

So maybe the formula isn't 2n+2?

Wait — let’s think again.

Perhaps the pattern is different.

Another way: In such problems, often the formula is derived as:

Each table has 4 sides, but when joined, shared sides don’t need chairs.

For n tables in a row:

- Number of exposed sides = 2 (ends) + 2*n (top and bottom) = 2 + 2n

Yes! That’s the same as before: 2n + 2

But for n=0, it gives 2, which is incorrect.

However, in context, n=0 is not practical, but mathematically the formula holds for n≥1.

But part (d) asks for 0 tables, so we must address it.

Perhaps the formula should be adjusted.

Wait — let’s list:

n | chairs
0 | ?
1 | 4
2 | 6
3 | 8
4 | 10
5 | 12

The sequence from n=1 onwards is linear with slope 2.

If we extend backwards:

From n=1 to n=0: decrease by 2 → 4 - 2 = 2

So mathematically, the line passes through (0,2)

But in real life, 0 tables should have 0 chairs.

So the formula works for n≥1, but not for n=0 in context.

For the purpose of this problem, we’ll use the mathematical formula.

So back to (b):

Chairs Needed = 2 x Tables + 2

To write as “minus”:

Chairs Needed = 2 x Tables - (-2)

But since the blank is for a number, and it says “- _____”, we put -2 in the blank? No.

Let me read the blank again:

“Chairs Needed = _____ x Tables - _____”

So first blank is multiplier, second blank is the number being subtracted.

So if Chairs = 2*T + 2, then to write as A*T - B, we have:

2T + 2 = 2T - (-2)

So B = -2

But typically, we don’t write minus negative. Perhaps the problem has a typo, or I need to think differently.

Another possibility: Maybe they want Chairs = 2*(Tables + 1) = 2*Tables + 2, same thing.

Or perhaps for the formula, they expect:

Notice that chairs = 2*Tables + 2 = 2*(Tables + 1)

But still.

Let’s look at part (c): Use formula for 8 tables.

If Chairs = 2*8 + 2 = 18

Similarly, 12 tables: 2*12 + 2 = 26

1000 tables: 2*1000 + 2 = 2002

Now for (d): 0 tables: 2*0 + 2 = 2 chairs — which does not make sense, as no tables mean no chairs needed.

So in context, it doesn't make sense.

But for the formula, we'll proceed.

Perhaps the intended formula is Chairs = 2*Tables + 2, and for (b), they want:

Chairs Needed = 2 x Tables + 2, but the blank says "minus", so maybe it's a trick.

Wait — another thought: Sometimes in these problems, they derive it as:

Total possible chairs if separate: 4 per table, but each join removes 2 chairs (since two sides are covered).

For n tables, number of joins = n-1

So chairs = 4n - 2(n-1) = 4n - 2n + 2 = 2n + 2

Same thing.

So I think we have to go with Chairs = 2*Tables + 2

For (b), to fit "minus", perhaps they allow:

Chairs Needed = 2 x Tables - (-2)

But since the blank is for a number, and it's " - _____ ", we can put -2 in the blank, meaning subtracting negative 2, which is adding 2.

In some contexts, it's accepted.

Perhaps the problem meant to say "+", but it says "-".

Let me double-check the user's image description.

The user wrote: "Chairs Needed = _____ x Tables - _____"

And in the image, it's likely printed as such.

Given that, and to move forward, I'll assume that the second blank is for the constant term with sign, so for +2, we put - (-2), but since it's a blank, perhaps they expect the magnitude and sign is handled by the minus.

Standard way in such worksheets is to fill the blanks with numbers, and the operation is fixed.

So for example, if it were Chairs = 3*Tables - 1, you'd put 3 and 1.

Here, Chairs = 2*Tables + 2 = 2*Tables - (-2), so the second blank is -2.

But that might confuse students.

Perhaps there's a different interpretation.

Let's calculate the difference.

From the table:

When tables increase by 1, chairs increase by 2, so slope is 2.

Intercept: when tables=0, chairs=2, as per extrapolation.

So y = 2x + 2

For the form y = mx - b, then 2x + 2 = 2x - (-2), so b = -2.

I think for accuracy, we'll use that.

So for (b): Chairs Needed = 2 x Tables - (-2)

But since the blank is for a number, and in many systems, they might expect the absolute value with the operation, but here the operation is fixed as minus.

Perhaps the problem has a mistake, but based on data, it's 2n+2.

Another idea: Maybe for the first table, it's 4, and each additional table adds 2, so for n tables, chairs = 4 + 2(n-1) = 4 + 2n - 2 = 2n + 2 — same thing.

I think we have to proceed.

So for (b): First blank: 2, second blank: -2 (since - (-2) = +2)

But let's see part (d): for 0 tables, 2*0 - (-2) = 0 +2 = 2 chairs.

Does it make sense? No, because with no tables, you don't need any chairs. So in context, it doesn't make sense.

For (c):

(i) 8 tables: 2*8 + 2 = 16 + 2 = 18

(ii) 12 tables: 2*12 + 2 = 24 + 2 = 26

(iii) 1000 tables: 2*1000 + 2 = 2000 + 2 = 2002

For (e) and (f): Plot points from table: (1,4), (2,6), (3,8), (4,10), (5,12)

On graph, x-axis is tables, y-axis is chairs.

Plot those points, join with straight line.

(g) Extend the line. When x=0, y=2, so the point (0,2) is on the line, which is the answer to (d): 2 chairs for 0 tables.

(h) Will a situation ever arise which requires 41 tables? Well, the question is probably: will there be a number of chairs that corresponds to 41 tables? Or perhaps, is 41 a possible number of chairs for some integer number of tables?

Let's read: "Will a situation ever arise which requires 41 tables?"

That seems odd. It says "requires 41 tables", but tables are chosen by us, not required by chairs.

Perhaps it's a typo, and it means "requires 41 chairs"?

Because in context, we find chairs for given tables, so likely it's "will there be a situation where 41 chairs are needed?"

Let me check the original: "(h) Will a situation ever arise which requires 41 tables?"

But that doesn't make much sense, because we can always have 41 tables; the question is probably about whether 41 is a possible number of chairs for some whole number of tables.

Perhaps "requires 41 chairs" — let's assume that, as it's common in such problems.

In many similar problems, they ask if a certain number of chairs is possible.

So I'll assume it's "requires 41 chairs".

So, is there an integer n such that chairs = 2n + 2 = 41?

2n + 2 = 41

2n = 39

n = 19.5

Not an integer, so no, you can't have exactly 41 chairs with whole tables.

If it really means 41 tables, then yes, you can have 41 tables, and chairs would be 2*41 + 2 = 84, so it arises, but that seems trivial.

Given the context, likely it's a misphrasing, and it's about 41 chairs.

In the initial problem, all parts are about finding chairs for tables, so (h) is probably asking if 41 chairs is possible.

I'll go with that.

So summary:

(a) Table filled as above.

(b) Chairs Needed = 2 x Tables - (-2) [but to write in blanks: first blank 2, second blank -2]

Since the second blank is after " - ", and we need to subtract a negative, but in practice, for the answer, we can write the numbers.

Perhaps the expected answer is Chairs = 2*Tables + 2, and for the formula with minus, they might have meant something else, but based on calculation, it's consistent.

Another way: Some sources write the formula as chairs = 2(tables + 1), but same.

I think for the sake of completing, we'll use:

For (b): Chairs Needed = 2 x Tables - (-2)

But since the blank might expect a positive number, and the operation is minus, perhaps it's not.

Let's calculate the constant.

From the pattern, the y-intercept is 2, so in y = mx + c, c=2.

For y = mx - b, then -b = c, so b = -c = -2.

So b = -2.

So in the blank for the subtraction, it's -2.

So when you do 2*Tables - (-2) = 2*Tables +2.

So for the answer, we'll put 2 and -2.

Now for the final answers.

Let's compile.

First, (a) table:

Tables | Chairs Needed
1 | 4
2 | 6
3 | 8
4 | 10
5 | 12

(b) Chairs Needed = 2 x Tables - (-2) → so blanks: 2 and -2

But to write it nicely, perhaps the problem allows, or maybe they want the absolute value, but I think mathematically it's correct.

Perhaps in some curricula, they teach it as chairs = 2n + 2, and for the formula with minus, it's not used, but here it is specified.

Another thought: Maybe " - " is a placeholder for the operation, but the user wrote " - ", so likely it's minus.

I recall that in some problems, they have chairs = 4 + 2(n-1) = 2n +2, same.

I think we have to box the answers as per calculation.

For (c):

(i) 8 tables: 2*8 + 2 = 18

(ii) 12 tables: 2*12 + 2 = 26

(iii) 1000 tables: 2*1000 + 2 = 2002

(d) For 0 tables: 2*0 + 2 = 2 chairs. Does it make sense? No, because with no tables, no chairs are needed. So it does not make sense in context.

(e) Plot points: (1,4), (2,6), (3,8), (4,10), (5,12) on the graph.

(f) Join with straight line.

(g) Extend the line; when x=0, y=2, so the point (0,2) is on the graph, which is the answer to (d).

(h) Assuming it means "requires 41 chairs": Solve 2n + 2 = 41 → 2n = 39 → n = 19.5, not integer, so no, it will not arise for whole number of tables.

If it means 41 tables, then yes, but that seems unlikely.

Given the context, I'll assume it's 41 chairs.

So for (h): No, because 41 is not of the form 2n+2 for integer n.

Now for the final answer section, since the user didn't specify which part, but the task has multiple parts, probably we need to provide all answers.

But the instruction says "Final Answer: [Provide only the final correct answer]", implying for the whole task, but it's multi-part.

Perhaps for each part, but the format suggests one final answer.

Looking back at the user's request: "Solve the problem accurately." and "Final Answer: [Provide only the final correct answer]"

But the problem has several subparts.

Perhaps the "final answer" refers to the key results, or maybe for the entire task.

To comply, I'll provide the answers for each part in the final answer section, summarized.

Since the user said "if the question has a final answer", and this is a multi-part question, perhaps we list them.

I think for clarity, in the final answer, we can state the completed table, the formula, etc.

But to keep it concise, perhaps the most important is the formula and the values.

Given that, I'll write the final answer as the completed items.

But let's see the response format.

Perhaps for this type, the final answer is the formula and the calculations.

I think I'll provide the answers for (a) to (h) in the final answer section.

So:

Final Answer:

(a) Table:
Tables | Chairs Needed
1 | 4
2 | 6
3 | 8
4 | 10
5 | 12

(b) Chairs Needed = 2 x Tables - (-2) [or simply 2 and -2 for the blanks]

To write it as per blanks: first blank 2, second blank -2

(c) (i) 18 (ii) 26 (iii) 2002

(d) 2 chairs; no, it does not make sense in context.

(e) Points plotted at (1,4), (2,6), (3,8), (4,10), (5,12)

(f) Line drawn through these points.

(g) Yes, the point (0,2) is on the extended line, which matches the answer to (d).

(h) No, because 41 chairs would require 19.5 tables, which is not possible.

For (b), to make it neat, perhaps write the formula as Chairs Needed = 2 \times \text{Tables} + 2, but the problem specifies the form with minus.

Since the problem says "Complete the formula: Chairs Needed = _____ x Tables - _____", and based on math, it's 2 and -2, so we'll use that.

In some contexts, they might expect the second blank to be the number to subtract, so for +2, it's not direct, but mathematically correct.

I think it's fine.

So for the final answer, I'll box the key numerical answers.

Perhaps the user expects the formula and the values for (c).

But to be comprehensive, I'll include all.

Since the assistant is to provide the final answer after reasoning, and the reasoning is done, here is the final answer.

Note: For (b), if the system expects positive numbers, it might be an issue, but based on calculation, it's correct.

Another way: Perhaps the formula is Chairs = 2*(Tables + 1), but still.

I recall that in some textbooks, for this exact problem, the formula is given as c = 2t + 2, and for the form with minus, it's not used, but here it is specified.

Perhaps " - " is a typo, and it should be "+", but we have to work with what's given.

So I'll proceed.

Final Answer:

(a)
| Tables | Chairs Needed |
|--------|---------------|
| 1 | 4 |
| 2 | 6 |
| 3 | 8 |
| 4 | 10 |
| 5 | 12 |

(b) Chairs Needed = 2 x Tables - (-2) → so the blanks are 2 and -2

(c) (i) 18 (ii) 26 (iii) 2002

(d) 2 chairs; no, it does not make sense because with no tables, no chairs are needed.

(e) Plot the points (1,4), (2,6), (3,8), (4,10), (5,12) on the graph.

(f) Draw a straight line through these points.

(g) Yes, when extended, the line passes through (0,2), which is the answer to (d).

(h) No, because 41 chairs would require 19.5 tables, which is not a whole number, so it cannot happen with complete tables.

For (h), if it literally means "requires 41 tables", then yes, you can have 41 tables, and chairs would be 84, so it arises, but that seems too straightforward, and the question is likely about chairs.

Given the context of the other parts, I think it's a misphrasing, and it's about 41 chairs.

So I'll go with that.

To match the format, perhaps in the final answer, list the numerical answers.

Since the user might expect the formula and the values, I'll emphasize those.

But for completeness, here is the final answer as per the task.

Final Answer:

- (a) Table completed as above.
- (b) 2 and -2
- (c) (i) 18, (ii) 26, (iii) 2002
- (d) 2; no
- (e) Points plotted at (1,4), (2,6), (3,8), (4,10), (5,12)
- (f) Line drawn
- (g) Yes, at (0,2)
- (h) No

For the boxed answer, perhaps the most critical is the formula and the calculation for (c).

Since the instruction is to provide the final correct answer, and it's a multi-part, I'll write:

The number of chairs for n tables is given by \( 2n + 2 \). For 8 tables: 18, for 12 tables: 26, for 1000 tables: 2002. For 0 tables, the formula gives 2, which does not make sense in context. 41 chairs is not possible for any whole number of tables.

But to be precise, let's output as per the parts.

I think for the response, in the final answer section, we can put:

\boxed{2} for the first blank in (b), \boxed{-2} for the second blank in (b), and for (c) \boxed{18}, \boxed{26}, \boxed{2002}, etc.

But that might be messy.

Perhaps the "final answer" refers to the overall solution, but I think for this platform, we can summarize.

Given the constraints, I'll provide the answers in a list.

So:

Final Answer:
- (a) See table above.
- (b) 2, -2
- (c) (i) 18 (ii) 26 (iii) 2002
- (d) 2, no
- (e) Points: (1,4), (2,6), (3,8), (4,10), (5,12)
- (f) Straight line through points.
- (g) Yes, at (0,2)
- (h) No

For (b), to make it clear, the formula is Chairs Needed = 2 × Tables - (-2)

But in the blank, it's 2 and -2.

I think it's acceptable.

So I'll go with that.