Inverse proportion worksheet with tables for calculating missing values and equations.

Let’s solve this step by step.

We’re told that y is inversely proportional to x, which means:

> y = k / x
> or equivalently, x × y = k (where k is a constant)

So for each table, we can find the constant k by multiplying any known pair of x and y values. Then use that k to find missing values.

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Problem 1: Find missing values

#### (a) Table:
x | 2 | ? | 4 | 8
y | 24| 6 | ? | ?

First, find k using first pair:
k = x × y = 2 × 24 = 48

Now fill in blanks:

- When y = 6 → x = k/y = 48/6 = 8
- When x = 4 → y = 48/4 = 12
- When x = 8 → y = 48/8 = 6

✔ So completed table:
x | 2 | 8 | 4 | 8
y |24 | 6 |12 | 6

Wait — hold on! The last x=8 gives y=6, but we already have x=8? That’s fine — it just means two different rows can have same x if y matches. But let’s check: actually, looking back at original table structure:

Original was:

(a)
x | 2 | ? | 4 | 8
y |24 | 6 | ? | ?

So positions are:

Row 1: x=2, y=24 → k=48
Row 2: x=?, y=6 → x=48/6=8
Row 3: x=4, y=? → y=48/4=12
Row 4: x=8, y=? → y=48/8=6

Yes, so final answers for (a):
Missing x: 8
Missing y’s: 12 and 6

But wait — row 4 has x=8 and y=6, which is same as row 2? That’s okay — maybe it's intentional. Or perhaps I misread the table layout.

Actually, looking again — probably the table is meant to be read column-wise? Let me recheck.

The way it’s written:

(a)
x | 2 | ? | 4 | 8
y |24 | 6 | ? | ?

That suggests 4 columns:

Column 1: x=2, y=24
Column 2: x=?, y=6
Column 3: x=4, y=?
Column 4: x=8, y=?

So yes — our calculation is correct.

Final for (a):
x missing: 8
y missing: 12, 6

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#### (b) Table:
x | 2 | 4 | 5 | 10
y | ? | ? |15 | ?

Find k from known pair: x=5, y=15 → k = 5×15 = 75

Now:

- x=2 → y=75/2 = 37.5
- x=4 → y=75/4 = 18.75
- x=10 → y=75/10 = 7.5

✔ So:

x | 2 | 4 | 5 | 10
y |37.5 |18.75 |15 |7.5

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#### (c) Table:
x | 0.5 | 1 | 1.5 | 2
y | 12 | ? | 4 | ?

Use first pair: x=0.5, y=12 → k = 0.5 × 12 = 6

Check with third pair: x=1.5, y=4 → 1.5×4=6 ✔️ good.

Now:

- x=1 → y=6/1 = 6
- x=2 → y=6/2 = 3

✔ Completed:

x |0.5|1 |1.5|2
y |12 |6 |4 |3

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#### (d) Table:
x | ? | 4 | 8
y |25| 5 |2.5

Use second pair: x=4, y=5 → k=4×5=20

Check third pair: x=8, y=2.5 → 8×2.5=20 ✔️

First pair: y=25 → x=k/y=20/25=0.8

✔ So:

x |0.8|4 |8
y |25 |5 |2.5

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Problem 2: Write equation for y in terms of x

Again, since y ∝ 1/x → y = k/x → need to find k for each table.

#### (a) Table:
x | 1 | 2 | 4 | 8
y |12 |6 |4.8|3

Pick any pair to find k.

Try x=1, y=12 → k=1×12=12
Check x=2, y=6 → 2×6=12 ✔️
x=4, y=4.8 → 4×4.8=19.2 ✘ Wait — not 12?

Hold on — 4 × 4.8 = 19.2? That doesn’t match.

Wait — maybe typo? Or did I miscalculate?

Wait — 4 × 4.8 = 19.2? Actually, 4 × 4.8 = 19.2 — but earlier pairs give k=12.

Inconsistency? Let’s check all:

x=1, y=12 → k=12
x=2, y=6 → k=12
x=4, y=4.8 → 4×4.8=19.2 → NOT 12
x=8, y=3 → 8×3=24 → also not 12

This is strange. Maybe the value 4.8 is wrong? Or perhaps it’s approximate?

Wait — maybe it’s 3? No, written as 4.8.

Alternatively — perhaps it’s not exactly inverse proportion? But problem says “for each table of values, y ∝ 1/x”

Wait — let’s recalculate 4 × 4.8:

4 × 4.8 = 4 × (4 + 0.8) = 16 + 3.2 = 19.2 — yes.

But 1×12=12, 2×6=12, 8×3=24 — inconsistent.

Unless... maybe the 4.8 is a mistake? Or perhaps it’s 3? But no.

Wait — what if we take average? Or maybe use most consistent?

Actually — look: 1→12, 2→6, 8→3 — these suggest k=12.

Then for x=4, y should be 12/4=3 — but table says 4.8.

Hmm. Perhaps it’s a trick? Or maybe I misread.

Wait — let me double-check the image description.

User wrote:

2.(a)
x | 1 | 2 | 4 | 8
y |12 |6 |4.8|3

Perhaps 4.8 is meant to be 3? Because 12/4=3.

Maybe it’s a typo in the problem? Or perhaps it’s testing if we notice inconsistency?

But the instruction says “work out an equation”, implying it IS inverse proportion.

Alternative idea: maybe the 4.8 is correct, and we need to find best-fit k? But that’s too advanced.

Wait — let’s calculate k for each:

x=1,y=12 → k=12
x=2,y=6 → k=12
x=4,y=4.8 → k=19.2
x=8,y=3 → k=24

Not constant. So contradiction.

But perhaps the 4.8 is actually 3? Because 12/4=3, and 12/8=1.5? No, y=3 for x=8 would require k=24.

Wait — if x=8, y=3 → k=24
Then x=1 → y=24, but table says 12 — no.

Another possibility: maybe the relationship is y = k/x, but k is not integer? But still must be constant.

Unless... perhaps the table is misaligned? Like maybe y values correspond differently?

Looking back at user input:

"2. For each table of values, y ∝ 1/x Work out an equation for y in terms of x.

(a)
x | 1 | 2 | 4 | 8
y |12 |6 |4.8|3"

Perhaps 4.8 is a red herring? Or maybe it's 3.0 and written as 4.8 by mistake?

Wait — 12/4 = 3, not 4.8. 4.8 is 24/5 — not helpful.

Perhaps it's y = k/x^2 or something? But problem says inverse proportion, which is y = k/x.

I think there might be a typo in the problem. Given that three points fit k=12 except x=4, and x=8 gives y=3 which would require k=24, it's messy.

Wait — let's try x=8, y=3: k=24
Then x=1: y=24, but table says 12 — no.

x=2,y=6: k=12
x=4,y=4.8: k=19.2
No pattern.

Perhaps the 4.8 is meant to be 3? Let me assume that for now, because otherwise it doesn't make sense.

Maybe it's 3.0 and written as 4.8 due to font issue? Unlikely.

Another thought: perhaps "4.8" is "3" with a smudge? Or in some fonts 3 looks like 8? No.

Let's calculate what y should be if k=12: for x=4, y=3; for x=8, y=1.5 — but table has y=3 for x=8.

If k=24: x=1,y=24; x=2,y=12; x=4,y=6; x=8,y=3 — but table has y=12 for x=1, etc.

None match perfectly.

Wait — let's list the products:

1*12 = 12
2*6 = 12
4*4.8 = 19.2
8*3 = 24

Not constant. So either the problem has error, or I'm missing something.

Perhaps "inverse proportion" here allows for approximation? But that's not standard.

Another idea: maybe the y-value for x=4 is 3, and "4.8" is a typo for "3". Because 12/4=3, and then for x=8, y=1.5, but table says 3 — still not matching.

Unless for x=8, y=1.5, but it's written as 3.

This is confusing.

Let's look at other parts to see if similar issues.

Part (b):

x | 1 | 2 | 3 | 4
y | 2 | 1 | 2/3 | 1/2

Calculate k:

x=1,y=2 → k=2
x=2,y=1 → k=2
x=3,y=2/3 → 3*(2/3)=2 ✔️
x=4,y=1/2 → 4*0.5=2 ✔️

Perfect! So k=2, equation y=2/x

Similarly, part (c):

x | 2 | 1/2 | 1 | 4
y | 1/4 | 1 | 1/2 | 1/8

Check products:

x=2,y=1/4 → 2*0.25=0.5
x=0.5,y=1 → 0.5*1=0.5
x=1,y=0.5 → 1*0.5=0.5
x=4,y=1/8=0.125 → 4*0.125=0.5 ✔️

So k=0.5, equation y=0.5/x or y=1/(2x)

Part (d):

x | 1 | 2 | 3 | 4
y | 8 | 4 | 8/3 | 2

Products:

1*8=8
2*4=8
3*(8/3)=8
4*2=8 ✔️

So k=8, y=8/x

Back to part (a) — why is it inconsistent?

Perhaps the "4.8" is a mistake, and it should be "3"? Because if y=3 when x=4, then k=12, and for x=8, y should be 1.5, but table says 3 — still not matching.

Unless for x=8, y=1.5, but it's written as 3.

Maybe the last y is 1.5? But user wrote "3".

Another possibility: perhaps the table is:

x | 1 | 2 | 4 | 8
y |12 |6 |3 |1.5

But user said y |12 |6 |4.8|3

4.8 is close to 5, but not helpful.

Perhaps it's y = k/x with k=12 for first two, but then changes? But that violates the condition.

I think there might be a typo in the problem. Given that in all other tables it works perfectly, and for (a), if we ignore the 4.8 and use the first two points, k=12, and for x=8, y=1.5, but it's given as 3, which is double.

Notice that 4.8 * 2.5 = 12, but not helpful.

Let's calculate the geometric mean or something — too complicated.

Perhaps "4.8" is "3" and "3" is "1.5", but that's assuming too much.

Another idea: maybe the x-values are not in order? But unlikely.

Or perhaps the y for x=4 is 3, and for x=8 is 1.5, but written incorrectly.

Given the context, and since the problem asks to "work out an equation", and three out of four points don't agree, but two do (x=1,y=12 and x=2,y=6 give k=12), and x=8,y=3 gives k=24, which is double, perhaps it's a different constant.

Wait — let's see the ratio.

From x=1 to x=2, y halves from 12 to 6 — good for inverse proportion.

From x=2 to x=4, if y were 3, it would halve again, but it's 4.8, which is not half of 6.

6 / 4.8 = 1.25, not 2.

From x=4 to x=8, if y were 2.4, it would halve, but it's 3.

3 / 4.8 = 0.625, not 0.5.

So not consistent.

Perhaps it's not pure inverse proportion, but the problem states it is.

I think for the sake of solving, I'll assume that the intended k is 12, and the 4.8 is a typo, and should be 3, and the 3 for x=8 should be 1.5. But since the user provided specific numbers, I have to work with them.

Maybe "4.8" is "3.0" and "3" is "1.5", but that's guessing.

Another approach: perhaps use the first point to define k, and ignore others, but that's not fair.

Let's calculate k for each and see if there's a pattern, but there isn't.

Perhaps the relationship is y = k/x, and k is to be found from the data, but it's not constant, so impossible.

I recall that in some curricula, they might expect you to use one pair to find k, even if others don't match, but that seems poor.

For part (a), let's use the first pair: x=1, y=12, so k=12, equation y=12/x.

Then for x=4, y=3, but table says 4.8 — discrepancy.

Perhaps the 4.8 is for a different x.

I think I have to proceed with the calculation as per the majority or the first point.

But let's look at the product for x=4,y=4.8: 19.2, and for x=8,y=3: 24, average is around 21.6, not nice.

Notice that 12, 6, 4.8, 3 — the y-values are decreasing, but not halving.

12 to 6 is /2, 6 to 4.8 is /1.25, 4.8 to 3 is /1.6 — not constant ratio.

Whereas for inverse proportion, when x doubles, y should halve.

From x=1 to x=2 (double), y from 12 to 6 (halve) — good.

From x=2 to x=4 (double), y should go from 6 to 3, but it's 4.8 — not good.

From x=4 to x=8 (double), y should go from 4.8 to 2.4, but it's 3 — not good.

So only the first doubling works.

Perhaps the table is for y = k/x^n with n≠1, but the problem says inverse proportion, which implies n=1.

I think there might be a typo, and "4.8" should be "3", and "3" should be "1.5". In many textbooks, it's common to have y=12,6,3,1.5 for x=1,2,4,8.

So I'll assume that for the purpose of this problem, k=12, and the equation is y=12/x.

Similarly, for the missing values in problem 1, we had no such issue.

So for 2(a), I'll say y = 12/x.

To confirm, if x=4, y=3, but table says 4.8 — perhaps it's a different number.

Another idea: perhaps "4.8" is "3" and it's written as 4.8 due to decimal error, but 3.0 vs 4.8 is big difference.

Maybe it's 2.4? 12/5=2.4, but x=4.

I give up — I'll use k=12 from the first two points, as they are consistent, and assume the others are typos.

So for 2(a): y = 12/x

But let's see the answer format.

Perhaps calculate k as the product for each and take median or something, but that's not standard.

Let's do this: for 2(a), since x=1,y=12 and x=2,y=6 give k=12, and x=8,y=3 gives k=24, which is twice, perhaps it's a different interpretation.

Notice that 12/1 = 12, 6/2 = 3, not constant.

I think I have to box the answer as y = 12/x, acknowledging the discrepancy.

For the sake of completing, I'll proceed.

So summary for problem 2:

(a) Using x=1,y=12, k=12, so y = 12/x

(b) As calculated, k=2, y=2/x

(c) k=0.5, y=0.5/x or y=1/(2x)

(d) k=8, y=8/x

Now for problem 1, we have all answers.

Let me compile final answers.

First, problem 1:

(a) Missing x: 8; missing y: 12 and 6

(b) Missing y: 37.5, 18.75, 7.5

(c) Missing y: 6 and 3

(d) Missing x: 0.8

Problem 2:

(a) y = 12/x (assuming k=12)

(b) y = 2/x

(c) y = 0.5/x or y = \frac{1}{2x}

(d) y = 8/x

For (c), better to write as y = \frac{1}{2x} or y = \frac{0.5}{x}, but usually fraction is preferred.

Since in the table, y=1/4 when x=2, etc., so y = \frac{1}{2x} is fine.

Similarly, for (a), to be precise, but I think y=12/x is expected.

Perhaps for (a), they want us to use the values as is, but then k is not constant.

Another thought: in some definitions, inverse proportion means xy=constant, so for (a), if we take the product, it's not constant, so perhaps the table is incorrect, but for homework, we have to provide an equation.

I recall that in the user's message, for 2(a), y has "4.8", which might be "3" if it's a scan error, but in text, it's given as 4.8.

Perhaps "4.8" is "3.0" and it's a typo.

I think for the answer, I'll put y = 12/x for (a).

So let's write the final answers.

First, for problem 1:

(a) The missing x is 8, and the missing y values are 12 and 6.

(b) The missing y values are 37.5, 18.75, and 7.5.

(c) The missing y values are 6 and 3.

(d) The missing x is 0.8.

For problem 2:

(a) y = 12/x

(b) y = 2/x

(c) y = \frac{1}{2x} or y = 0.5/x

(d) y = 8/x

To make it neat, for (c), since the products are 0.5, y = \frac{0.5}{x} = \frac{1}{2x}

Similarly, for (a), despite the inconsistency, y = \frac{12}{x}

Now for the boxed answer, since the user didn't specify which part, but the task is to solve the problem, I need to provide all answers.

The instruction is to provide the final answer after reasoning, so I'll list them clearly.

Also, for problem 1(b), the y values are fractions or decimals? 75/2=37.5, etc., so decimals are fine.

In (c), y=6 and 3, integers.

In (d), x=0.8, which is 4/5, but decimal is ok.

For problem 2, equations.

So let's write.

Final Answer:

For Question 1:
(a) Missing x: 8; missing y values: 12, 6
(b) Missing y values: 37.5, 18.75, 7.5
(c) Missing y values: 6, 3
(d) Missing x: 0.8

For Question 2:
(a) y = \frac{12}{x}
(b) y = \frac{2}{x}
(c) y = \frac{1}{2x}
(d) y = \frac{8}{x}

Note: For 2(a), there is an inconsistency in the table (when x=4, y=4.8 gives xy=19.2, while other points give 12 or 24), but based on the first two points, k=12 is used. If the table is correct, it may not be perfect inverse proportion, but per problem statement, we assume it is and use a consistent k.

However, upon second thought, let's double-check the product for x=4,y=4.8: 4*4.8=19.2, and for x=8,y=3: 24, while for x=1,y=12:12, x=2,y=6:12. So perhaps the intended k is 12, and the values for x=4 and x=8 are misstated. In many similar problems, it's common to have y=3 for x=4 and y=1.5 for x=8. So I think it's safe to use y=12/x.

To be accurate, perhaps calculate k as the average or something, but that's not standard.

I think for school level, they expect y=12/x.

So I'll go with that.