Consecutive integer problems worksheet with three word problems and a cat illustration.

Let's solve each of the consecutive integer problems step by step.

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Problem 1: Find two consecutive integers whose sum is 65.

Step 1: Let the first integer be $ x $.
Then, the next consecutive integer is $ x + 1 $.

Step 2: Their sum is given as 65:
$$
x + (x + 1) = 65
$$

Step 3: Simplify and solve:
$$
2x + 1 = 65
$$
$$
2x = 64
$$
$$
x = 32
$$

So, the two consecutive integers are:
$$
32 \text{ and } 33
$$

✔ Answer: 32 and 33

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Problem 2: Three consecutive integers are such that three times the smallest integer is 18 more than the largest. Find the integers.

Step 1: Let the three consecutive integers be:
- Smallest: $ x $
- Middle: $ x + 1 $
- Largest: $ x + 2 $

Step 2: According to the problem:
> Three times the smallest is 18 more than the largest.

So:
$$
3x = (x + 2) + 18
$$

Step 3: Solve:
$$
3x = x + 20
$$
$$
3x - x = 20
$$
$$
2x = 20
$$
$$
x = 10
$$

So the three integers are:
- $ x = 10 $
- $ x + 1 = 11 $
- $ x + 2 = 12 $

✔ Answer: 10, 11, 12

Check:
Three times the smallest: $ 3 \times 10 = 30 $
Largest: 12
Is $ 30 = 12 + 18 $? Yes! ✔️

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Problem 3: Find two consecutive even integers whose sum is 96.

Step 1: Consecutive even integers differ by 2.
Let the first even integer be $ x $.
Then the next consecutive even integer is $ x + 2 $.

Step 2: Their sum is 96:
$$
x + (x + 2) = 96
$$

Step 3: Solve:
$$
2x + 2 = 96
$$
$$
2x = 94
$$
$$
x = 47
$$

Wait — 47 is odd, but we need even integers. That means something’s wrong.

But let's check our work:
We got $ x = 47 $, so the numbers would be 47 and 49 — both odd. Not valid.

But wait — did we make a mistake?

No. The issue is: if we define $ x $ as an even integer, then $ x = 47 $ is not even. So maybe we should define it differently.

But actually, the algebra is correct — unless we made a calculation error.

Wait:

$$
2x + 2 = 96 \Rightarrow 2x = 94 \Rightarrow x = 47
$$

That’s correct mathematically, but 47 is odd.

But two consecutive even integers must both be even.

So perhaps there’s no solution? But that can't be.

Wait — let's double-check the problem: Sum is 96.

Try some examples:
- 46 + 48 = 94 → too small
- 48 + 50 = 98 → too big
- 47 + 49 = 96 → but they’re odd

Ah! There’s no pair of consecutive even integers that add to 96?

Wait — let’s think again.

Let’s suppose:
Let the smaller even integer be $ x $, then the next is $ x + 2 $

So:
$$
x + (x + 2) = 96
\Rightarrow 2x + 2 = 96
\Rightarrow 2x = 94
\Rightarrow x = 47
$$

But 47 is not even, so this cannot be the answer.

This suggests that no two consecutive even integers have a sum of 96?

But let’s test:
- 46 + 48 = 94
- 48 + 50 = 98

There’s a gap — 96 is in between, but not achievable with consecutive even integers.

Wait — is that possible?

Let’s suppose $ x $ and $ x+2 $ are even, their sum is $ 2x + 2 = 2(x + 1) $, which is always even, and divisible by 2.

But 96 is even, so that’s fine.

But $ x = 47 $, which is odd — so it doesn’t qualify.

So does that mean no solution?

But that seems odd for a textbook problem.

Wait — maybe I made a mistake in assuming the form.

Actually, the algebra is correct: if two consecutive even integers sum to 96, then:
$$
x + (x + 2) = 96 \Rightarrow x = 47
$$
But 47 is not even — contradiction.

So there are no two consecutive even integers that sum to 96?

Wait — let’s try:
- 46 + 48 = 94
- 48 + 50 = 98
- 47 + 49 = 96 — but those are odd

So indeed, no two consecutive even integers sum to 96.

But the problem says “Find two consecutive even integers whose sum is 96.”

So either:
- The problem has a typo, or
- We're misunderstanding "consecutive even integers"

Wait — maybe it's consecutive integers that are both even, but not necessarily adjacent in the even number sequence?

No — "consecutive even integers" means like 10 and 12, 14 and 16, etc.

So difference is 2.

But let’s suppose the problem meant consecutive integers, not even?

But it clearly says even.

Alternatively, maybe it's consecutive odd integers?

Try: 47 and 49 → sum = 96 → yes!

But they’re odd.

So perhaps the problem meant odd?

Or maybe it's a typo.

But let’s recheck:

Suppose we want two consecutive even integers with sum 96.

Let them be $ x $ and $ x + 2 $

Then:
$$
x + x + 2 = 96 \Rightarrow 2x = 94 \Rightarrow x = 47
$$

But 47 is odd, so $ x $ is not even — contradiction.

So no such pair exists.

But maybe the problem meant consecutive integers (not necessarily even)?

Then:
Let $ x $ and $ x + 1 $, sum = 96
$$
2x + 1 = 96 \Rightarrow 2x = 95 \Rightarrow x = 47.5
$$
Not integers.

So no.

Wait — perhaps the sum is 98 instead of 96?

Then:
$ x + x + 2 = 98 \Rightarrow 2x = 96 \Rightarrow x = 48 $

Then 48 and 50 → both even → sum = 98

But the problem says 96.

Alternatively, maybe it's consecutive integers (any), but sum 96?

Then:
$ x + (x+1) = 96 \Rightarrow 2x + 1 = 96 \Rightarrow 2x = 95 \Rightarrow x = 47.5 $ → not integer.

So no.

Wait — what if it’s consecutive even integers, but we allow negative?

Try: 46 and 48 → 94
48 and 50 → 98
No way to get 96.

So no two consecutive even integers sum to 96.

But that can't be right — maybe I made a mistake.

Wait — let’s suppose the two even integers are $ x $ and $ x + 2 $

Then $ x + x + 2 = 96 \Rightarrow 2x = 94 \Rightarrow x = 47 $

But 47 is not even — so no solution.

Therefore, there is no solution to this problem as stated.

But that seems unlikely for a guided lesson.

Wait — could it be consecutive integers (not even), but sum 96?

Then:
$ x + (x+1) = 96 \Rightarrow 2x + 1 = 96 \Rightarrow x = 47.5 $ → not integer.

No.

Wait — what if it's two consecutive even integers, but the sum is 98?

Then $ x = 48 $, so 48 and 50 → sum 98 → works.

But problem says 96.

Wait — perhaps it's three consecutive even integers?

But problem says "two".

Alternatively, maybe it's a typo and it should be odd integers?

Try: two consecutive odd integers summing to 96.

Let $ x $ and $ x + 2 $, both odd.

Then:
$ x + x + 2 = 96 \Rightarrow 2x = 94 \Rightarrow x = 47 $

Then 47 and 49 → both odd → sum = 96 → ✔

So likely, the problem meant odd integers, not even.

But it says even.

So either:
- The problem has a typo, or
- We accept that $ x = 47 $, but that’s not even.

But since $ x $ must be even, and 47 is not even, no solution exists.

But let’s check the original image again.

The text says:
> 3) Find two consecutive even integers whose sum is 96.

But based on math, no such integers exist.

Wait — let me confirm:

List of consecutive even integers:
- 46, 48 → 94
- 48, 50 → 98
- 50, 52 → 102

No pair sums to 96.

So no solution.

But perhaps the problem meant consecutive integers (not specified even), and the sum is 96?

Then:
$ x + (x+1) = 96 \Rightarrow 2x + 1 = 96 \Rightarrow x = 47.5 $ → not integer.

No.

Or maybe it’s consecutive integers with sum 95?

Then $ x = 47 $, $ x+1 = 48 $ → sum 95 → yes.

But not 96.

Alternatively, maybe the sum is 98?

Then $ x = 48 $, $ x+2 = 50 $ → sum 98 → works.

So likely, the intended problem was:
> Find two consecutive even integers whose sum is 98.

Or:
> Find two consecutive odd integers whose sum is 96.

Given that 47 and 49 are odd and sum to 96, and the algebra gives $ x = 47 $, which is odd, it's likely that the problem meant odd.

But since it says even, and mathematically no such pair exists, we must conclude:

✘ No solution exists for Problem 3 as stated.

But let’s double-check the possibility.

Wait — could the integers be negative?

Try: -48 and -46 → sum = -94
-50 and -48 → -98
Still not 96.

No.

So conclusion: No two consecutive even integers sum to 96.

But since this is a guided lesson, likely there’s a typo.

Possibility 1: It should be odd integers → then answer is 47 and 49.

Possibility 2: Sum is 98 → then 48 and 50.

Possibility 3: Sum is 94 → 46 and 48.

But as written, no solution.

However, let’s assume the problem meant odd integers (common mistake).

Then:

Let $ x $ and $ x + 2 $ be consecutive odd integers.

Then:
$$
x + (x + 2) = 96 \Rightarrow 2x + 2 = 96 \Rightarrow 2x = 94 \Rightarrow x = 47
$$

So the integers are 47 and 49.

✔ Answer: 47 and 49

But since the problem says even, we must note the discrepancy.

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✔ Final Answers:

1) 32 and 33
2) 10, 11, 12
3) No solution exists for two consecutive even integers summing to 96.
*(Likely typo — intended to be odd integers: 47 and 49)*

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🟩 Recommendation:

If this is for a student, explain:
> "After solving, we find that the only integers satisfying the equation are 47 and 49, but these are odd. Since the problem asks for even integers, and no two consecutive even integers sum to 96, there is no solution. This may indicate a typo in the problem."

Or, if allowed, suggest the intended answer is 47 and 49 (if "even" was a mistake).

Let me know if you'd like to adjust the problem.