Solve the Fives Challenge by using exactly four fives and arithmetic operations to reach the target numbers.

Let's solve the "Fives Challenge" step by step.

We are given five equations, each using exactly four 5s and any combination of arithmetic operations:
+ (plus), − (minus), × (times), ÷ (divide), and parentheses/brackets for grouping.
Our goal is to make each expression equal the target number on the right.

---

1. 5 5 5 5 = 3

We need to make 3 using four 5s.

Let’s think:
- We can use division to get smaller numbers.
- Try: $ \frac{5 + 5 + 5}{5} = \frac{15}{5} = 3 $

✔ That works!

Answer:
$$
\frac{5 + 5 + 5}{5} = 3
$$

---

2. 5 5 5 5 = 5

We want to get 5.

One idea: Use operations that cancel out or balance.

Try:
$ 5 + 5 - 5 - 5 = 0 $ → no
$ 5 \times (5 - 5) + 5 = 5 $ → $ 5 \times 0 + 5 = 5 $

✔ Yes!

Answer:
$$
5 \times (5 - 5) + 5 = 5
$$

Alternatively:
$ (5 + 5 + 5 + 5)/4 = 5 $, but we can't use 4 — only 5s.

So stick with:
$$
5 \times (5 - 5) + 5 = 5
$$

---

3. 5 5 5 5 = 6

Need to get 6.

Try using fractions or combinations.

Idea: $ 5 + \frac{5 + 5}{5} = 5 + \frac{10}{5} = 5 + 2 = 7 $ → too big.

Try: $ \frac{5 \times 5 - 5}{5} = \frac{25 - 5}{5} = \frac{20}{5} = 4 $ → too small.

Try: $ 5 + \frac{5}{5} + \frac{5}{5} = 5 + 1 + 1 = 7 $ → again too big.

Wait: What if we do:

$ \frac{5 + 5}{5} + 5 = 2 + 5 = 7 $

Still not 6.

Try: $ 5 + \frac{5 + 5}{5} = 5 + 2 = 7 $ → same.

Wait: How about:
$ (5 + 5 + 5) / 5 = 15/5 = 3 $ → no.

Try this:

What if we do:
$ 5 + \frac{5}{5} + \frac{5}{5} = 5 + 1 + 1 = 7 $ → too much.

Wait — maybe subtraction?

Try: $ (5 \times 5 - 5) / 5 = (25 - 5)/5 = 20/5 = 4 $

Still not 6.

Alternative idea: Use decimal? But we can’t.

Wait — here’s a known trick:

How about:
$$
(5 + 5) / 5 + 5 = 2 + 5 = 7
$$

No.

Wait — what if we do:

$$
5 + \left( \frac{5 + 5}{5} \right) = 5 + 2 = 7
$$

Still 7.

Hmm.

Wait — let’s reverse: Can we make 6 from 5s?

Try:
$$
5 + \frac{5}{5} = 6 \quad \text{but that's only three 5s}
$$

We need four 5s.

So we must use all four.

Idea: Add zero.

$ 5 + \frac{5}{5} + (5 - 5) = 5 + 1 + 0 = 6 $

But that uses five 5s: 5, 5, 5, 5, 5 → no.

Wait: Let's count:

- First 5
- Then 5 in numerator
- Then 5 in denominator
- Then 5 and 5 in (5 - 5)

That’s five 5s.

Too many.

Can we do:
$$
5 + \frac{5}{5} + 0 \quad \text{and make 0 from two 5s?}
$$

Yes: $ 5 - 5 = 0 $

But then total 5s: 5 (first), 5 (num), 5 (denom), 5 (sub), 5 (sub) → five 5s.

No — we need exactly four.

So how about:

Use:
$$
(5 \times 5 - 5) / 5 = 20/5 = 4
$$

No.

Wait — another idea:

What if we do:
$$
\frac{5 \times 5 + 5}{5} = \frac{25 + 5}{5} = 30/5 = 6
$$

✔ Yes! That’s four 5s.

Answer:
$$
\frac{5 \times 5 + 5}{5} = 6
$$

Breakdown:
- 5 × 5 = 25
- 25 + 5 = 30
- 30 ÷ 5 = 6

Perfect.

---

4. 5 5 5 5 = 26

Target: 26

We know 5 × 5 = 25, which is close.

So: 5 × 5 = 25, then add 1 → 26

Can we make 1 from two 5s? Yes: $ \frac{5}{5} = 1 $

So:
$$
5 \times 5 + \frac{5}{5} = 25 + 1 = 26
$$

✔ Uses four 5s.

Answer:
$$
5 \times 5 + \frac{5}{5} = 26
$$

---

5. 5 5 5 5 = 30

Target: 30

We know: 5 × 5 = 25, 5 × 6 = 30

Or: 5 × (5 + 1) = 30 → need 1

Again: $ \frac{5}{5} = 1 $

So:
$$
5 \times (5 + \frac{5}{5}) = 5 \times (5 + 1) = 5 \times 6 = 30
$$

✔ Uses four 5s.

Answer:
$$
5 \times \left(5 + \frac{5}{5}\right) = 30
$$

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✔ Final Answers:

1. $ \frac{5 + 5 + 5}{5} = 3 $
2. $ 5 \times (5 - 5) + 5 = 5 $
3. $ \frac{5 \times 5 + 5}{5} = 6 $
4. $ 5 \times 5 + \frac{5}{5} = 26 $
5. $ 5 \times \left(5 + \frac{5}{5}\right) = 30 $

All use exactly four 5s and valid operations.

Let me know if you'd like them written more neatly!