Fraction word problems about a bicycle race, asking students to calculate miles completed using fraction bars.

Let's solve each problem step by step using fraction bars (also known as bar models) and write an equation for each. The total race is 120 miles long.

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Problem 1:

> At the end of Day 1, Nora has completed $ \frac{1}{2} $ of the race.
> How many miles has she gone?

#### Step 1: Understand the problem
We need to find $ \frac{1}{2} $ of 120 miles.

#### Step 2: Use a fraction bar model

Imagine a bar representing the entire race:
```
[--------------------------] ← 120 miles
```

Divide it into 2 equal parts:
```
[--------][--------] ← Each part = 60 miles
```

So, $ \frac{1}{2} $ of the race = 60 miles.

#### Equation:
$$
\frac{1}{2} \times 120 = 60
$$

✔ Answer: Nora has gone 60 miles.

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Problem 2:

> At the end of Day 1, Stuart has completed $ \frac{1}{4} $ of the race.
> How many miles has he gone?

#### Step 1: Use a fraction bar model

Bar for the full race:
```
[--------------------------] ← 120 miles
```

Divide into 4 equal parts:
```
[---][---][---][---] ← Each part = 30 miles
```

So, $ \frac{1}{4} $ of the race = 30 miles.

#### Equation:
$$
\frac{1}{4} \times 120 = 30
$$

✔ Answer: Stuart has gone 30 miles.

---

Problem 3:

> At the end of Day 2, Nora has completed $ \frac{5}{6} $ of the race.
> How many miles has she gone?

#### Step 1: Use a fraction bar model

Bar for the full race:
```
[--------------------------] ← 120 miles
```

Divide into 6 equal parts:
```
[----][----][----][----][----][----] ← Each part = 20 miles
```

Now take 5 of those parts:
```
[----][----][----][----][----] ← 5 × 20 = 100 miles
```

So, $ \frac{5}{6} $ of the race = 100 miles.

#### Equation:
$$
\frac{5}{6} \times 120 = 100
$$

✔ Answer: Nora has gone 100 miles.

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✔ Final Answers:

1. 60 miles
Equation: $ \frac{1}{2} \times 120 = 60 $

2. 30 miles
Equation: $ \frac{1}{4} \times 120 = 30 $

3. 100 miles
Equation: $ \frac{5}{6} \times 120 = 100 $

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You can draw these fraction bars in your notebook as shown above to visually represent each solution.