Solved Lesson 4 Homework Practice Proportional and | Chegg.com - Free Printable

Let’s solve each problem one by one, explaining the concept of proportional relationships as we go.

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🔹 Problem 1: Sailfish Swimming

> The world’s fastest fish, a sailfish, swims at a rate of 69 miles per hour. Is the distance a sailfish swims proportional to the number of hours it swims?

✔ Answer: YES

#### 💡 Explanation:

A relationship is proportional if two quantities vary in such a way that their ratio is always constant.

Here:
- Distance = Rate × Time → `d = 69 × t`
- So, for every hour (`t`), the distance (`d`) increases by exactly 69 miles.
- The ratio `distance / time = 69` (constant).

This is a classic example of a directly proportional relationship:
> As time increases, distance increases at a constant rate.

✔️ Therefore, yes, the distance is proportional to the number of hours.

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🔹 Problem 2: Fossils in August

> In July, a paleontologist found 368 fossils at a dig. In August, she found about 14 fossils per day. Is the number of fossils the paleontologist found in August proportional to the number of days she spent looking for fossils that month?

✔ Answer: YES

#### 💡 Explanation:

In August, she found 14 fossils per day — that’s a constant rate.

So:
- Number of fossils found in August = 14 × number of days
- Ratio: `fossils / days = 14` (constant)

This means the more days she worked in August, the more fossils she found — and always at the same rate.

✔️ So, yes, the number of fossils found in August is proportional to the number of days she spent digging in August.

*(Note: The July data is irrelevant here — the question is only about August.)*

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🔹 Problem 3: Total Fossils in July and August vs. Days in August

> Is the total number of fossils found during July and August proportional to the number of days the paleontologist spent looking for fossils in August?

✘ Answer: NO

#### 💡 Explanation:

Let’s break this down.

- Total fossils = Fossils in July + Fossils in August
= 368 + (14 × days in August)

Let’s call the number of days in August = `d`

Then:
> Total fossils = 368 + 14d

Now, is this proportional to `d`?

For proportionality, the relationship must be of the form:
> Total fossils = k × d (where k is a constant)

But here, we have:
> Total fossils = 14d + 368

This is a linear relationship, but not proportional, because it has a non-zero y-intercept (the "+368").

✔ Proportional relationships must pass through the origin (0,0). That is, when `d = 0`, total fossils should be 0.

But if she spent 0 days in August, she still found 368 fossils in July → so total fossils = 368 ≠ 0.

Therefore, the ratio `(total fossils) / (days in August)` is not constant — it changes depending on `d`.

Example:
- If d = 1 → total = 368 + 14 = 382 → ratio = 382/1 = 382
- If d = 2 → total = 368 + 28 = 396 → ratio = 396/2 = 198
- Not the same → not proportional

✔️ So, no, the total number of fossils found in July and August is not proportional to the number of days spent in August.

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## ✔ Final Answers:

1. Yes — Distance is proportional to hours (constant rate of 69 mph).
2. Yes — Fossils found in August are proportional to days in August (14 per day).
3. No — Total fossils (July + August) are NOT proportional to days in August (because of the fixed 368 from July).

Let me know if you’d like a graph or table to visualize these! 📊