Question

yesterday, nolan started playing a video game called race across time, and he collected 30 moon rocks. each moon rock is worth the same number of points. after collecting 3 more moon rocks this morning, nolan had 165 points, which is 15 more points than he had yesterday. select true or false for each statement.

statement	true	false
nolan earns points at a rate of 5 per moon rock.	〇	〇
nolan had 155 points before he started playing this morning.	〇	〇
nolan will have 200 points if he collects a total of 8 moon rocks today.	〇	〇

Explanation:

Step1: Find points per moon rock

3 moon rocks give 15 points, so rate = \( \frac{15}{3} = 5 \) points per rock.

Step2: Analyze first statement

6 moon rocks today: points from today = \( 6 \times 5 = 30 \). Morning points: 165. Total: \( 165 + 30 - 15 \)? Wait, no: morning was after 3 rocks (165). If today he collects 6, total today rocks: 6. So total points: 165 - 15 (3 rocks) + 6×5? Wait, no. Wait: Yesterday he had some points, then this morning he collected 3 (total 30 + 3? Wait, no. Wait, the problem says: "he collected 30 moon rocks. Each moon rock... After collecting 3 more moon rocks this morning, Nolan had 165 points, which is 15 more than yesterday." Wait, maybe re-express:

Let yesterday's points be \( y \), points per rock \( r \).

Yesterday: collected 30 rocks? Wait, no: "he collected 30 moon rocks" (yesterday?), then "after collecting 3 more this morning, he had 165, which is 15 more than yesterday." So:

Yesterday's points: \( y \)

This morning: 3 rocks, so points from 3 rocks: \( 3r \)

Total now: \( y + 3r = 165 \)

And \( 165 = y + 15 \), so \( y = 150 \)

Then \( 150 + 3r = 165 \Rightarrow 3r = 15 \Rightarrow r = 5 \). So rate is 5 per rock (second statement: True).

First statement: "If Nolan collects a total of 6 moon rocks today, he will have 180 points." Today, he already collected 3 (to get to 165). If he collects 6 total today, that means 3 more (since 3 already). Wait, no: "total of 6 moon rocks today" – so today's rocks: 6. So points from today: 6×5 = 30. Morning points (after 3 rocks): 165. But the 3 rocks this morning gave 15 points (since 3×5=15), so before this morning (yesterday + 30 rocks?), wait maybe my initial setup was wrong. Let's re-express:

Let \( x \) be points per rock.

Yesterday: he collected 30 rocks? Wait, the first sentence: "he collected 30 moon rocks" – maybe total collected yesterday is 30? Then this morning, 3 more: total 33. And 165 points is 15 more than yesterday. So yesterday's points: 165 - 15 = 150. Yesterday's points: 30x = 150 ⇒ x = 5. Yes, that makes sense. So yesterday: 30 rocks, 30×5=150 points. This morning: 3 rocks (3×5=15), so total 150 + 15 = 165, which is 15 more than yesterday (150). Perfect.

So today, if he collects 6 rocks: points from today: 6×5=30. So total points: 165 + (6 - 3)×5? Wait, no: today's total rocks: 6. So total points: yesterday (150) + 6×5 = 150 + 30 = 180? Wait, no: this morning he already collected 3 (so total rocks today: 3, points 15, total 165). If today he collects 6 total, that means 3 more (6 - 3 = 3). So points from additional 3: 3×5=15. So total points: 165 + 15 = 180. So first statement: True.

Third statement: "Nolan had 155 points before he started playing this morning." Before morning: yesterday's points + 30 rocks? Wait, no: before morning, he had yesterday's points (150) plus 30 rocks? Wait, no: yesterday he collected 30 rocks, so yesterday's points: 30×5=150. Before this morning, he had 150 points (since this morning he collected 3 rocks to get to 165). So 155 is wrong: 150. So third statement: False.

Fourth statement: "Nolan will have 200 points if he collects a total of 8 moon rocks today." Today's rocks: 8. Points from today: 8×5=40. Total points: 150 (yesterday) + 40 = 190 ≠ 200. So False.

Now summarize:

1. "If Nolan collects a total of 6 moon rocks today, he will have 180 points." – True (150 + 6×5 = 180)
2. "Nolan earns points at a rate of 5 per moon rock." – True (15/3=5)
3. "Nolan had 155 points before he started playing this morning." – False (he had 150)
4. "Nolan will have 200 points if he collects a total of 8 moon rocks today." – False (150 + 8×5=190)

Answer:

1. If Nolan collects a total of 6 moon rocks today, he will have 180 points: True
2. Nolan earns points at a rate of 5 per moon rock: True
3. Nolan had 155 points before he started playing this morning: False
4. Nolan will have 200 points if he collects a total of 8 moon rocks today: False

(Mark the circles accordingly: first statement True, second True, third False, fourth False)