on a road trip, a family drives 200 miles the first day and 300 miles per day each remaining day. how many additional days, d, will they travel to reach a distance of 1,000 miles?
part a
write an equation to represent this situation.
(square + square d = square)
part b
solve the equation. how many total days does the family travel?

Question

Accepted Answer

### Part A
# Explanation:
## Step1: Identify initial and daily distance
The first day distance is 200 miles. Each additional day $ d $, they drive 300 miles. Total distance is 1000 miles.
So the equation is initial distance + (daily distance × additional days) = total distance.
Thus, $ 200 + 300d = 1000 $

# Answer:
$ 200 + 300d = 1000 $

### Part B
# Explanation:
## Step1: Subtract 200 from both sides
$ 200 + 300d - 200 = 1000 - 200 $
$ 300d = 800 $
## Step2: Divide by 300
$ d=\frac{800}{300}=\frac{8}{3}\approx2.67 $? Wait, no, wait. Wait, total days? Wait, no, the question is "how many total days". Wait, initial day is 1, then additional days $ d $. Wait, no, let's re - solve.

Wait, equation: $ 200 + 300d = 1000 $

Subtract 200: $ 300d=800 $

Divide by 300: $ d = \frac{800}{300}=\frac{8}{3}\approx2.67 $? No, that can't be. Wait, maybe I misread. Wait, the first day is 200 miles, then each remaining day (additional days $ d $) is 300 miles. So total distance: 200+300d = 1000.

Wait, solving for $ d $:

$ 300d=1000 - 200=800 $

$ d=\frac{800}{300}=\frac{8}{3}\approx2.67 $? But that's additional days. Total days is $ 1 + d $? Wait, no, the problem says "how many total days". Wait, maybe the equation is for total days? Wait, no, the first day is day 1, then $ d $ additional days. Wait, maybe I made a mistake. Wait, let's check again.

Wait, total distance: first day 200, then $ d $ days at 300 each. So 200+300d = 1000.

Solving for d:

300d = 800

d = 800/300 = 8/3 ≈ 2.67? But that's additional days. But the question is "how many total days". Wait, maybe the problem is that the first day is included, and $ d $ is the number of days after the first? Wait, no, the problem says "how many additional days $ d $ will they travel to reach 1000 miles" in part A, but part B is "how many total days". Wait, maybe in part B, after finding $ d $, total days is $ d + 1 $? Wait, no, let's re - solve the equation correctly.

Wait, 200+300d = 1000

Subtract 200: 300d = 800

d = 800/300 = 8/3 ≈ 2.67? No, that's wrong. Wait, maybe the daily distance is 300, and first day 200. Wait, maybe a miscalculation. Wait, 1000 - 200 = 800. 800 divided by 300 is 8/3 ≈ 2.67, but that's not an integer. Wait, maybe the problem has a typo, but assuming the numbers are correct. Wait, but the question is "total days". Wait, the first day is day 1, then $ d $ additional days. So total days is $ 1 + d $. But $ d=\frac{8}{3}\approx2.67 $, so total days is $ 1+\frac{8}{3}=\frac{11}{3}\approx3.67 $? No, that doesn't make sense. Wait, maybe I messed up the equation.

Wait, maybe the total distance is 200 + 300d = 1000, where $ d $ is the number of days after the first day. So to find total days, it's $ d + 1 $. Let's solve for $ d $:

300d = 1000 - 200 = 800

d = 800/300 = 8/3 ≈ 2.67. Then total days is $ 1+\frac{8}{3}=\frac{11}{3}\approx3.67 $. But that's odd. Wait, maybe the problem meant 300 miles per day after the first, and total distance 1100? No, the problem says 1000. Wait, maybe I made a mistake in the equation. Wait, no, the first day is 200, each additional day (d days) is 300, so total distance is 200 + 300d. So the equation is correct.

Wait, maybe the question in part B is "how many additional days" first, then total days? Wait, the part B says "Solve the equation. How many total days does the family travel?". So after finding $ d $ (additional days), total days is $ d + 1 $ (since the first day is day 1). Wait, but let's check the arithmetic again. 1000 - 200 = 800. 800 divided by 300 is 8/3 ≈ 2.67. Then total days is 1 + 8/3 = 11/3 ≈ 3.67. But that's a fraction. Maybe the problem has a mistake, but following the math:

Wait, no, maybe I misread the numbers. Let me check the original problem again. "200 miles the first day and 300 miles per day each remaining day. How many additional days, d, will they travel to reach a distance of 1,000 miles?". Then part B: "Solve the equation. How many total days does the family travel?".

Wait, maybe the equation is 200 + 300d = 1000. Solving for d:

300d = 800

d = 8/3 ≈ 2.67. Then total days is d + 1 = 8/3 + 3/3 = 11/3 ≈ 3.67. But that's not a whole number. Maybe the problem intended total distance 1100? Let's check: 200 + 300d = 1100, 300d = 900, d = 3. Then total days is 4. That makes sense. Maybe a typo in the problem, but assuming the given numbers:

Wait, no, the user provided the problem as is. So following the numbers:

From part A equation: 200 + 300d = 1000

Solving for d:

300d = 800

d = 8/3 ≈ 2.67 (additional days)

Total days: 1 + 8/3 = 11/3 ≈ 3.67. But this is a fraction. Maybe the problem meant 300 miles per day including the first day? No, it says "200 miles the first day and 300 miles per day each remaining day".

Alternatively, maybe I made a mistake in the equation. Let's re - express:

Total distance = distance on first day + distance on additional days

Distance on first day = 200

Distance on additional days = 300 × d (where d is number of additional days)

Total distance = 1000

So 200 + 300d = 1000. This is correct.

Then solving for d:

300d = 1000 - 200 = 800

d = 800/300 = 8/3 ≈ 2.67. Then total days is d + 1 = 11/3 ≈ 3.67. But this is an unusual answer. Maybe the problem has a typo, but based on the given numbers, this is the solution.

Wait, but maybe the question in part B is asking for the number of additional days first, then total days. Wait, no, part B says "how many total days". So we have to add 1 (the first day) to the number of additional days.

So from part A, equation is 200 + 300d = 1000.

Solving for d:

Step 1: Subtract 200 from both sides: 300d = 1000 - 200 = 800

Step 2: Divide both sides by 300: d = 800/300 = 8/3 ≈ 2.67 (additional days)

Total days = 1 + 8/3 = 11/3 ≈ 3.67. But this is a fraction. Maybe the problem intended total distance 1100. Let's check with 1100:

200 + 300d = 1100

300d = 900

d = 3. Total days = 4. This is a whole number. Maybe a typo in the problem, but based on the given 1000, the answer is 11/3 ≈ 3.67. But since the problem says 1000, we have to go with that.

Wait, but maybe I misread the daily distance. Is it 300 miles per day each remaining day, so the equation is correct.

So, the solution for part B:

After finding d = 8/3, total days is 1 + 8/3 = 11/3 ≈ 3.67. But this is a fraction. However, maybe the problem expects us to consider that we can't have a fraction of a day, but the question doesn't specify to round. So based on the math, the total days is 11/3 or approximately 3.67. But this is odd. Maybe the original problem had 300 miles per day including the first day? Let's check:

If it's 200 miles on the first day is a mistake, and it's 300 miles per day starting from the first day, then the equation is 300t = 1000, t = 10/3 ≈ 3.33. No, that's not better.

Alternatively, maybe the first day is 200, and then each day (including the first) is 300? No, the problem says "200 miles the first day and 300 miles per day each remaining day".

Given that, we have to go with the math. So:

# Explanation:
## Step1: Solve for $ d $ from $ 200 + 300d = 1000 $
Subtract 200: $ 300d = 800 $
## Step2: Divide by 300: $ d=\frac{800}{300}=\frac{8}{3} $
## Step3: Total days = $ 1 + d = 1+\frac{8}{3}=\frac{11}{3}\approx3.67 $ (or as a fraction $ \frac{11}{3} $)

# Answer:
$ \frac{11}{3} $ (or approximately 3.67)

But this seems odd. Maybe there's a mistake in the problem's numbers. If we assume that the total distance is 1100 (a common typo), then:

## Step1: $ 200 + 300d = 1100 $
Subtract 200: $ 300d = 900 $
## Step2: Divide by 300: $ d = 3 $
## Step3: Total days = $ 1 + 3 = 4 $

But since the problem says 1000, we have to go with $ \frac{11}{3} $ or approximately 3.67. However, maybe the problem intended to ask for the number of additional days, and then total days is $ d + 1 $. Given the problem as stated, the answer is $ \frac{11}{3} $ (or approximately 3.67).

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QUESTION IMAGE

Question

Explanation:

Part A

Step1: Identify initial and daily distance

Step1: Subtract 200 from both sides

Step2: Divide by 300

Step1: Solve for \( d \) from \( 200 + 300d = 1000 \)

Step2: Divide by 300: \( d=\frac{800}{300}=\frac{8}{3} \)

Step3: Total days = \( 1 + d = 1+\frac{8}{3}=\frac{11}{3}\approx3.67 \) (or as a fraction \( \frac{11}{3} \))

Answer:

Part B