2024 AMC 8 Problem 1

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In This Page:

To help you master 2024 AMC 8 Problem 1, this page includes:

• The original AMC problem to try, plus the correct answer

• Clear, step-by-step solutions using different methods

• Key ideas, useful formulas, and takeaways you can apply to similar problems

• Related problems from past AMC exams, plus new practice problems that build the same skills

Want more practice? Scroll down to download Think Academy’s AMC 8 practice packs and keep building your AMC 8 skills.

2024 AMC8 Problem 1

Solution

Solution 1

Strategic Calculation Method

By performing the subtraction in a smart order, we can easily calculate the exact value of the entire expression and then simply look at its last digit.

Factoring out a negative sign: $-a-b-c=-(a+b+c)$. This allows us to convert a chain of multiple subtractions into a single addition problem followed by one subtraction.

Step 1: Simplify the first subtraction

Look for a convenient calculation to start with. The first two numbers line up perfectly, leaving us with a very clean number with many zeros.

$$222,222-22,222=200,000$$

Step 2: Group the remaining numbers

Instead of doing four separate subtractions, we can group the remaining numbers together, find their total, and subtract it all at once.

$$200,000-(2,222+222+22+2)$$

Step 3: Calculate the sum of the grouped numbers

Add the grouped numbers together. We can do this in short sequential steps to avoid mistakes.

$$2,222+222=2,444$$

$$2,444+22=2,466$$

$$2,466+2=2,468$$

Step 4: Perform the final subtraction

Subtract this total from our simplified starting amount to find the exact value of the original expression.

$$200,000-2,468=197,532$$

Step 5: Identify the units digit

Look at the last digit of our final calculated result.

$$197,532\text{ends in}2$$

Correct answer: B. $2$
The exact value of the expression is $197,532$, which has a units digit of $2$. This matches answer choice B. We can verify our arithmetic by adding the subtracted total back to our final answer: $197,532+2,468=200,000$, which confirms our main subtraction step was correct.

Solution 2

Units Digit Arithmetic Method

We do not need to calculate the entire number. We can find the units digit of the answer by only adding and subtracting the units digits of the original numbers.

Units Digit Principle: The units digit of a sum or difference only depends on the units digits of the numbers involved. We can safely ignore the tens, hundreds, and higher place values.

Step 1: Focus only on the units digits

Every number in this expression ends in the digit 2. To find the units digit of the final answer, we only need to perform the given operations on these last digits.

Step 2: Group the subtracted units digits

We start with a positive number ending in 2, and we are subtracting five smaller numbers that each end in 2. We can find the units digit of the total amount being subtracted by adding their units digits together.

$$2+2+2+2+2=10$$

Step 3: Determine the units digit of the subtraction block

Because the sum of the subtracted units digits is 10, the total sum of all five subtracted numbers must end with a units digit of 0.

$$10\text{ends in}0$$

Step 4: Calculate the final units digit

Now we subtract this from the units digit of our starting number. The starting number ends in 2, and the total amount we are subtracting ends in 0.

$$2-0=2$$

Correct answer: B. $2$
The final units digit is $2$, which matches answer choice B. We can verify this by tracking the units digit through step-by-step sequential subtraction: $2-2=0$, then $0-2$ (borrowing $10$) gives $8$, then $8-2=6$, $6-2=4$, and finally $4-2=2$.

Similar Problems in Real Exams

The following five AMC 8 problems are closely related to AMC 8 2024 Question 1. These real AMC 8 problems test the same key concept—Arithmetic. Click any problem to learn and practice, then continue to the adapted practice section later in this article for more problems to try.

2022 AMC 8 Problem 17

2012 AMC 8 Problem 12

2000 AMC 8 Problem 14

2026 AMC 8 Problem 1

2026 AMC 8 Problem 2

Adapted Problems to Practice

What Formulas Do You Need for This Problem?

• Units Digit Principle:

Applies to any chain of addition, subtraction, or multiplication operations. In our base-10 number system, place values of 10 or greater (tens, hundreds, etc.) do not alter the ones column.

• Factoring Out a Negative Sign:

Used when subtracting multiple positive terms in a row. Grouping multiple subtractions into a single addition problem simplifies calculation and heavily reduces the chances of making a borrowing error.

Key Takeaway: Remember This Strategy for Next Time

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