2024 AMC 8 Problem 2

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To help you master 2024 AMC 8 Problem 2, 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

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2024 AMC8 Problem 2

Solution

Solution 1

Individual Simplification and Conversion Method

By simplifying each fraction on its own, the denominators become very small or perfectly suited for decimal conversion. We can then convert each to a decimal and add them together.

Step 1: Simplify and evaluate the first fraction

Divide the numerator by the denominator. Because the numerator is a direct multiple of the denominator, this fraction simplifies to an integer.

\frac{44}{11} = 4

Step 2: Simplify and evaluate the second fraction

Observe that the numerator and denominator share a greatest common divisor of 22. Factoring out 22 leaves a simple fraction, which converts easily to a decimal.

\frac{110}{44} = \frac{22 \times 5}{22 \times 2}

\frac{5}{2} = 2.5

Step 3: Simplify and evaluate the third fraction

Both the numerator and denominator are multiples of 11. Factoring out 11 leaves a fraction with a denominator of 100, which perfectly translates into a decimal.

\frac{44}{1100} = \frac{11 \times 4}{11 \times 100}

\frac{4}{100} = 0.04

Step 4: Add the decimal values together

Align the decimal points of the three evaluated terms and compute the final sum.

4 + 2.5 + 0.04 = 6.54

Correct answer: C. $6.54$
The final sum is

6.54

, which corresponds to answer choice C. We can verify this by checking the order of magnitude: adding the first two terms yields

6.5

, and adding a very small amount to it keeps the answer just above

6.5

, leaving

6.54

as the only reasonable multiple choice option.

Solution 2

Fraction Addition with a Convenient Common Denominator

After simplifying the fractions, we can give them a convenient common denominator of 100, add them together as a single fraction, and then convert the result to a decimal.

Step 1: Simplify the fractions

As seen before, we reduce each fraction to its simplest form or a form with a small denominator.

\frac{44}{11} = 4

\frac{110}{44} = \frac{5}{2}

\frac{44}{1100} = \frac{4}{100}

Step 2: Find a convenient common denominator

Since our goal is to convert the final sum to a decimal, a denominator of 100 is ideal. We scale the first two terms so they have a denominator of 100.

4 = \frac{400}{100}

\frac{5}{2} = \frac{5 \times 50}{2 \times 50} = \frac{250}{100}

Step 3: Add the fractions together

Now that all fractions share the same denominator, we simply add their numerators.

\frac{400}{100} + \frac{250}{100} + \frac{4}{100} = \frac{400 + 250 + 4}{100}

\frac{654}{100}

Step 4: Convert the final fraction to a decimal

Dividing by 100 corresponds to shifting the decimal point of the numerator two places to the left.

\frac{654}{100} = 6.54

Correct answer: C. $6.54$
The result is

6.54

, choice C. Calculating the sum entirely in fraction form avoids the risk of misaligning decimal places during addition, confirming our previous result.

Adapted Problems to Practice

1 / 10

What Formulas Do You Need for This Problem?

Fraction Simplification Strategy:

Always look for common factors to reduce fractions before performing operations like finding a common denominator.

Applies whenever numerators and denominators share common divisors. Reducing fractions first keeps numbers small, minimizes arithmetic errors, and frequently reveals immediate paths to decimal conversion.

Base-10 Denominators for Decimals:

Fractions with denominators of $10, 100, 1000$ , etc., can be written directly as decimals by placing the numerator's digits in the corresponding decimal places.

Requires a fraction in the form of $\frac{a}{1 0 ^{n}}$ . Scaling terms to fractions over $100$ allowed an immediate conversion to decimals without performing long division.

Key Takeaway: Remember This Strategy for Next Time

A sum of fractions where the denominators are large but share obvious factors with their numerators.

Try: Simplify each fraction individually before attempting to combine them.

Why: Finding a least common multiple for large denominators can create huge, error-prone numbers. Simplifying first often makes the problem collapse into basic mental math.

A multiple-choice question asking for a decimal sum.

Try: Evaluate the largest parts of the sum first, then use the answer choices to guide you.

Why: Computing just the largest components (like $4 + 2.5 = 6.5$ ) is frequently enough to confidently eliminate most or all of the incorrect answer choices.

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2024 AMC 8 Problem 2

In This Page:

2024 AMC8 Problem 2