AMC 8 · 2024 · #2

Grade 5 algebra

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

fraction-arithmeticfraction-decimal-conversionmulti-digit-arithmetic identify-subproblems ↑ Prerequisites: fraction-arithmeticmulti-digit-arithmetic

📏 Long solution 💡 2 insights

📘 View easy version →

Problem

We need the value of

\frac{44}{11}

\frac{110}{44}

\frac{44}{1100}

expressed as a decimal, and we have to match it to one of the five answer choices.

Pick an answer.

(A)

6.4

(B)

6.504

(C)

6.54

(D)

6.9

(E)

6.94

AMC 8 2024 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

Adding the three fractions with a common denominator at once would give a messy denominator (e.g. 1100) and ugly arithmetic. Instead we split the sum into three independent subproblems (Tool #7): convert each fraction to a decimal by itself, then add. Each subproblem is small enough for an elementary student. Since the problem is multiple-choice, we finish by comparing to the answer choices (Tool #3) as a verification.

1STEP 1

First term: since 44 = 11 × 4, the fraction $\frac{44}{11}$ divides evenly and equals 4.

\frac{44}{11}

= 44 ÷ 11 = 4

2STEP 2

Second term: divide $\frac{110}{44}$ by 11 to get $\frac{10}{4}$ , then by 2 to reach the simpler equivalent $\frac{5}{2}$ .

\frac{110}{44}

\frac{110÷ 11}{44÷ 11}

\frac{10}{4}

\frac{5}{2}

3STEP 3

A fraction means numerator divided by denominator, so the simplified $\frac{5}{2}$ = 5 ÷ 2 = 2.5.

\frac{5}{2}

= 5 ÷ 2 = 2.5

4STEP 4

Third term: $\frac{44}{1100}$ divided by 11 is $\frac{4}{100}$ , and a denominator of 100 means two decimal places, so 0.04.

\frac{44}{1100}

\frac{44÷ 11}{1100÷ 11}

\frac{4}{100}

= 0.04

5STEP 5

Line up the decimal points and add the three pieces: 4 + 2.5 + 0.04 = 6.54.

4.00 + 2.50 + 0.04 = 6.54

6STEP 6

Compare with the choices: our value matches (C), while the others miss the hundredths digit or add a stray one.

6.54 → (C)

Answer

6.54

Quick magnitude check: the first term is 4, the second is between 2 and 3, and the third is a tiny number close to 0, so the sum should land near 4 + 2.5 + 0 ≈ 6.5. The value 6.54 sits exactly where we expect, with the small extra 0.04 from the third term accounted for. Choices like 6.4 and 6.9 ignore the hundredths digit, and 6.504 has a thousandths digit that our calculation cannot produce.

💡Key takeaway

This AMC 8 problem only needs Grade 5 fraction-to-decimal conversion and decimal addition you already know!

More like this