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

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Problem

What is the value of this expression in decimal form?
$\frac{44}{11} + \frac{110}{44} + \frac{44}{1100}$

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.

Try it yourself first — the explanation is most useful after you’ve attempted it.

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Toolkit + CCSS Solution

Understand

Restated: 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.

Givens: A sum of three fractions: $\frac{44}{11} + \frac{110}{44} + \frac{44}{1100}$; Every fraction is built from the numbers 11, 44, 110, and 1100, so 11 (and powers of 10) cancel cleanly; Five answer choices: (A) 6.4, (B) 6.504, (C) 6.54, (D) 6.9, (E) 6.94

Unknowns: The decimal value of the entire sum

Understand

Restated: 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.

Plan

Primary tool: #7 Identify Subproblems

Secondary: #3 Eliminate Possibilities

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.

Execute — Answer: C

#7 Identify Subproblems 3.OA.C.7 Step 1

Handle the first term $\frac{44}{11}$.
Because $44 = 11 \times 4$, the division comes out exactly and the fraction equals the whole number 4.

$$\dfrac{44}{11} = 44 \div 11 = 4$$

💡 Fluency with multiplication and division within 100 is a Grade 3 standard, so $44 \div 11$ is immediate.

#7 Identify Subproblems 4.NF.A.1 Step 2

Handle the second term $\frac{110}{44}$.
Dividing numerator and denominator both by 11 gives $\frac{10}{4}$, and dividing by 2 again gives $\frac{5}{2}$.
This is just rewriting the fraction in an equivalent, simpler form.

$$\dfrac{110}{44} = \dfrac{110\div 11}{44\div 11} = \dfrac{10}{4} = \dfrac{5}{2}$$

💡 Dividing the numerator and denominator by a common factor to get an equivalent fraction is taught in Grade 4.

#7 Identify Subproblems 5.NF.B.3 Step 3

Now convert the simplified $\frac{5}{2}$ to a decimal.
A fraction means "numerator divided by denominator," so $5 \div 2 = 2.5$.

$$\dfrac{5}{2} = 5 \div 2 = 2.5$$

💡 Interpreting a fraction as numerator-divided-by-denominator and writing the result as a decimal is the heart of Grade 5 fractions.

#7 Identify Subproblems 4.NF.C.6 Step 4

Handle the third term $\frac{44}{1100}$.
Dividing top and bottom by 11 gives $\frac{4}{100}$.
A fraction with denominator 100 translates directly into two decimal places, so the value is $0.04$.

$$\dfrac{44}{1100} = \dfrac{44\div 11}{1100\div 11} = \dfrac{4}{100} = 0.04$$

💡 Rewriting a fraction with denominator 10 or 100 in decimal notation is exactly the Grade 4 fraction-to-decimal standard.

#7 Identify Subproblems 5.NBT.B.7 Step 5

Finally, add the three decimals, lining up the decimal points.
The ones column gives $4+2=6$, the tenths column gives $0+5=5$, and the hundredths column gives $0+0+4=4$, so the sum is $6.54$.

$$4.00 + 2.50 + 0.04 = 6.54$$

💡 Adding decimals to hundredths is a Grade 5 standard; once the place values are aligned it behaves just like whole-number addition.

#3 Eliminate Possibilities 5.NBT.A.3 Step 6

Cross-check against the answer choices.
The candidates are 6.4, 6.504, 6.54, 6.9, and 6.94.
Our computed value $6.54$ matches choice **(C)** exactly.
Choices 6.4 and 6.9 are missing the hundredths digit 4, 6.504 has a thousandths digit that our calculation never produces, and 6.94 has the wrong tenths digit.

$$6.54 \;\Rightarrow\; \textbf{(C)}$$

💡 Reading and comparing decimals to thousandths place-by-place is the Grade 5 standard that makes the elimination safe.

[1] #7 3.OA.C.7 Handle the first term $\frac{44}{11}$. Because $44 = 11 \times 4$, the division

[2] #7 4.NF.A.1 Handle the second term $\frac{110}{44}$. Dividing numerator and denominator both

[3] #7 5.NF.B.3 Now convert the simplified $\frac{5}{2}$ to a decimal. A fraction means "numerat

[4] #7 4.NF.C.6 Handle the third term $\frac{44}{1100}$. Dividing top and bottom by 11 gives $\f

[5] #7 5.NBT.B.7 Finally, add the three decimals, lining up the decimal points. The ones column g

[6] #3 5.NBT.A.3 Cross-check against the answer choices. The candidates are 6.4, 6.504, 6.54, 6.9

Review

Reasonableness: 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 $4 + 2.5 + 0 \approx 6.5$. 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.

Alternative: An alternative is Tool #13 (Convert to Algebra): give all three fractions a common denominator of 1100, add the numerators, then convert. $\frac{4400}{1100} + \frac{2750}{1100} + \frac{44}{1100} = \frac{7194}{1100} = 6.54$. Same answer, but the arithmetic is heavier, so splitting term-by-term with Tool #7 is friendlier.

CCSS standards used (min grade 5)

3.OA.C.7 Fluently multiply and divide within 100 (Computing $44 \div 11 = 4$ in one step.)
4.NF.A.1 Explain why a fraction is equivalent to another fraction (Reducing $\frac{110}{44}$ to the equivalent fraction $\frac{5}{2}$ by dividing numerator and denominator by common factors.)
4.NF.C.6 Use decimal notation for fractions with denominators 10 or 100 (Rewriting $\frac{4}{100}$ as the decimal $0.04$.)
5.NF.B.3 Interpret a fraction as division of the numerator by the denominator (Converting $\frac{5}{2}$ to the decimal $2.5$ via $5 \div 2$.)
5.NBT.A.3 Read, write, and compare decimals to thousandths (Comparing the computed value to the five decimal answer choices $6.4$, $6.504$, $6.54$, $6.9$, $6.94$ digit by digit.)
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths (Performing the final aligned-decimal addition $4 + 2.5 + 0.04 = 6.54$.)

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

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Review

CCSS standards used (min grade 5)

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