Sensim Math Original · sm-2

SM Original Grade 5 arithmeticrate-ratio

Inspired by AMC 8 2024 #10

Archetype Arithmetic Expression Decomposition

ratemulti-digit-arithmeticfraction-decimal-conversion dimensional-analysis ↑ Prerequisites: multi-digit-arithmeticfraction-decimal-conversion

📏 Medium solution 💡 3 insights

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Problem

The finance office at the Lakeside Conservatory of Music has been tracking how its annual membership dues drift upward over time. Long-term data show a steady linear trend: each year, dues climb by exactly $\$148.75$ more than the year before, with no compounding. If a new member joining today pays $\$6{,}400$ for the first year, what should the conservatory bill a member $25$ years from now, rounded to the nearest whole dollar?

Pick an answer.

(A)

9919

(B)

10019

(C)

10119

(D)

10219

(E)

10319

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Toolkit + CCSS Solution

Understand

Restated: The Lakeside Conservatory of Music charges $\$6{,}400$ in annual membership dues today, and dues rise by a fixed $\$148.75$ every year with no compounding. We need to find what the conservatory should bill a member $25$ years from now, rounded to the nearest whole dollar.

Givens: Starting point: today's annual membership dues = $\$6{,}400$; Annual increase (rate): $\$148.75$ per year; Time horizon: $25$ years from now; Five answer choices: (A) 9919, (B) 10019, (C) 10119, (D) 10219, (E) 10319

Unknowns: The expected membership dues 25 years from now (in whole dollars, rounded to the nearest dollar)

Understand

Plan

Primary tool: #8 Analyze the Units

Secondary: #9 Solve an Easier Related Problem, #3 Eliminate Possibilities

This is a rate problem: (dollars per year) $\times$ (years) $=$ dollars of increase, which then adds to the starting membership dues. Tool #8 (Analyze Units) keeps the recipe honest — "years $\times$ (\$/year) = \$," and "\$ + \$ = \$." The only tricky arithmetic is $148.75 \times 25$, where Tool #9 (Easier Related Problem) lets us replace it with a friendlier breakdown using $148.75 = 148 + 0.75$ (or equivalently, $148.75 \times 25 = 148.75 \times 100 / 4$). Finally, since the question is multiple-choice, Tool #3 (Eliminate Possibilities) confirms the answer against the five choices.

Execute — Answer: C

#8 Analyze the Units 4.NBT.B.4 Step 1

Read the time horizon directly from the problem: the conservatory wants to bill a member $25$ years from now.
The problem states this span explicitly, so no calculation is needed here.
Units: the number $25$ carries the unit 'years,' which is exactly what we need to pair with the rate in Step 2.

$$t = 25 \text{ years}$$

💡 Identifying the given time span and labeling its unit before multiplying is the first move in any rate problem — it keeps the unit bookkeeping clean from the start.

#9 Solve an Easier Related Problem 5.NBT.B.7 Step 2

Now apply the annual increase rate to the time span.
Multiply $\$148.75 \text{ per year} \times 25 \text{ years}$ to find the total rise in membership dues over 25 years. To make the arithmetic friendlier, solve the easier related problem first: $148.75 \times 100 = 14{,}875$, then divide by $4$ (because $25 = 100 \div 4$). We get $14{,}875 \div 4 = 3{,}718.75$. Alternatively, break it up: $148 \times 25 = 3{,}700$ and $0.75 \times 25 = 18.75$, so the total is $3{,}700 + 18.75 = 3{,}718.75$. The units cancel correctly: $\tfrac{\$}{\text{year}} \times \text{year} = \$$.

$$148.75 \times 25 = (148 \times 25) + (0.75 \times 25) = 3{,}700 + 18.75 = 3{,}718.75 \text{ dollars}$$

💡 Multiplying a decimal by a whole number — and splitting the decimal into easier pieces — is exactly the decimal-arithmetic move from Grade 5.

#8 Analyze the Units 5.NBT.B.7 Step 3

Add the total increase to today's membership dues to get the dues 25 years from now.
Both quantities are in dollars, so the units agree: dollars $+$ dollars $=$ dollars.
The expected dues 25 years from now are $6{,}400 + 3{,}718.75 = 10{,}118.75$ dollars.

$$6{,}400 + 3{,}718.75 = 10{,}118.75 \text{ dollars}$$

💡 Adding today's dues (a whole number) to the accumulated increase (a decimal) by lining up the decimal point is a standard Grade 5 decimal-addition skill.

#3 Eliminate Possibilities 5.NBT.A.4 Step 4

The problem asks for the answer rounded to the nearest whole dollar.
The tenths digit of $10{,}118.75$ is $7$ (which is $\geq 5$), so we round up: $10{,}118.75 \approx 10{,}119$.
Cross-check against the choices: $10{,}119$ matches (C).
Choice (A) $9919$ would mean the membership dues rose less than $\$3{,}600$ over 25 years; (B) $10{,}019$ is $\$100$ too low; (D) $10{,}219$ and (E) $10{,}319$ are too high by $\$100$ and $\$200$ — only (C) is consistent with the calculation.

$$10{,}118.75 \approx 10{,}119 \;\Rightarrow\; \textbf{(C)}$$

💡 Rounding a decimal to the nearest whole number by looking at the tenths digit is the rounding-decimals skill from Grade 5.

[1] #8 4.NBT.B.4 Read the time horizon directly from the problem: the conservatory wants to bill

[2] #9 5.NBT.B.7 Now apply the annual increase rate to the time span. Multiply $\$148.75 \text{ p

[3] #8 5.NBT.B.7 Add the total increase to today's membership dues to get the dues 25 years from

[4] #3 5.NBT.A.4 The problem asks for the answer rounded to the nearest whole dollar. The tenths

Review

Reasonableness: Sanity check the size: over 25 years, membership dues go up by about $\$150$ per year $\times 25 = \$3{,}750$, so dues 25 years from now should be roughly $\$6{,}400 + \$3{,}750 \approx \$10{,}150$. Our answer $\$10{,}119$ sits right inside that ballpark, just a tiny bit lower because the actual rate $\$148.75$ is a hair below $\$150$. Units are right (dollars), and the answer sits cleanly between the too-low (A) $9919$ / (B) $10019$ and the too-high (D) $10219$ / (E) $10319$ options.

Alternative: An alternative is Tool #13 (Convert to Algebra): write the membership dues as $D(t) = 6400 + 148.75 t$, where $t$ is the number of years from today. Plug in $t = 25$: $D(25) = 6400 + 3718.75 = 10118.75 \approx 10119$. Same answer, but for an elementary student the rate-and-units approach is more intuitive and easier to check.

CCSS standards used (min grade 5)

4.NBT.B.4 Fluently add and subtract multi-digit whole numbers (Identifying and labeling the given 25-year time span before multiplying by the rate.)
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths (Multiplying the decimal rate $148.75 \times 25$ and adding $6400 + 3718.75$ to combine the initial membership dues with the total increase.)
5.NBT.A.4 Round decimals to any place (Rounding the final decimal result $10{,}118.75$ to the nearest whole dollar to match an answer choice.)

⭐ This problem about music conservatory dues only needs Grade 5 decimal arithmetic and rounding you already know!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Review

CCSS standards used (min grade 5)

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