Sensim Math Original · sm-2
SM Original Grade 5 arithmeticrate-ratioProblem
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.
Try it yourself first — the explanation is most useful after you’ve attempted it.
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
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
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
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.
💡 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.
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: $\tfrac{\$}{\text{year}} \times \text{year} = \$$.
💡 Multiplying a decimal by a whole number — and splitting the decimal into easier pieces — is exactly the decimal-arithmetic move from Grade 5.
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.
💡 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.
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.
💡 Rounding a decimal to the nearest whole number by looking at the tenths digit is the rounding-decimals skill from Grade 5.
4.NBT.B.4 Read the time horizon directly from the problem: the conservatory wants to bill 5.NBT.B.7 Now apply the annual increase rate to the time span. Multiply $\$148.75 \text{ p 5.NBT.B.7 Add the total increase to today's membership dues to get the dues 25 years from 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.4Fluently add and subtract multi-digit whole numbers (Identifying and labeling the given 25-year time span before multiplying by the rate.)5.NBT.B.7Add, 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.4Round 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!
⭐ This problem about music conservatory dues only needs Grade 5 decimal arithmetic and rounding you already know!
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