Estimation

First, estimate the numerator: $19.87\times 403.1$

• Round $19.87$ to $20$.

• Round $403.1$ to $400$.

• Numerator estimate: $20\times 400=8000$.

Next, estimate the denominator: $0.24\times 101$

• Round $0.24$. It is exactly $\frac{6}{25}$ or $0.24$. $0.24$ is closer to $0.25$ ($\frac{1}{4}$) than it is to $0.2$ ($\frac{1}{5}$). So, round $0.24$ to $0.25$.

• Round $101$ to $100$.

• Denominator estimate: $0.25\times 100=25$.

Now, perform the division with the estimated values:

$\frac{8000}{25}$

To simplify $\frac{8000}{25}$, we can multiply both numerator and denominator by $4$ to get $\frac{32000}{100}=320$.

The best estimate is $320$. Among the given choices, $330$ is the closest value.

First term: $39.8\%\text{ of }1201$

• Round $39.8\%$ to $40\%$. $40\%$ is equivalent to the fraction $\frac{2}{5}$.

• Round $1201$ to $1200$.

• Calculate: $40\%\text{ of }1200=0.40\times 1200=480$.

(Alternatively: $(\frac{2}{5})\times 1200=2\times 240=480$).

Second term: $14.9\%\text{ of }280.3$

• Round $14.9\%$ to $15\%$. $15\%$ is equivalent to the fraction $\frac{3}{20}$.

• Round $280.3$ to $280$.

• Calculate: $15\%\text{ of }280=0.15\times 280=42$.

(Alternatively: $(\frac{3}{20})\times 280=3\times 14=42$).

Now, add the approximate values of the two terms:

$480+42=522$.

The approximate value is $522$. Among the given choices, $520$ is the closest value.

• Reservoir Capacity: $1,100,000,000$ gallons

• Daily Consumption: $31,500,000$ gallons

Set up the division: $1,100,000,\frac{000}{31},500,000$

First, cancel out the common zeros. There are 6 zeros in $1,100,000,000$ and 5 zeros in $31,500,000$. So, we can cancel 5 zeros from both numbers:

$11,\frac{000}{31}.5$

Now, we need to approximate $11,\frac{000}{31}.5$.

Round $31.5$ to a convenient number. $31.5$ is exactly in the middle of $31$ and $32$. However, for estimation, rounding to $32$ makes the calculation $11,\frac{000}{32}$ easier to evaluate for accuracy than $11,\frac{000}{30}$ which might introduce more error since $31.5$ is closer to $32$ than $30$ (difference of $0.5$ vs $1.5$).

Let's use $32$ as the divisor:

$11,\frac{000}{32}$

We can simplify this by dividing by $2$ multiple times:

$11,\frac{000}{32}=5,\frac{500}{16}=2,\frac{750}{8}=1,\frac{375}{4}$

Now perform the division: $\frac{1375}{4}$

$1375÷4=343.75$.

Wait, there was a mistake in the zero cancellation. Let's re-do.

$1,100,000,\frac{000}{31},500,000$

Cancel 5 zeros from both:

$11,\frac{000}{315}$

Now approximate $11,\frac{000}{315}$.

Round $315$ to $300$ or $320$.

• If we round $315$ to $300$:

$11,\frac{000}{300}=\frac{110}{3}\approx 36.67$.

• If we round $315$ to $320$ (since $315$ is closer to $320$ than $300$):

$11,\frac{000}{320}=\frac{1100}{32}$

$\frac{1100}{32}=\frac{550}{16}=\frac{275}{8}=34.375$.

The estimate $34.375$ is closer to $35$ days than $30$ or $40$ days.

The estimate $36.67$ is also closest to $35$ days.

Both reasonable rounding approaches point to approximately $35$ days. The closest option is C. 35 days.