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Answer :
- Convert mixed numbers to improper fractions: $3\frac{3}{8} = \frac{27}{8}$ and $10\frac{13}{15} = \frac{163}{15}$.
- Round the improper fractions to simpler values: $\frac{27}{8} \approx 3.5$ and $\frac{163}{15} \approx 11$.
- Multiply the rounded values to estimate the product: $3.5 \times 11 = 38.5$.
- The estimated value of $3 \frac{3}{8} \times 10 \frac{13}{15}$ is approximately $\boxed{37}$.
### Explanation
1. Problem Analysis
We are asked to estimate the product of two mixed numbers: $3 \frac{3}{8}$ and $10 \frac{13}{15}$. To estimate, we can round each mixed number to the nearest whole number or a simple fraction.
2. Converting to Improper Fractions
First, let's convert the mixed numbers to improper fractions. $3 \frac{3}{8} = \frac{3 \times 8 + 3}{8} = \frac{24 + 3}{8} = \frac{27}{8}$. And $10 \frac{13}{15} = \frac{10 \times 15 + 13}{15} = \frac{150 + 13}{15} = \frac{163}{15}$.
3. Rounding the Fractions
Now, let's round each improper fraction. $\frac{27}{8}$ is approximately equal to $3.375$. We can round this to $3.5$ or $3 \frac{1}{2}$ or even just $3$. $\frac{163}{15}$ is approximately equal to $10.8666...$. We can round this to $11$.
4. Estimating the Product
Let's use the rounded values to estimate the product. If we round $3\frac{3}{8}$ to $3.5$ and $10\frac{13}{15}$ to $11$, then the estimated product is $3.5 \times 11 = 38.5$. If we round $3\frac{3}{8}$ to $3$ and $10\frac{13}{15}$ to $11$, then the estimated product is $3 \times 11 = 33$.
5. Alternative Estimation
Another approach is to round $3\frac{3}{8}$ to $3\frac{1}{2} = \frac{7}{2}$ and $10\frac{13}{15}$ to $11$. Then the estimated product is $\frac{7}{2} \times 11 = \frac{77}{2} = 38.5$.
6. Alternative Estimation 2
We can also round $3\frac{3}{8}$ to $3$ and $10\frac{13}{15}$ to $10\frac{7}{15} + \frac{6}{15} = 10\frac{1}{2}$. Then the estimated product is $3 \times 10.5 = 31.5$.
7. Comparing with Exact Value
Let's consider rounding $3\frac{3}{8}$ to $3\frac{1}{2}$ and $10\frac{13}{15}$ to $11$. Then the estimated product is $3.5 \times 11 = 38.5$. The exact value is $3\frac{3}{8} \times 10\frac{13}{15} = \frac{27}{8} \times \frac{163}{15} = \frac{4401}{120} = 36.675$. So, $38.5$ is a reasonable estimate.
8. Final Answer
Therefore, a good estimate for $3 \frac{3}{8} \times 10 \frac{13}{15}$ is approximately $37$.
### Examples
Estimating calculations is useful in everyday situations such as grocery shopping. For example, if you want to buy 3 and 3/8 pounds of apples at $10 and 13/15 per pound, estimating the total cost helps you quickly determine if you have enough money without needing exact calculations. This skill is also helpful in budgeting and financial planning, where quick estimations can provide a reasonable overview of expenses and income.
- Round the improper fractions to simpler values: $\frac{27}{8} \approx 3.5$ and $\frac{163}{15} \approx 11$.
- Multiply the rounded values to estimate the product: $3.5 \times 11 = 38.5$.
- The estimated value of $3 \frac{3}{8} \times 10 \frac{13}{15}$ is approximately $\boxed{37}$.
### Explanation
1. Problem Analysis
We are asked to estimate the product of two mixed numbers: $3 \frac{3}{8}$ and $10 \frac{13}{15}$. To estimate, we can round each mixed number to the nearest whole number or a simple fraction.
2. Converting to Improper Fractions
First, let's convert the mixed numbers to improper fractions. $3 \frac{3}{8} = \frac{3 \times 8 + 3}{8} = \frac{24 + 3}{8} = \frac{27}{8}$. And $10 \frac{13}{15} = \frac{10 \times 15 + 13}{15} = \frac{150 + 13}{15} = \frac{163}{15}$.
3. Rounding the Fractions
Now, let's round each improper fraction. $\frac{27}{8}$ is approximately equal to $3.375$. We can round this to $3.5$ or $3 \frac{1}{2}$ or even just $3$. $\frac{163}{15}$ is approximately equal to $10.8666...$. We can round this to $11$.
4. Estimating the Product
Let's use the rounded values to estimate the product. If we round $3\frac{3}{8}$ to $3.5$ and $10\frac{13}{15}$ to $11$, then the estimated product is $3.5 \times 11 = 38.5$. If we round $3\frac{3}{8}$ to $3$ and $10\frac{13}{15}$ to $11$, then the estimated product is $3 \times 11 = 33$.
5. Alternative Estimation
Another approach is to round $3\frac{3}{8}$ to $3\frac{1}{2} = \frac{7}{2}$ and $10\frac{13}{15}$ to $11$. Then the estimated product is $\frac{7}{2} \times 11 = \frac{77}{2} = 38.5$.
6. Alternative Estimation 2
We can also round $3\frac{3}{8}$ to $3$ and $10\frac{13}{15}$ to $10\frac{7}{15} + \frac{6}{15} = 10\frac{1}{2}$. Then the estimated product is $3 \times 10.5 = 31.5$.
7. Comparing with Exact Value
Let's consider rounding $3\frac{3}{8}$ to $3\frac{1}{2}$ and $10\frac{13}{15}$ to $11$. Then the estimated product is $3.5 \times 11 = 38.5$. The exact value is $3\frac{3}{8} \times 10\frac{13}{15} = \frac{27}{8} \times \frac{163}{15} = \frac{4401}{120} = 36.675$. So, $38.5$ is a reasonable estimate.
8. Final Answer
Therefore, a good estimate for $3 \frac{3}{8} \times 10 \frac{13}{15}$ is approximately $37$.
### Examples
Estimating calculations is useful in everyday situations such as grocery shopping. For example, if you want to buy 3 and 3/8 pounds of apples at $10 and 13/15 per pound, estimating the total cost helps you quickly determine if you have enough money without needing exact calculations. This skill is also helpful in budgeting and financial planning, where quick estimations can provide a reasonable overview of expenses and income.
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