How many pounds of a liquid are contained in a barrel if the barrel holds 11 and four sevenths
 ​gallons, and a gallon of the liquid weighs 9 and two thirds
 ​lb?

To solve this problem, we need to calculate the total weight of the liquid in pounds given the volume in gallons and the weight per gallon.

### Step-by-Step Solution

1. **Convert Mixed Numbers to Improper Fractions:**

   - The volume of the liquid in the barrel is \(11 \frac{4}{7}\) gallons.
   - Convert \(11 \frac{4}{7}\) to an improper fraction:
     \[
     11 \frac{4}{7} = \frac{11 \times 7 + 4}{7} = \frac{77 + 4}{7} = \frac{81}{7} \text{ gallons.}
     \]

   - The weight of the liquid per gallon is \(9 \frac{2}{3}\) pounds.
   - Convert \(9 \frac{2}{3}\) to an improper fraction:
     \[
     9 \frac{2}{3} = \frac{9 \times 3 + 2}{3} = \frac{27 + 2}{3} = \frac{29}{3} \text{ pounds per gallon.}
     \]

2. **Calculate the Total Weight in Pounds:**

   The total weight \(W\) of the liquid is the product of the volume in gallons and the weight per gallon:
   \[
   W = \left( \frac{81}{7} \right) \times \left( \frac{29}{3} \right).
   \]

   Multiply the fractions:
   \[
   W = \frac{81 \times 29}{7 \times 3} = \frac{2349}{21}.
   \]

3. **Simplify the Fraction:**

   To simplify \(\frac{2349}{21}\), we divide the numerator and the denominator by their greatest common divisor (GCD). The GCD of 2349 and 21 is 3.

   \[
   W = \frac{2349 \div 3}{21 \div 3} = \frac{783}{7}.
   \]

4. **Convert to a Mixed Number:**

   To convert \(\frac{783}{7}\) to a mixed number:

   - Divide 783 by 7:  
     \[
     783 \div 7 = 111 \quad \text{with a remainder of} \quad 6.
     \]
   - So, \(\frac{783}{7} = 111 \frac{6}{7}\).

### Answer

The total weight of the liquid in the barrel is **111 \(\frac{6}{7}\) pounds**.

Would you like more details or have any questions about this solution?

### Related Questions

1. How do you convert a mixed number to an improper fraction?
2. How can you find the GCD of two numbers?
3. What is the process for converting an improper fraction back into a mixed number?
4. How do you multiply two fractions together?
5. How can you simplify a fraction by finding the GCD?

### Tip

When dealing with mixed numbers in calculations, it's often easiest to first convert them to improper fractions to perform arithmetic operations more easily.

Learn how to convert and calculate the weight of a liquid in pounds from gallons using fractions and mixed numbers. This problem involves converting mixed numbers to improper fractions, multiplying fractions, and simplifying fractions. Suitable for Grades 6-8.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation