Aryson is playing green beans. He has already picked $1\frac{1}{4}$ bushels and picks at a rate of $\frac{7}{8}$ of a bushel each hour. The equation $\frac{7}{8}h + 1\frac{1}{4} = 6$ can be used to represent $h$, the number of hours it will take him to pick 6 bushels. What is the value of h?
A. $4\frac{8}{7}$ hours
B. $5\frac{3}{7}$ hours
C. $5\frac{17}{28}$ hours
D. $5\frac{2}{7}$ hours
Answer (1)
Convert the mixed number to an improper fraction: 1 4 1 = 4 5 .
Subtract 4 5 from both sides: 8 7 h = 6 − 4 5 = 4 19 .
Multiply both sides by 7 8 : h = 4 19 × 7 8 = 7 38 .
Convert to a mixed number: h = 5 7 3 . The final answer is 5 7 3 .
Explanation
Understanding the Equation We are given the equation 8 7 h + 1 4 1 = 6 which represents the number of hours, h , it takes Aryson to pick 6 bushels of green beans. We need to solve for h .
Converting to Improper Fraction First, convert the mixed number 1 4 1 to an improper fraction: 1 4 1 = 4 4 + 4 1 = 4 5 . So the equation becomes 8 7 h + 4 5 = 6 .
Subtracting Fractions Next, subtract 4 5 from both sides of the equation: 8 7 h = 6 − 4 5 . To subtract, we need a common denominator, which is 4. So, 6 = 4 24 . Thus, 8 7 h = 4 24 − 4 5 = 4 19 .
Multiplying by the Reciprocal Now, multiply both sides of the equation by the reciprocal of 8 7 , which is 7 8 : h = 4 19 × 7 8 .
Simplifying the Fraction Simplify the expression: h = 4 × 7 19 × 8 = 1 × 7 19 × 2 = 7 38 .
Converting to Mixed Number Convert the improper fraction 7 38 to a mixed number: 7 38 = 5 7 3 . Therefore, h = 5 7 3 hours.
Final Answer The value of h is 5 7 3 hours.
Examples
Understanding how to solve linear equations like this is useful in many real-life scenarios. For example, if you are saving money for a new bicycle that costs $200 and you already have $50 saved, and you earn $10 per hour, you can use a similar equation to determine how many hours you need to work to save enough money. The equation would be 10 h + 50 = 200 , where h is the number of hours you need to work. Solving this equation helps you plan your time and reach your financial goals.
