Roni and Allie are mowing the grass at the soccer field.
Roni has a riding lawn mower and can mow the field in 30 minutes. Allie is pushing a lawn mower and can mow the field in 75 minutes.
If Roni and Allie work together to mow the field, what part of the field would Roni mow?
A. [$\frac{1}{30}$]
B. about 0.52 of the field
C. about 0.64 of the field
D. about 0.71 of the field
E. about 0.87 of the field
Answer (1)
Calculate Roni's rate: 30 1 .
Calculate Allie's rate: 75 1 .
Calculate the combined rate: 30 1 + 75 1 = 150 7 .
Calculate the fraction of the field Roni mows: 30 1 × 7 150 = 7 5 ≈ 0.71 .
The part of the field Roni would mow is ab o u t 0.71 o f t h e f i e l d .
Explanation
Problem Analysis Let's analyze the problem. Roni can mow the entire field in 30 minutes, and Allie can mow the same field in 75 minutes. We need to find the fraction of the field Roni mows when they work together.
Individual Rates First, we need to determine the rate at which each person mows the field. Roni's rate is 30 1 fields per minute, and Allie's rate is 75 1 fields per minute.
Combined Rate Next, we find their combined rate by adding their individual rates: 30 1 + 75 1 . To add these fractions, we need a common denominator. The least common multiple of 30 and 75 is 150. So, we rewrite the fractions: 150 5 + 150 2 = 150 7 . This means together they mow 150 7 of the field per minute.
Time Working Together Now, we find the time it takes for them to mow the field together. Since rate times time equals work (and the work is 1 field), we have: Time = Combined Rate 1 = 150 7 1 = 7 150 minutes.
Fraction of Field Roni Mows Finally, we calculate the fraction of the field Roni mows. This is Roni's rate multiplied by the time they work together: 30 1 × 7 150 = 210 150 = 7 5 .
Final Calculation To express this fraction as a decimal, we divide 5 by 7: 7 5 ≈ 0.7142857 . Therefore, Roni mows approximately 0.71 of the field.
Examples
Imagine you and a friend are painting a room. You can paint the entire room in 4 hours, and your friend can paint the same room in 6 hours. Working together, you want to know what fraction of the room you painted. This problem is similar; by calculating individual rates, combining them, and then multiplying by the time worked together, you can determine the portion of work each person contributes.
