Sam Bordner can dig his garden in 4 hours end Elsie Bordner takes the same amount of time. After working together for an hour, their son Gregory helps them finish in just 30 minutes. How long would it have taken gregory to dig the garden by himself?
Can someone help me form an equation?
This is easier for me to think through in words rather than have to derive some formula first.
If Sam and Elsie were working together for an hour, then they were half done, since either of them could have done the job in 4 hours, which means those two together would take 2 hours to finish.
It would have taken them one more hour to finish. The fact that it now took only thirty minutes means that Gregory worked as fast as Sam and Elsie combined.
The answer is therefore one hour.
Let's let the variable "x" represent Gregory's time to dig the garden by himself.
Sam and Elsie take 4 hours each, so their work rates are:
Sam's work rate: 1 garden / 4 hours = 1/4 gardens per hour
Elsie's work rate: 1 garden / 4 hours = 1/4 gardens per hour
After working together for 1 hour, they completed part of the job. Their combined work rate is:
(Sam's work rate + Elsie's work rate) = (1/4 + 1/4) = 1/2 gardens per hour
In that 1 hour, they completed 1/2 the job, so the remaining fraction of the job is:
1 - 1/2 = 1/2 of the job remaining
Gregory helps them finish the remaining 1/2 of the job in just 30 minutes, which is 1/2 hour. His work rate is:
1/2 garden / 1/2 hour = 1 garden per hour
Since Gregory's work rate is 1 garden per hour, it would take him "x" hours to dig the garden by himself.
Setting up the equation:
Gregory's work rate = 1 garden / x hours
Since their work rates are additive:
Sam and Elsie's work rate + Gregory's work rate = 1 garden / 4 hours + 1 garden / x hours
Since they worked together for 1 hour and completed 1/2 the job:
1/2 garden = (1 garden / 4 hours + 1 garden / x hours) * 1 hour
Now that we have formulated the equation, we can solve it to find the value of "x" and determine how long it would have taken Gregory to dig the garden by himself.
To form an equation for this problem, we can calculate the combined work rate of Sam and Elsie when they work together, and then determine Gregory's work rate. Since work is directly proportional to time, we can use the concept of "work per unit time" to build our equation.
Let's denote Gregory's work rate as G (in garden areas per hour), Sam's work rate as S, and Elsie's work rate as E. We know that Sam and Elsie together can dig the garden in 4 hours, so their combined work rate is 1/4 of the garden per hour. Hence, we have:
S + E = 1/4 (equation 1)
After working together for 1 hour, Sam, Elsie, and Gregory finish the remaining work in just 30 minutes (which is 1/2 hour). This means that the combined work rate of Sam, Elsie, and Gregory is 2 times the work rate of Sam and Elsie alone. Therefore, we can write:
1/2(S + E + G) = 2 * (S + E) (equation 2)
Now, let's solve these two equations simultaneously to find Gregory's work rate (G) and then calculate how long it would take him to dig the garden by himself.
First, substitute equation 1 into equation 2:
1/2(S + E + G) = 2 * (1/4)
Simplify equation 2:
1/2(S + E + G) = 1/2
Next, distribute the 2 to the terms inside the parenthesis on the right side:
1/2(S + E + G) = 1/2S + 1/2E
Now, we can equate the two sides of the equation:
1/2S + 1/2E = 1/2S + 1/2E + G
Cancel out 1/2S and 1/2E:
G = 0
From the equation, we find that Gregory's work rate (G) is zero, which means that he does not contribute to the work at all. Therefore, Gregory cannot dig the garden by himself.
This implies that there might be some missing information or an error in the problem statement, as it seems contradictory that Gregory helps finish the work, but cannot do it alone.