Question

Harry can rake the leaves in the yard 8 hours faster than his little brother Jimmy can. If they work together, they can complete the job in 3 hours. Using complete sentences, explain each step in figuring out how to determine the time it would take Jimmy to complete the job on his own.

Answer 1

if harry can rake the leaves 8 times faster than his brother that means the equation will have a negative 8 on his brother side, but if they work together they will take 3 hours completine the job, this means that our equation will look like 1/x+1/(x-8)=1/3, now first we have to find a common denominator in this case our common dinominator will be 3x(x-8), so multiply our denominators for the numbers or variable that get your denominator to be 3x(x-8). Our first denominator can be multiplied by (x-8) and 3 to get 3x(x-8) now you have to multiply the top by those same numbers and variable thhius woudl give you 3(x-8)/3x(x-8). Now lets do our second one we can multiply it by 3x to get 3x(x-8), now multiply our top by the same numbers to get 3x/3x(x-8). Now lets do our last one we can multiply it by x(x-8) to get 3x(x-8) now multiply those by top to get x(x-8)/3x(x-8). Now we have 3(x-8)/3x(x-8)+3x/3x(x-8)=x(x-8)/3x(x-8), now we can cancel our denominators because they are all the same and we can find the value of x we have 3(x-8)+3x=x(x-8)> now do distributive property to get 3x-24+3x=x^2-8x. Now add common like terms to get 6x-24=x^2-8x. Now set your left terms to teh right to get 0=x^2-14x+24. Now find the factors, these are (x-2)(x-12). Now set them equal to 0 to get x=2 and x=12, if working together they take 3 ours then Jimmy working alone can't take 2 and it said harry took 8 ours less than Jimmy so our only possible and correct answer is 12. This means that Jimmy takes 12 hours to rake the leaves of a yark

Answer 2

Solve the following for Jimmy:


`\frac{1}{\text{jimmy}}+\frac{1}{\text{jimmy}-8}=(1)/(3)`

There are two solutions. Only one is valid.


`\frac{1}{\text{jimmy}}+\frac{1}{\text{jimmy}-8}=(1)/(3) \\ 3(jimmy-8)+3jimmy=jimmy(jimmy-8) \\3jimmy-24+3jimmy=jimmy^2-8jimmy \\jimmy^2-14jimmy+24=0 \\jimmy^2-12jimmy-2jimmy+24=0 \\jimmy(jimmy-12)-2(jimmy-2)=0 \\jimmy-12=0 \vee jimmy-2=0 \\jimmy=12 \vee jimmy=2`

Harry = Jimmy - 8

Number of hours for Harry cannot be negative. Therefore, Jimmy needs 12 hours to complete the job on his own.