Anonymous
Anonymous asked in Science & MathematicsMathematics · 1 month ago

Twin brothers Billy and Bobby, can mow their grandparents lawn together in 59 minutes. Billy could mow the lawn by himself in 20 minutes less  time that it would take Bobby. How long would it take Bobby to mow the lawn by himself?

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• 1 month ago

Twin brothers Billy and Bobby can mow

their grandparents' lawn together in 59 minutes.

Billy could mow the lawn by himself

in 20 minutes less time than it would take Bobby.

How long would it take Bobby to mow the lawn by himself?

1/(x - 20) + 1/x = 1/59

(x - 69)^2/3581 = 1 (for x!=0 and x!=20)

Solutions:

x = 69 - sqrt(3581)

x = 69 + sqrt(3581)

• 1 month ago

Let Bi & Bo be the time they need to mow the

lawn individually, then

1/Bi+1/Bo=1/59----(1)

Bi+20=Bo------(2)

=>

1/(Bo-20)+1/Bo=1/59

=>

Bo^2-138Bo+1180=0

=>

Bo=128.84 or 9.16 (rejected)

=>

Bobby needs 128.84 min. approximately

to mow the lawn alone.

• Anonymous
1 month ago

Sorry, my answer is "Anonymous" as your post is Anonymous. There is no need for an Anonymous question in this category, Most 'Anonymous' posts are trolls, maybe yours is not, but the high likelihood is there. If you post as yourself, I would be glad to help you with any problem you have.

I spend a lot of time on each problem, I just want to make sure it's for a worthwhile cause.

• 1 month ago

SHUT UP LIBERAL #GodBless

Source(s): Ben Shapiro
• 1 month ago

Let o be the number of minutes it takes for Bobby to mow the lawn.

Let (o - 20) be the number of minutes it takes for Billy to mow the lawn.

Then:

Bobby's rate to cut the lawn is (1 / o) lawns per minute.

Billy's rate to cut the lawn is (1 / (o - 20)) lawns per minute.

Finally:

59(1 / o) + 59(1 / (o - 20)) = 1 Lawn

59(o - 20) + 59o = o(o - 20)

59o - 1180 + 59o = o^2 - 20o

118o -1180 = o^2 - 20o

o^2 - 138o + 1180 = 0

We must throw out the second one as extraneous because that would make Billy's time to cut the lawn negative.

So, it takes Bobby 128.84 minutes to cut the lawn by himself. That's 128 minutes and 50 seconds, approximately.

• 1 month ago

b = Bobby's time

b - 20 = Billy's time

---------------------

1/b + 1/(b - 20) = 1/59

multiply both sides by 59(b)(b - 20)

59(b - 20) + 59b = b(b - 20)

59b - 1180 + 59b = b² - 20b

b² - 138b = - 1180

Complete the square using -138/2= -69 as the complete the square term

(b - 69)² = -1180 + (-69)²

(b - 69)² = 3581

Take the square root of both sides

b - 69 = ±59.84

b = 69 ± 59.84

b = {9.16, 128.84}

Disregard 9.16 since Billy's time = 9.16 - 20  = -10.84, would be negative.

b = 128.84 minutes <––––––

• Anonymous
1 month ago

What would this be if it has to be solved by hours and minutes together  ?

• 1 month ago

This problem can be solved in just the same way as your other problem. I suggest rereading my answer to that question and trying it yourself.

STEP 1 - Define the variables.

Let x be the time it takes Bobby to mow the lawn by himself (in minutes)

Let x-20 be the time it takes Billy to mow the lawn by himself (in minutes)

STEP 2 - Convert those to rates:

Bobby's rate --> 1/x of the lawn per minute

Billy's rate --> 1/(x-20) of the lawn per minute

Total rate --> 1/59 of the lawn per minute

STEP 3 - Create an equation

1/x + 1/(x - 20) = 1/59

STEP 4 - Solve the equation for x.

[This step is left to you using the example of your other problem]

• 1 month ago

work = rate * time

When two people are working together, you add their rates together.

So I'll call "b" the rate of Billy and "o" the rate of Bobby.  So when they work together the rate is the sum or (b + o).

Work here is the number of lawns, so we'll use 1 for each case since we're talking about the same lawn every time.

When they work together it takes 59 minutes so we can set up this equation:

w = rt

1 = (b + o) * 59

simplified to:

1 = 59(b + o)

Let's leave it like this for now and look at the next statement.

Billy could mow the lawn by himself in 20 min less than it would take Bobby.   We can make two new equations with unknown rates and times:

w = rt

1 = bt₁ and 1 = ot₂

Each time is currently unknown, but since billy can do it 20 minutes faster, we have this as a fourth equation:

t₁ = t₂ - 20

We now have a system of four equations and four unknowns.  We want to solve for t₂ (Bobby's time doing this job alone).

Let's start with substituting the expression from the fourth equation into the second.  Then I'll show the three remaining equations:

1 = bt₁

1 = b(t₂ - 20)

Now our equations are:

1 = 59(b + o) and 1 = b(t₂ - 20) and 1 = ot₂

Since we only have one "t" variable left, I'll drop the subscript and now we solve for "t":

1 = 59(b + o) and 1 = b(t - 20) and 1 = ot

Let's solve the last two equations for b and o in terms of t, then susbstitute into the first equation:

1 = b(t - 20) and 1 = ot

1/(t - 20) = b and 1/t = o

Susbtitute now to get:

1 = 59(b + o)

1 = 59[[1/(t - 20) + 1/t]

Let's start with dividing 59 from both sides and then getting a common denominator on the right side:

1/59 = 1/(t - 20) + 1/t

1/59 = t/[t(t - 20)] + (t - 20)/[t(t - 20)]

Now we can add the numerators:

1/59 = [t + (t - 20)]/[t(t - 20)]

Let's simpify both halves of the fraction, then mutiply both sides by 59 again:

1/59 = (t + t - 20) / (t² - 20t)

1/59 = (2t - 20) / (t² - 20t)

1 = 59(2t - 20) / (t² - 20t)

Now multiply both sides by the denominator and simplify to a quadratic:

t² - 20t = 59(2t - 20)

t² - 20t = 118t - 1180

t² - 138t + 1180 = 0

t = [ -b ± √(b² - 4ac)] / (2a)

t = [ -(-138) ± √((-138)² - 4(1)(1180))] / (2 * 1)

t = [ 138 ± √(19044 - 4720)] / 2

t = [ 138 ± √(14324)] / 2

t = [ 138 ± √(4 * 3581)] / 2

t = [ 138 ± 2√(3581)] / 2

t = 69 ± √(3581)

This gives us two possible values for t, which have decimal approximations of:

9.16 and 128.84

Since the other time is 20 seconds less than this, the first one results in negative time which doesn't make sense so we can throw that out.

So Bobby will take about 128.84 min to do the lawn by himself.