Working together, two ants named Aran and Beatrice can build an ant hill in 10 hours. Aran and Charlie can build the ant hill in 12 hours. Beatrice and Charlie can build the ant hill in 15 hours. How long, in hours, will it take to build the ant hill if Aran, Beatrice, and Charlie work together?

Here we have a problem of rates of work, we want to know how long will take the 3 ants to build an ant hill if they work together.

We will find that working together, they need to work for 8 hours.

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

Let's start by defining the variables that we will use:

Now let's see the given information:

"Aran and Beatrice can build an ant hill in 10 hours"

This can be written as:

(A + B)*10 h = 1 ant hill

"Aran and Charlie can build the ant hill in 12 hours"

This can be written as:

(A + C)*12 h = 1 ant hill

"Beatrice and Charlie can build the ant hill in 15 hours"

This can be written as:

(B + C)*15 h = 1 ant hill.

Then we have 3 equations:

[tex](A + B)*10 h = 1[/tex]

[tex](A + C)*12 h = 1[/tex]

[tex](B + C)*15 h = 1[/tex]

Where for commodity, I removed the "ant hill" unit.

Now we need to solve this system of equations, to do it, we need to isolate one variable in one of the equations and then replace it on another equation.

I will isolate A in the first equation:

[tex]A = \frac{1}{10h} - B[/tex]

Now we can replace it in the second equation to get:

[tex](\frac{1}{10h} - B + C)*12h = 1[/tex]

Now we can isolate C in the last equation to get:

[tex]C = \frac{1}{15h} - B[/tex]

Replacing that in the above equation we get:

[tex](\frac{1}{10h} - B + \frac{1}{15h} - B)*12h = 1\\\\\frac{1}{6h} - \frac{1}{12h}= 2*B\\\\\frac{1}{2}*\frac{1}{12h} = \frac{1}{24h} = B[/tex]

This means that Beatrice alone would construct the hill in 24 hours.

Now that we know the value of B, we can use:

[tex]C = \frac{1}{15h} - B = \frac{1}{15h} - \frac{1}{24h} = \frac{1}{40h}[/tex]

This means that Charlie needs 40 hours to finish the hill alone.

Finally, we can use [tex]A = \frac{1}{10h} - B[/tex] to find the value of A:

[tex]A = \frac{1}{10h} - B = \frac{1}{10h} - \frac{1}{24h} = \frac{7}{120h}[/tex]

Now if the 3 ants work together, the total rate at which they work is:

[tex](A + B + C) = (\frac{7}{120h} + \frac{1}{40h} + \frac{1}{24h} ) = 0.125 h^{-1}[/tex]

This means that they need to work for the inverse of that rate

[tex]\frac{1}{0.125h^{-1}} = 8h[/tex]

Working together, they need to work for 8 hours to complete an ant hill

If you want to learn more, you can read:

https://brainly.com/question/12895249

PLS HELP I'LL MARK BRAINLEST

Determine the intercepts of the line x-intercept

What is the value of j?

___°​

can someone please help me?

What is the image of the point (3,2)(3,2) after a rotation of 90^{\circ}90 ∘ counterclockwise about the origin?

Need Help with this please!!

at a concession stand three hotdogs in for hamburgers cost $13.50 for hotdogs and three hamburgers cost $12.75 find the cost of one hotdog and find the cost of one hamburger

Number seven plz helppp

What hazard is present in this image?

Select the best option.


A mat that has upturned edges


Hair clippings on the floor


Tool cord left out


Spills on floor

which of the following numbers is irrational

14 is 33 1/2% of what number?

Solve the system.... 2y+8x+58 and y-5x=11

pls dont say random stuff to get points

Juan received a bank account statement reporting the recent changes
to his account. The statement showed an overall increase in the
account balance of $1,909.15. During the time shown on the
statement, Juan was paid twice and withdrew a total of $1,437.99 to
cover all his bills and expenses. How much (x) is each of Juan's
paychecks?

Kim drew a map of the Mississippi River. The scale was 5 centimeters represents 120 miles. The actual length of the Mississippi River is 360 miles. Approximately what is the length in centimeters of the Mississippi River on the map?
50 points to who ever answer it corectly

If GEOM is a square, find m2GQE.

2x + y = 11

3x - y = 4

-x + 7y = 8 if x equals 6

HURRY
1/12 is the same as 1÷12. What is the exact quotient in decimal form of 1÷12?

need help plz help me

Help me with this circle question

Guys help I’ve been trying to solve this problem for the past 40 minutes.

Re write the statement 19+7 divided by 2-5 by including one pair of brackets to make the total equal to 8

What is the measure of angle P?

How many times will the graph of this equation cross the x-
axis?
y= 22^2– 7x - 9

Help pleas brainliest promise

hello My name es Liliana y arigato bye​

please solve for b

a=b/c

I need to write an equation for the triangle and solve for x

reduce to the simplest form[tex]\frac{286}{858}[/tex]