Question

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?

Answer 1

Answer:

They complete the hill in 8 hours

Step-by-step explanation:

Equations:

Let's call the variables:

Aran can make build the ant hill in A hours

Beatrice can make build the ant hill in B hours

Charlie can make build the ant hill in C hours

In one hour, Aran makes 1/A of the ant hill.

In one hour, Beatrice makes 1/B of the ant hill.

In one hour, Charlie makes 1/C of the ant hill.

Aran and Beatrice build it in 10 hours, thus:

$\displaystyle \frac{1}{A}+\frac{1}{B}=\frac{1}{10}\qquad\qquad[1]$

Similarly:

$\displaystyle \frac{1}{A}+\frac{1}{C}=\frac{1}{12}\qquad\qquad[2]$

$\displaystyle \frac{1}{B}+\frac{1}{C}=\frac{1}{15}\qquad\qquad[3]$

We need to find the time taken for the three ants to build the anthill:

$\displaystyle \frac{1}{A}+\frac{1}{B}+\frac{1}{C}=$

Adding [1], [2], and [3]:

$\displaystyle \frac{2}{A}+\frac{2}{B}+\frac{2}{C}=\frac{1}{10}+\frac{1}{12}+\frac{1}{15}$

Adding the fractions (LCM=60):

$\displaystyle \frac{2}{A}+\frac{2}{B}+\frac{2}{C}=\frac{6}{60}+\frac{5}{60}+\frac{4}{60}$

$\displaystyle \frac{2}{A}+\frac{2}{B}+\frac{2}{C}=\frac{6+5+4}{60}=\frac{15}{60}=\frac{1}{4}$

Dividing by 2:

$\displaystyle \frac{1}{A}+\frac{1}{B}+\frac{1}{C}=\frac{1}{8}$

All ants together make 1/8 of the hill, thus they complete the hill in 8 hours