Anton has two species of ants, Species A and Species B, in his ant farm. The two species are identical in appearance, but Anton knows that every day, there are twice as many ants of Species A than before, while there are three times as many ants of Species B. On Day 0, Anton counts that there are 30 ants in his ant farm. On Day 5, Anton counts that there are 3281 ants in his ant farm. How many of these are of Species A?

Guest Sep 28, 2017

#1**+2 **

Let a be the number of species A ants he had to begin with.

Let b be the number of species B ants he had to begin with.

On day 0 there are a + b ants.

On day 1 there are 2a + 3b ants.

On day 2 there are 2 * 2a + 3 * 3b ants.

On day 3 there are 2 * 2 * 2 * a + 3 * 3 * 3 * b ants.

On day 4 there are 2^{4} * a + 3^{4} * b ants.

On day *n* there are 2^{n} * a + 3^{n} * b

So...

On day 5 there are 2^{5} * a + 3^{5} * b

a + b = 30 → b = 30 - a

2^{5} * a + 3^{5} * b = 3281

2^{5} * a + 3^{5} * (30 - a) = 3281

2^{5}a + 3^{5} * 30 - 3^{5}a = 3281

2^{5}a - 3^{5}a = 3281 - 3^{5} * 30

a(2^{5} - 3^{5}) = 3281 - 3^{5} * 30

a = (3281 - 3^{5} * 30) / (2^{5} - 3^{5})

a = 19

So...he started with 19 ants of species A.

After 5 days, he will have 2^{5 }* 19 = 608 ants of species A.

hectictar
Sep 28, 2017

#1**+2 **

Best Answer

Let a be the number of species A ants he had to begin with.

Let b be the number of species B ants he had to begin with.

On day 0 there are a + b ants.

On day 1 there are 2a + 3b ants.

On day 2 there are 2 * 2a + 3 * 3b ants.

On day 3 there are 2 * 2 * 2 * a + 3 * 3 * 3 * b ants.

On day 4 there are 2^{4} * a + 3^{4} * b ants.

On day *n* there are 2^{n} * a + 3^{n} * b

So...

On day 5 there are 2^{5} * a + 3^{5} * b

a + b = 30 → b = 30 - a

2^{5} * a + 3^{5} * b = 3281

2^{5} * a + 3^{5} * (30 - a) = 3281

2^{5}a + 3^{5} * 30 - 3^{5}a = 3281

2^{5}a - 3^{5}a = 3281 - 3^{5} * 30

a(2^{5} - 3^{5}) = 3281 - 3^{5} * 30

a = (3281 - 3^{5} * 30) / (2^{5} - 3^{5})

a = 19

So...he started with 19 ants of species A.

After 5 days, he will have 2^{5 }* 19 = 608 ants of species A.

hectictar
Sep 28, 2017