Question

Two playful kittens working together can unwind all the balls of yarn in a basket 9

minutes faster than if the first kitten is working alone, and 4 minutes faster than if the
second kitten is working alone. How long does it take each kitten to unwind all the
balls of yarn alone?

Answer 1

The time it takes the first kitten to unwind the balls of yarn alone is 15 minutes

The time it takes the second kitten to unwind the balls of yarn alone is 10 minutes

Explanation:

The time it takes the two kitten to unwind the balls of yarn = 9 minutes faster than the first kitten working alone

The time it takes the two kitten to unwind the balls of yarn = 4 minutes faster than the second kitten working alone

Let 'x' represent the time it takes the first kitten to unwind the balls of yarn alone, let 'y' represent the time it takes the second kitten to unwind the balls of yarn alone, and let 't' represent the time it takes the two kitten to unwind the balls of yarn we have;

x = t + 9

y = t + 4

The rate of working, per ball of yarn, of the first kitten = 1/x = 1/(t + 9)

The rate of working, per ball of yarn, of the second kitten = 1/y = 1/(t + 4)

The amount of work each kitten does in the time 't', added together to unwind 1 ball of yarn is given as follows;

t/(t + 9) + t/(t + 4) = 1

(t·(t +4) + t·(t + 9))/(t + 4)×(t + 9) = 1

Using a graphing calculator, we have;

(2·t²+13·t)/(t² + 13·t + 36) = 1

∴ (2·t² + 13·t) = (t² + 13·t + 36)

(2·t² + 13·t) - (t² + 13·t + 36) = 0

t² - 36 = 0

t = √36 = 6

t = 6

x = t + 9 = 6 + 9 = 15

The time it takes the first kitten to unwind the balls of yarn alone, x = 15 minutes

y = t + 4 = 6 + 4 = 10

The time it takes the second kitten to unwind the balls of yarn alone, y = 10 minutes