Answer:
Here, James took the right decision. I will clarify this with few points.
1st - If he purchases a car at high interest rate, he will still have all his debt on previous credit card standing as it is. He will have to pay car loan plus his older dues thus paying at double places.
2nd - It is likely that he can still default on loans as paying double money each month can create problems and James can again stop making payments.
3rd - If James starts paying his debts now, he can be free in a few years time and his credit score will again become good. Then he will get the regular rate of interest for his car as he will be debt free.
So, we can say, he made the right decision.
X - the number of cats
The ratio of dogs to cats is 3:5, and there are 21 dogs.
![\frac{21}{x}=\frac{3}{5} \\ 3x=21 \times 5 \\ x=7 \times 5 \\ x=35](https://tex.z-dn.net/?f=%5Cfrac%7B21%7D%7Bx%7D%3D%5Cfrac%7B3%7D%7B5%7D%20%5C%5C%0A3x%3D21%20%5Ctimes%205%20%5C%5C%0Ax%3D7%20%5Ctimes%205%20%5C%5C%0Ax%3D35)
There are 35 cats in the pet store.
$240
Since a square yard is 3ft x 3ft, a square yard is equal to 9 square feet. Dividing 180 square feet by 9 square feet, we find that the room is 20 square yards.
Since carpet is $12 per square yard, and we have 20 square yards to cover, by multiplying 20 by $12, we find that the room will cost $240 to cover.
To divide these complex numbers you have to multiply by the conjugate of the denominator. That will look like this:
![\frac{5i}{4+3i}* \frac{4-3i}{4-3i}](https://tex.z-dn.net/?f=%20%5Cfrac%7B5i%7D%7B4%2B3i%7D%2A%20%5Cfrac%7B4-3i%7D%7B4-3i%7D%20%20)
. Multiply straight across the top and straight across the bottom to get
![\frac{20i-15i^2}{16-12i+12i-9i^2}](https://tex.z-dn.net/?f=%20%5Cfrac%7B20i-15i%5E2%7D%7B16-12i%2B12i-9i%5E2%7D%20)
. In the denominator, the 12i and -12i cancel each other out, which is nice. Now, in both the numerator and the denominator we have an i-squared. i-squared is equal to -1, so we will make that substitution in our solution:
![\frac{20i-15(-1)}{16-9(-1)}](https://tex.z-dn.net/?f=%20%5Cfrac%7B20i-15%28-1%29%7D%7B16-9%28-1%29%7D%20)
. Doing the math on that we have
![\frac{20i+15}{25}](https://tex.z-dn.net/?f=%20%5Cfrac%7B20i%2B15%7D%7B25%7D%20)
. We can simplify that as well as write it in standard form:
![\frac{15+20i}{25}](https://tex.z-dn.net/?f=%20%5Cfrac%7B15%2B20i%7D%7B25%7D%20)
. Now we will get it into legit standard form, separating the real part of the complex number from the imaginary part.
![\frac{15}{25}+ \frac{20}{25}i](https://tex.z-dn.net/?f=%20%5Cfrac%7B15%7D%7B25%7D%2B%20%5Cfrac%7B20%7D%7B25%7Di%20%20)
. That can be reduced to its final answer of
![\frac{3}{5}+ \frac{4}{5}i](https://tex.z-dn.net/?f=%20%5Cfrac%7B3%7D%7B5%7D%2B%20%5Cfrac%7B4%7D%7B5%7Di%20%20)
. There you go!