1. The ratio of the number of dimes to the number of pennies in a purse is 3 to 4. If 3 pennies are taken out of the purse, the number of dimes and the number of pennies in the purse will be equal. How many dimes are in the purse? (Answer key says 9- and if I multiply the ratio by three it makes sense. But how do I do this without guess and test method?)

d/p = 3/4

p-3 = d

p = 4d/3, so

4d/3 - 3 = d
d/3 = 3
d = 9

To solve this problem without using guess and test, you can set up a system of equations based on the given information. Let's use "d" to represent the number of dimes and "p" to represent the number of pennies in the purse.

From the problem statement, we know that the ratio of dimes to pennies is 3 to 4, which can be expressed as:

d/p = 3/4

We are also given that if 3 pennies are removed from the purse, the number of dimes and pennies will be equal. This means that the new ratio of dimes to pennies will be 1 to 1, which can be expressed as:

(d)/(p-3) = 1/1

Now we have a system of two equations:

1) d/p = 3/4
2) (d)/(p-3) = 1/1

To solve this system, we can use substitution. Rearrange equation 1) to solve for "d" in terms of "p":

d = (3/4)p

Substitute this expression for "d" into equation 2):

(3/4)p / (p-3) = 1/1

Now we can cross multiply and solve for "p":

3(p-3) = 4p

3p - 9 = 4p

4p - 3p = 9

p = 9

Now that we have the number of pennies (p=9), we can substitute this value back into equation 1) to find the number of dimes:

d = (3/4)(9)

d = 6.75

Since we cannot have a fractional number of dimes, we round this value down to the nearest whole number:

d = 6

Therefore, there are 6 dimes in the purse.