Directions: write and solve an algebraic equation or inequality for each problem.

1. Mabel has a total of 54 beads. Some are black and some are white. The ratio of the number of black beads to the number of white beads is 7:2. How many more black beads than white beads are there?

54/(7+2) = 6

Multiply the 7 and 2 by 6.

To solve this problem, let's assign variables to the unknown quantities. Let's say the number of black beads is represented by "x," and the number of white beads is represented by "y."

According to the problem, the total number of beads is 54. So, we can set up an equation based on the given information:

x + y = 54 (Equation 1)

We are also given that the ratio of the number of black beads to the number of white beads is 7:2. This can be expressed as:

x/y = 7/2

To remove the ratios, we can cross-multiply to get:

2x = 7y

Now, we have a system of equations. We can solve this system using substitution or elimination.

Let's solve it using substitution. From the second equation, we can express x in terms of y:

x = (7y) / 2

Now, substitute this expression for x into Equation 1:

(7y) / 2 + y = 54

Multiply both sides by 2 to eliminate the fraction:

7y + 2y = 108

Combine like terms:

9y = 108

Divide both sides by 9:

y = 12

Now that we know the value of y is 12, substitute this value back into the expression for x:

x = (7 * 12) / 2
x = 42

Therefore, there are 42 black beads and 12 white beads.

To find how many more black beads than white beads there are, we subtract the number of white beads from the number of black beads:

Black beads - White beads = 42 - 12 = 30

So, there are 30 more black beads than white beads.