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?
To solve this problem, we can set up an algebraic equation. Let's call the number of black beads B and the number of white beads W.
According to the problem, the ratio of black beads to white beads is 7:2. This can be written as:
B/W = 7/2
We also know that the total number of beads is 54, so we can set up another equation:
B + W = 54
Now, we can solve this system of equations to find the values of B and W.
First, let's rewrite the ratio equation so that B is isolated:
B = (7/2)W
Now, substitute this expression for B in the second equation:
(7/2)W + W = 54
To simplify this equation, we can multiply both sides by 2 to eliminate the fraction:
7W + 2W = 108
Combining like terms, we get:
9W = 108
Divide both sides by 9 to solve for W:
W = 12
Now, substitute this value back into the second equation to solve for B:
B + 12 = 54
B = 54 - 12
B = 42
So, 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:
42 - 12 = 30
Therefore, there are 30 more black beads than white beads.
7/9 = x/54
Cross multiply and solve for x.