Please help me also with this problem, thanks.
Quiana had some beads. The ratio of the number of black beads to the number of white beads was 1:5. After she added 2 more than twice the number of black beads, she still had 18 fewer black beads than white beads. How many black beads and how many white beads did Quiana have to begin with?
w = 5b
b + 2b + 2 + 18 = 5b
if this is the equation, I will simplify this as:
2+18=5b-3b
20=2b
20/2 =b
b=10
white beads is 20
black beads is 10
Let's start by assigning variables to the unknown quantities in the problem.
Let:
- the number of black beads be "b"
- the number of white beads be "w"
According to the problem, the ratio of black beads to white beads is 1:5. This can be written as:
b/w = 1/5
We can simplify this ratio by multiplying both sides by 5w:
5b = w
After Quiana added 2 more than twice the number of black beads, she still had 18 fewer black beads than white beads. We can express this information mathematically:
(2 + 2b) = (w - 18)
Now we have a system of equations with two unknowns. Let's solve it step-by-step:
1. Substitute the value of "w" from the first equation into the second equation:
(2 + 2b) = (5b - 18)
2. Simplify the equation:
2 + 2b = 5b - 18
3. Move the variable terms to one side of the equation and the constant terms to the other side:
2 + 18 = 5b - 2b
4. Simplify on both sides:
20 = 3b
5. Divide both sides by 3 to solve for "b":
20/3 = b
Therefore, the number of black beads, b, is approximately 6.67.
6. Substitute the value of "b" into the first equation to solve for "w":
5b = w
5(6.67) = w
w ≈ 33.33
Therefore, the number of white beads, w, is approximately 33.33.
Since we can't have a fraction of a bead, we can round the values to the nearest whole numbers. So, Quiana initially had 7 black beads and 33 white beads.
To solve this problem, let's break it down step by step.
Step 1: Set up the equations based on the given information.
Let's assume that Quiana initially had x black beads and y white beads.
Given information:
The ratio of black beads to white beads is 1:5.
This can be written as: x/y = 1/5
After adding 2 more than twice the number of black beads, Quiana had 18 fewer black beads than white beads.
This can be written as: (2x + 2) + 18 = y - x
Step 2: Simplify and solve the equations.
We can simplify the second equation by combining like terms:
2x + 20 = y - x - 18
3x = y - 38
Now, we can substitute the ratio equation (x/y = 1/5) into the second equation:
3x = (5x) - 38
Simplifying further:
3x - 5x = -38
-2x = -38
x = -38 / -2
x = 19
Now that we know the value of x (the number of black beads), we can substitute it back into the ratio equation to find y (the number of white beads):
19/y = 1/5
Cross-multiply:
5 * 19 = y * 1
95 = y
So, Quiana had 19 black beads and 95 white beads initially.